complexOTHERauxiliary - Man Page

complex

Synopsis

Functions

subroutine clabrd (M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
CLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
subroutine clacgv (N, X, INCX)
CLACGV conjugates a complex vector.
subroutine clacn2 (N, V, X, EST, KASE, ISAVE)
CLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.
subroutine clacon (N, V, X, EST, KASE)
CLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.
subroutine clacp2 (UPLO, M, N, A, LDA, B, LDB)
CLACP2 copies all or part of a real two-dimensional array to a complex array.
subroutine clacpy (UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
subroutine clacrm (M, N, A, LDA, B, LDB, C, LDC, RWORK)
CLACRM multiplies a complex matrix by a square real matrix.
subroutine clacrt (N, CX, INCX, CY, INCY, C, S)
CLACRT performs a linear transformation of a pair of complex vectors.
complex function cladiv (X, Y)
CLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.
subroutine claein (RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK, EPS3, SMLNUM, INFO)
CLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.
subroutine claev2 (A, B, C, RT1, RT2, CS1, SN1)
CLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.
subroutine clags2 (UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, SNV, CSQ, SNQ)
CLAGS2
subroutine clagtm (TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX, BETA, B, LDB)
CLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1.
subroutine clahqr (WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, INFO)
CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.
subroutine clahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
CLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.
subroutine claic1 (JOB, J, X, SEST, W, GAMMA, SESTPR, S, C)
CLAIC1 applies one step of incremental condition estimation.
real function clangt (NORM, N, DL, D, DU)
CLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix.
real function clanhb (NORM, UPLO, N, K, AB, LDAB, WORK)
CLANHB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian band matrix.
real function clanhp (NORM, UPLO, N, AP, WORK)
CLANHP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix supplied in packed form.
real function clanhs (NORM, N, A, LDA, WORK)
CLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix.
real function clanht (NORM, N, D, E)
CLANHT returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian tridiagonal matrix.
real function clansb (NORM, UPLO, N, K, AB, LDAB, WORK)
CLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.
real function clansp (NORM, UPLO, N, AP, WORK)
CLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form.
real function clantb (NORM, UPLO, DIAG, N, K, AB, LDAB, WORK)
CLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix.
real function clantp (NORM, UPLO, DIAG, N, AP, WORK)
CLANTP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form.
real function clantr (NORM, UPLO, DIAG, M, N, A, LDA, WORK)
CLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix.
subroutine clapll (N, X, INCX, Y, INCY, SSMIN)
CLAPLL measures the linear dependence of two vectors.
subroutine clapmr (FORWRD, M, N, X, LDX, K)
CLAPMR rearranges rows of a matrix as specified by a permutation vector.
subroutine clapmt (FORWRD, M, N, X, LDX, K)
CLAPMT performs a forward or backward permutation of the columns of a matrix.
subroutine claqhb (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)
CLAQHB scales a Hermitian band matrix, using scaling factors computed by cpbequ.
subroutine claqhp (UPLO, N, AP, S, SCOND, AMAX, EQUED)
CLAQHP scales a Hermitian matrix stored in packed form.
subroutine claqp2 (M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2, WORK)
CLAQP2 computes a QR factorization with column pivoting of the matrix block.
subroutine claqps (M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, VN2, AUXV, F, LDF)
CLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3.
subroutine claqr0 (WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)
CLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
subroutine claqr1 (N, H, LDH, S1, S2, V)
CLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.
subroutine claqr2 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SH, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
CLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
subroutine claqr3 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SH, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
CLAQR3 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
subroutine claqr4 (WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)
CLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.
subroutine claqr5 (WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, S, H, LDH, ILOZ, IHIZ, Z, LDZ, V, LDV, U, LDU, NV, WV, LDWV, NH, WH, LDWH)
CLAQR5 performs a single small-bulge multi-shift QR sweep.
subroutine claqsb (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)
CLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ.
subroutine claqsp (UPLO, N, AP, S, SCOND, AMAX, EQUED)
CLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppequ.
subroutine clar1v (N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK)
CLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.
subroutine clar2v (N, X, Y, Z, INCX, C, S, INCC)
CLAR2V applies a vector of plane rotations with real cosines and complex sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices.
subroutine clarcm (M, N, A, LDA, B, LDB, C, LDC, RWORK)
CLARCM copies all or part of a real two-dimensional array to a complex array.
subroutine clarf (SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
CLARF applies an elementary reflector to a general rectangular matrix.
subroutine clarfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, T, LDT, C, LDC, WORK, LDWORK)
CLARFB applies a block reflector or its conjugate-transpose to a general rectangular matrix.
subroutine clarfb_gett (IDENT, M, N, K, T, LDT, A, LDA, B, LDB, WORK, LDWORK)
CLARFB_GETT
subroutine clarfg (N, ALPHA, X, INCX, TAU)
CLARFG generates an elementary reflector (Householder matrix).
subroutine clarfgp (N, ALPHA, X, INCX, TAU)
CLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
subroutine clarft (DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
CLARFT forms the triangular factor T of a block reflector H = I - vtvH
subroutine clarfx (SIDE, M, N, V, TAU, C, LDC, WORK)
CLARFX applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10.
subroutine clarfy (UPLO, N, V, INCV, TAU, C, LDC, WORK)
CLARFY
subroutine clargv (N, X, INCX, Y, INCY, C, INCC)
CLARGV generates a vector of plane rotations with real cosines and complex sines.
subroutine clarnv (IDIST, ISEED, N, X)
CLARNV returns a vector of random numbers from a uniform or normal distribution.
subroutine clarrv (N, VL, VU, D, L, PIVMIN, ISPLIT, M, DOL, DOU, MINRGP, RTOL1, RTOL2, W, WERR, WGAP, IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ, WORK, IWORK, INFO)
CLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.
subroutine clartg (F, G, CS, SN, R)
CLARTG generates a plane rotation with real cosine and complex sine.
subroutine clartv (N, X, INCX, Y, INCY, C, S, INCC)
CLARTV applies a vector of plane rotations with real cosines and complex sines to the elements of a pair of vectors.
subroutine clascl (TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
CLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
subroutine claset (UPLO, M, N, ALPHA, BETA, A, LDA)
CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
subroutine clasr (SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA)
CLASR applies a sequence of plane rotations to a general rectangular matrix.
subroutine classq (N, X, INCX, SCALE, SUMSQ)
CLASSQ updates a sum of squares represented in scaled form.
subroutine claswp (N, A, LDA, K1, K2, IPIV, INCX)
CLASWP performs a series of row interchanges on a general rectangular matrix.
subroutine clatbs (UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO)
CLATBS solves a triangular banded system of equations.
subroutine clatdf (IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, JPIV)
CLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.
subroutine clatps (UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM, INFO)
CLATPS solves a triangular system of equations with the matrix held in packed storage.
subroutine clatrd (UPLO, N, NB, A, LDA, E, TAU, W, LDW)
CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.
subroutine clatrs (UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM, INFO)
CLATRS solves a triangular system of equations with the scale factor set to prevent overflow.
subroutine clauu2 (UPLO, N, A, LDA, INFO)
CLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm).
subroutine clauum (UPLO, N, A, LDA, INFO)
CLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm).
subroutine crot (N, CX, INCX, CY, INCY, C, S)
CROT applies a plane rotation with real cosine and complex sine to a pair of complex vectors.
subroutine cspmv (UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
CSPMV computes a matrix-vector product for complex vectors using a complex symmetric packed matrix
subroutine cspr (UPLO, N, ALPHA, X, INCX, AP)
CSPR performs the symmetrical rank-1 update of a complex symmetric packed matrix.
subroutine csrscl (N, SA, SX, INCX)
CSRSCL multiplies a vector by the reciprocal of a real scalar.
subroutine ctprfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, A, LDA, B, LDB, WORK, LDWORK)
CTPRFB applies a real or complex 'triangular-pentagonal' blocked reflector to a real or complex matrix, which is composed of two blocks.
integer function icmax1 (N, CX, INCX)
ICMAX1 finds the index of the first vector element of maximum absolute value.
integer function ilaclc (M, N, A, LDA)
ILACLC scans a matrix for its last non-zero column.
integer function ilaclr (M, N, A, LDA)
ILACLR scans a matrix for its last non-zero row.
integer function izmax1 (N, ZX, INCX)
IZMAX1 finds the index of the first vector element of maximum absolute value.
real function scsum1 (N, CX, INCX)
SCSUM1 forms the 1-norm of the complex vector using the true absolute value.

Detailed Description

This is the group of complex other auxiliary routines

Function Documentation

subroutine clabrd (integer M, integer N, integer NB, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, complex, dimension( * ) TAUQ, complex, dimension( * ) TAUP, complex, dimension( ldx, * ) X, integer LDX, complex, dimension( ldy, * ) Y, integer LDY)

CLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.  

Purpose:

 CLABRD reduces the first NB rows and columns of a complex general
 m by n matrix A to upper or lower real bidiagonal form by a unitary
 transformation Q**H * A * P, and returns the matrices X and Y which
 are needed to apply the transformation to the unreduced part of A.

 If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
 bidiagonal form.

 This is an auxiliary routine called by CGEBRD
Parameters

M

          M is INTEGER
          The number of rows in the matrix A.

N

          N is INTEGER
          The number of columns in the matrix A.

NB

          NB is INTEGER
          The number of leading rows and columns of A to be reduced.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the m by n general matrix to be reduced.
          On exit, the first NB rows and columns of the matrix are
          overwritten; the rest of the array is unchanged.
          If m >= n, elements on and below the diagonal in the first NB
            columns, with the array TAUQ, represent the unitary
            matrix Q as a product of elementary reflectors; and
            elements above the diagonal in the first NB rows, with the
            array TAUP, represent the unitary matrix P as a product
            of elementary reflectors.
          If m < n, elements below the diagonal in the first NB
            columns, with the array TAUQ, represent the unitary
            matrix Q as a product of elementary reflectors, and
            elements on and above the diagonal in the first NB rows,
            with the array TAUP, represent the unitary matrix P as
            a product of elementary reflectors.
          See Further Details.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

D

          D is REAL array, dimension (NB)
          The diagonal elements of the first NB rows and columns of
          the reduced matrix.  D(i) = A(i,i).

E

          E is REAL array, dimension (NB)
          The off-diagonal elements of the first NB rows and columns of
          the reduced matrix.

TAUQ

          TAUQ is COMPLEX array, dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the unitary matrix Q. See Further Details.

TAUP

          TAUP is COMPLEX array, dimension (NB)
          The scalar factors of the elementary reflectors which
          represent the unitary matrix P. See Further Details.

X

          X is COMPLEX array, dimension (LDX,NB)
          The m-by-nb matrix X required to update the unreduced part
          of A.

LDX

          LDX is INTEGER
          The leading dimension of the array X. LDX >= max(1,M).

Y

          Y is COMPLEX array, dimension (LDY,NB)
          The n-by-nb matrix Y required to update the unreduced part
          of A.

LDY

          LDY is INTEGER
          The leading dimension of the array Y. LDY >= max(1,N).
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  The matrices Q and P are represented as products of elementary
  reflectors:

     Q = H(1) H(2) . . . H(nb)  and  P = G(1) G(2) . . . G(nb)

  Each H(i) and G(i) has the form:

     H(i) = I - tauq * v * v**H  and G(i) = I - taup * u * u**H

  where tauq and taup are complex scalars, and v and u are complex
  vectors.

  If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
  A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

  If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
  A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
  A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).

  The elements of the vectors v and u together form the m-by-nb matrix
  V and the nb-by-n matrix U**H which are needed, with X and Y, to apply
  the transformation to the unreduced part of the matrix, using a block
  update of the form:  A := A - V*Y**H - X*U**H.

  The contents of A on exit are illustrated by the following examples
  with nb = 2:

  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):

    (  1   1   u1  u1  u1 )           (  1   u1  u1  u1  u1  u1 )
    (  v1  1   1   u2  u2 )           (  1   1   u2  u2  u2  u2 )
    (  v1  v2  a   a   a  )           (  v1  1   a   a   a   a  )
    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    (  v1  v2  a   a   a  )           (  v1  v2  a   a   a   a  )
    (  v1  v2  a   a   a  )

  where a denotes an element of the original matrix which is unchanged,
  vi denotes an element of the vector defining H(i), and ui an element
  of the vector defining G(i).

Definition at line 210 of file clabrd.f.

subroutine clacgv (integer N, complex, dimension( * ) X, integer INCX)

CLACGV conjugates a complex vector.  

Purpose:

 CLACGV conjugates a complex vector of length N.
Parameters

N

          N is INTEGER
          The length of the vector X.  N >= 0.

X

          X is COMPLEX array, dimension
                         (1+(N-1)*abs(INCX))
          On entry, the vector of length N to be conjugated.
          On exit, X is overwritten with conjg(X).

INCX

          INCX is INTEGER
          The spacing between successive elements of X.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 73 of file clacgv.f.

subroutine clacn2 (integer N, complex, dimension( * ) V, complex, dimension( * ) X, real EST, integer KASE, integer, dimension( 3 ) ISAVE)

CLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.  

Purpose:

 CLACN2 estimates the 1-norm of a square, complex matrix A.
 Reverse communication is used for evaluating matrix-vector products.
Parameters

N

          N is INTEGER
         The order of the matrix.  N >= 1.

V

          V is COMPLEX array, dimension (N)
         On the final return, V = A*W,  where  EST = norm(V)/norm(W)
         (W is not returned).

X

          X is COMPLEX array, dimension (N)
         On an intermediate return, X should be overwritten by
               A * X,   if KASE=1,
               A**H * X,  if KASE=2,
         where A**H is the conjugate transpose of A, and CLACN2 must be
         re-called with all the other parameters unchanged.

EST

          EST is REAL
         On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be
         unchanged from the previous call to CLACN2.
         On exit, EST is an estimate (a lower bound) for norm(A).

KASE

          KASE is INTEGER
         On the initial call to CLACN2, KASE should be 0.
         On an intermediate return, KASE will be 1 or 2, indicating
         whether X should be overwritten by A * X  or A**H * X.
         On the final return from CLACN2, KASE will again be 0.

ISAVE

          ISAVE is INTEGER array, dimension (3)
         ISAVE is used to save variables between calls to SLACN2
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  Originally named CONEST, dated March 16, 1988.

  Last modified:  April, 1999

  This is a thread safe version of CLACON, which uses the array ISAVE
  in place of a SAVE statement, as follows:

     CLACON     CLACN2
      JUMP     ISAVE(1)
      J        ISAVE(2)
      ITER     ISAVE(3)
Contributors:

Nick Higham, University of Manchester

References:

N.J. Higham, 'FORTRAN codes for estimating the one-norm of
 a real or complex matrix, with applications to condition estimation', ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

Definition at line 132 of file clacn2.f.

subroutine clacon (integer N, complex, dimension( n ) V, complex, dimension( n ) X, real EST, integer KASE)

CLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vector products.  

Purpose:

 CLACON estimates the 1-norm of a square, complex matrix A.
 Reverse communication is used for evaluating matrix-vector products.
Parameters

N

          N is INTEGER
         The order of the matrix.  N >= 1.

V

          V is COMPLEX array, dimension (N)
         On the final return, V = A*W,  where  EST = norm(V)/norm(W)
         (W is not returned).

X

          X is COMPLEX array, dimension (N)
         On an intermediate return, X should be overwritten by
               A * X,   if KASE=1,
               A**H * X,  if KASE=2,
         where A**H is the conjugate transpose of A, and CLACON must be
         re-called with all the other parameters unchanged.

EST

          EST is REAL
         On entry with KASE = 1 or 2 and JUMP = 3, EST should be
         unchanged from the previous call to CLACON.
         On exit, EST is an estimate (a lower bound) for norm(A).

KASE

          KASE is INTEGER
         On the initial call to CLACON, KASE should be 0.
         On an intermediate return, KASE will be 1 or 2, indicating
         whether X should be overwritten by A * X  or A**H * X.
         On the final return from CLACON, KASE will again be 0.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

Originally named CONEST, dated March 16, 1988.
Last modified: April, 1999

Contributors:

Nick Higham, University of Manchester

References:

N.J. Higham, 'FORTRAN codes for estimating the one-norm of
 a real or complex matrix, with applications to condition estimation', ACM Trans. Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.

Definition at line 113 of file clacon.f.

subroutine clacp2 (character UPLO, integer M, integer N, real, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB)

CLACP2 copies all or part of a real two-dimensional array to a complex array.  

Purpose:

 CLACP2 copies all or part of a real two-dimensional matrix A to a
 complex matrix B.
Parameters

UPLO

          UPLO is CHARACTER*1
          Specifies the part of the matrix A to be copied to B.
          = 'U':      Upper triangular part
          = 'L':      Lower triangular part
          Otherwise:  All of the matrix A

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

N

          N is INTEGER
          The number of columns of the matrix A.  N >= 0.

A

          A is REAL array, dimension (LDA,N)
          The m by n matrix A.  If UPLO = 'U', only the upper trapezium
          is accessed; if UPLO = 'L', only the lower trapezium is
          accessed.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

B

          B is COMPLEX array, dimension (LDB,N)
          On exit, B = A in the locations specified by UPLO.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,M).
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 103 of file clacp2.f.

subroutine clacpy (character UPLO, integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB)

CLACPY copies all or part of one two-dimensional array to another.  

Purpose:

 CLACPY copies all or part of a two-dimensional matrix A to another
 matrix B.
Parameters

UPLO

          UPLO is CHARACTER*1
          Specifies the part of the matrix A to be copied to B.
          = 'U':      Upper triangular part
          = 'L':      Lower triangular part
          Otherwise:  All of the matrix A

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

N

          N is INTEGER
          The number of columns of the matrix A.  N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          The m by n matrix A.  If UPLO = 'U', only the upper trapezium
          is accessed; if UPLO = 'L', only the lower trapezium is
          accessed.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

B

          B is COMPLEX array, dimension (LDB,N)
          On exit, B = A in the locations specified by UPLO.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,M).
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 102 of file clacpy.f.

subroutine clacrm (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, complex, dimension( ldc, * ) C, integer LDC, real, dimension( * ) RWORK)

CLACRM multiplies a complex matrix by a square real matrix.  

Purpose:

 CLACRM performs a very simple matrix-matrix multiplication:
          C := A * B,
 where A is M by N and complex; B is N by N and real;
 C is M by N and complex.
Parameters

M

          M is INTEGER
          The number of rows of the matrix A and of the matrix C.
          M >= 0.

N

          N is INTEGER
          The number of columns and rows of the matrix B and
          the number of columns of the matrix C.
          N >= 0.

A

          A is COMPLEX array, dimension (LDA, N)
          On entry, A contains the M by N matrix A.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >=max(1,M).

B

          B is REAL array, dimension (LDB, N)
          On entry, B contains the N by N matrix B.

LDB

          LDB is INTEGER
          The leading dimension of the array B. LDB >=max(1,N).

C

          C is COMPLEX array, dimension (LDC, N)
          On exit, C contains the M by N matrix C.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >=max(1,N).

RWORK

          RWORK is REAL array, dimension (2*M*N)
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 113 of file clacrm.f.

subroutine clacrt (integer N, complex, dimension( * ) CX, integer INCX, complex, dimension( * ) CY, integer INCY, complex C, complex S)

CLACRT performs a linear transformation of a pair of complex vectors.  

Purpose:

 CLACRT performs the operation

    (  c  s )( x )  ==> ( x )
    ( -s  c )( y )      ( y )

 where c and s are complex and the vectors x and y are complex.
Parameters

N

          N is INTEGER
          The number of elements in the vectors CX and CY.

CX

          CX is COMPLEX array, dimension (N)
          On input, the vector x.
          On output, CX is overwritten with c*x + s*y.

INCX

          INCX is INTEGER
          The increment between successive values of CX.  INCX <> 0.

CY

          CY is COMPLEX array, dimension (N)
          On input, the vector y.
          On output, CY is overwritten with -s*x + c*y.

INCY

          INCY is INTEGER
          The increment between successive values of CY.  INCY <> 0.

C

          C is COMPLEX

S

          S is COMPLEX
          C and S define the matrix
             [  C   S  ].
             [ -S   C  ]
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 104 of file clacrt.f.

complex function cladiv (complex X, complex Y)

CLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.  

Purpose:

 CLADIV := X / Y, where X and Y are complex.  The computation of X / Y
 will not overflow on an intermediary step unless the results
 overflows.
Parameters

X

          X is COMPLEX

Y

          Y is COMPLEX
          The complex scalars X and Y.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 63 of file cladiv.f.

subroutine claein (logical RIGHTV, logical NOINIT, integer N, complex, dimension( ldh, * ) H, integer LDH, complex W, complex, dimension( * ) V, complex, dimension( ldb, * ) B, integer LDB, real, dimension( * ) RWORK, real EPS3, real SMLNUM, integer INFO)

CLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by inverse iteration.  

Purpose:

 CLAEIN uses inverse iteration to find a right or left eigenvector
 corresponding to the eigenvalue W of a complex upper Hessenberg
 matrix H.
Parameters

RIGHTV

          RIGHTV is LOGICAL
          = .TRUE. : compute right eigenvector;
          = .FALSE.: compute left eigenvector.

NOINIT

          NOINIT is LOGICAL
          = .TRUE. : no initial vector supplied in V
          = .FALSE.: initial vector supplied in V.

N

          N is INTEGER
          The order of the matrix H.  N >= 0.

H

          H is COMPLEX array, dimension (LDH,N)
          The upper Hessenberg matrix H.

LDH

          LDH is INTEGER
          The leading dimension of the array H.  LDH >= max(1,N).

W

          W is COMPLEX
          The eigenvalue of H whose corresponding right or left
          eigenvector is to be computed.

V

          V is COMPLEX array, dimension (N)
          On entry, if NOINIT = .FALSE., V must contain a starting
          vector for inverse iteration; otherwise V need not be set.
          On exit, V contains the computed eigenvector, normalized so
          that the component of largest magnitude has magnitude 1; here
          the magnitude of a complex number (x,y) is taken to be
          |x| + |y|.

B

          B is COMPLEX array, dimension (LDB,N)

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).

RWORK

          RWORK is REAL array, dimension (N)

EPS3

          EPS3 is REAL
          A small machine-dependent value which is used to perturb
          close eigenvalues, and to replace zero pivots.

SMLNUM

          SMLNUM is REAL
          A machine-dependent value close to the underflow threshold.

INFO

          INFO is INTEGER
          = 0:  successful exit
          = 1:  inverse iteration did not converge; V is set to the
                last iterate.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 147 of file claein.f.

subroutine claev2 (complex A, complex B, complex C, real RT1, real RT2, real CS1, complex SN1)

CLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.  

Purpose:

 CLAEV2 computes the eigendecomposition of a 2-by-2 Hermitian matrix
    [  A         B  ]
    [  CONJG(B)  C  ].
 On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
 eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
 eigenvector for RT1, giving the decomposition

 [ CS1  CONJG(SN1) ] [    A     B ] [ CS1 -CONJG(SN1) ] = [ RT1  0  ]
 [-SN1     CS1     ] [ CONJG(B) C ] [ SN1     CS1     ]   [  0  RT2 ].
Parameters

A

          A is COMPLEX
         The (1,1) element of the 2-by-2 matrix.

B

          B is COMPLEX
         The (1,2) element and the conjugate of the (2,1) element of
         the 2-by-2 matrix.

C

          C is COMPLEX
         The (2,2) element of the 2-by-2 matrix.

RT1

          RT1 is REAL
         The eigenvalue of larger absolute value.

RT2

          RT2 is REAL
         The eigenvalue of smaller absolute value.

CS1

          CS1 is REAL

SN1

          SN1 is COMPLEX
         The vector (CS1, SN1) is a unit right eigenvector for RT1.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  RT1 is accurate to a few ulps barring over/underflow.

  RT2 may be inaccurate if there is massive cancellation in the
  determinant A*C-B*B; higher precision or correctly rounded or
  correctly truncated arithmetic would be needed to compute RT2
  accurately in all cases.

  CS1 and SN1 are accurate to a few ulps barring over/underflow.

  Overflow is possible only if RT1 is within a factor of 5 of overflow.
  Underflow is harmless if the input data is 0 or exceeds
     underflow_threshold / macheps.

Definition at line 120 of file claev2.f.

subroutine clags2 (logical UPPER, real A1, complex A2, real A3, real B1, complex B2, real B3, real CSU, complex SNU, real CSV, complex SNV, real CSQ, complex SNQ)

CLAGS2  

Purpose:

 CLAGS2 computes 2-by-2 unitary matrices U, V and Q, such
 that if ( UPPER ) then

           U**H *A*Q = U**H *( A1 A2 )*Q = ( x  0  )
                             ( 0  A3 )     ( x  x  )
 and
           V**H*B*Q = V**H *( B1 B2 )*Q = ( x  0  )
                            ( 0  B3 )     ( x  x  )

 or if ( .NOT.UPPER ) then

           U**H *A*Q = U**H *( A1 0  )*Q = ( x  x  )
                             ( A2 A3 )     ( 0  x  )
 and
           V**H *B*Q = V**H *( B1 0  )*Q = ( x  x  )
                             ( B2 B3 )     ( 0  x  )
 where

   U = (   CSU    SNU ), V = (  CSV    SNV ),
       ( -SNU**H  CSU )      ( -SNV**H CSV )

   Q = (   CSQ    SNQ )
       ( -SNQ**H  CSQ )

 The rows of the transformed A and B are parallel. Moreover, if the
 input 2-by-2 matrix A is not zero, then the transformed (1,1) entry
 of A is not zero. If the input matrices A and B are both not zero,
 then the transformed (2,2) element of B is not zero, except when the
 first rows of input A and B are parallel and the second rows are
 zero.
Parameters

UPPER

          UPPER is LOGICAL
          = .TRUE.: the input matrices A and B are upper triangular.
          = .FALSE.: the input matrices A and B are lower triangular.

A1

          A1 is REAL

A2

          A2 is COMPLEX

A3

          A3 is REAL
          On entry, A1, A2 and A3 are elements of the input 2-by-2
          upper (lower) triangular matrix A.

B1

          B1 is REAL

B2

          B2 is COMPLEX

B3

          B3 is REAL
          On entry, B1, B2 and B3 are elements of the input 2-by-2
          upper (lower) triangular matrix B.

CSU

          CSU is REAL

SNU

          SNU is COMPLEX
          The desired unitary matrix U.

CSV

          CSV is REAL

SNV

          SNV is COMPLEX
          The desired unitary matrix V.

CSQ

          CSQ is REAL

SNQ

          SNQ is COMPLEX
          The desired unitary matrix Q.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 156 of file clags2.f.

subroutine clagtm (character TRANS, integer N, integer NRHS, real ALPHA, complex, dimension( * ) DL, complex, dimension( * ) D, complex, dimension( * ) DU, complex, dimension( ldx, * ) X, integer LDX, real BETA, complex, dimension( ldb, * ) B, integer LDB)

CLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or -1.  

Purpose:

 CLAGTM performs a matrix-vector product of the form

    B := alpha * A * X + beta * B

 where A is a tridiagonal matrix of order N, B and X are N by NRHS
 matrices, and alpha and beta are real scalars, each of which may be
 0., 1., or -1.
Parameters

TRANS

          TRANS is CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  No transpose, B := alpha * A * X + beta * B
          = 'T':  Transpose,    B := alpha * A**T * X + beta * B
          = 'C':  Conjugate transpose, B := alpha * A**H * X + beta * B

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

NRHS

          NRHS is INTEGER
          The number of right hand sides, i.e., the number of columns
          of the matrices X and B.

ALPHA

          ALPHA is REAL
          The scalar alpha.  ALPHA must be 0., 1., or -1.; otherwise,
          it is assumed to be 0.

DL

          DL is COMPLEX array, dimension (N-1)
          The (n-1) sub-diagonal elements of T.

D

          D is COMPLEX array, dimension (N)
          The diagonal elements of T.

DU

          DU is COMPLEX array, dimension (N-1)
          The (n-1) super-diagonal elements of T.

X

          X is COMPLEX array, dimension (LDX,NRHS)
          The N by NRHS matrix X.

LDX

          LDX is INTEGER
          The leading dimension of the array X.  LDX >= max(N,1).

BETA

          BETA is REAL
          The scalar beta.  BETA must be 0., 1., or -1.; otherwise,
          it is assumed to be 1.

B

          B is COMPLEX array, dimension (LDB,NRHS)
          On entry, the N by NRHS matrix B.
          On exit, B is overwritten by the matrix expression
          B := alpha * A * X + beta * B.

LDB

          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(N,1).
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 143 of file clagtm.f.

subroutine clahqr (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, complex, dimension( ldh, * ) H, integer LDH, complex, dimension( * ) W, integer ILOZ, integer IHIZ, complex, dimension( ldz, * ) Z, integer LDZ, integer INFO)

CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.  

Purpose:

    CLAHQR is an auxiliary routine called by CHSEQR to update the
    eigenvalues and Schur decomposition already computed by CHSEQR, by
    dealing with the Hessenberg submatrix in rows and columns ILO to
    IHI.
Parameters

WANTT

          WANTT is LOGICAL
          = .TRUE. : the full Schur form T is required;
          = .FALSE.: only eigenvalues are required.

WANTZ

          WANTZ is LOGICAL
          = .TRUE. : the matrix of Schur vectors Z is required;
          = .FALSE.: Schur vectors are not required.

N

          N is INTEGER
          The order of the matrix H.  N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
          It is assumed that H is already upper triangular in rows and
          columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1).
          CLAHQR works primarily with the Hessenberg submatrix in rows
          and columns ILO to IHI, but applies transformations to all of
          H if WANTT is .TRUE..
          1 <= ILO <= max(1,IHI); IHI <= N.

H

          H is COMPLEX array, dimension (LDH,N)
          On entry, the upper Hessenberg matrix H.
          On exit, if INFO is zero and if WANTT is .TRUE., then H
          is upper triangular in rows and columns ILO:IHI.  If INFO
          is zero and if WANTT is .FALSE., then the contents of H
          are unspecified on exit.  The output state of H in case
          INF is positive is below under the description of INFO.

LDH

          LDH is INTEGER
          The leading dimension of the array H. LDH >= max(1,N).

W

          W is COMPLEX array, dimension (N)
          The computed eigenvalues ILO to IHI are stored in the
          corresponding elements of W. If WANTT is .TRUE., the
          eigenvalues are stored in the same order as on the diagonal
          of the Schur form returned in H, with W(i) = H(i,i).

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER
          Specify the rows of Z to which transformations must be
          applied if WANTZ is .TRUE..
          1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

Z

          Z is COMPLEX array, dimension (LDZ,N)
          If WANTZ is .TRUE., on entry Z must contain the current
          matrix Z of transformations accumulated by CHSEQR, and on
          exit Z has been updated; transformations are applied only to
          the submatrix Z(ILOZ:IHIZ,ILO:IHI).
          If WANTZ is .FALSE., Z is not referenced.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z. LDZ >= max(1,N).

INFO

          INFO is INTEGER
           = 0:  successful exit
           > 0:  if INFO = i, CLAHQR failed to compute all the
                  eigenvalues ILO to IHI in a total of 30 iterations
                  per eigenvalue; elements i+1:ihi of W contain
                  those eigenvalues which have been successfully
                  computed.

                  If INFO > 0 and WANTT is .FALSE., then on exit,
                  the remaining unconverged eigenvalues are the
                  eigenvalues of the upper Hessenberg matrix
                  rows and columns ILO through INFO of the final,
                  output value of H.

                  If INFO > 0 and WANTT is .TRUE., then on exit
          (*)       (initial value of H)*U  = U*(final value of H)
                  where U is an orthogonal matrix.    The final
                  value of H is upper Hessenberg and triangular in
                  rows and columns INFO+1 through IHI.

                  If INFO > 0 and WANTZ is .TRUE., then on exit
                      (final value of Z)  = (initial value of Z)*U
                  where U is the orthogonal matrix in (*)
                  (regardless of the value of WANTT.)
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

     02-96 Based on modifications by
     David Day, Sandia National Laboratory, USA

     12-04 Further modifications by
     Ralph Byers, University of Kansas, USA
     This is a modified version of CLAHQR from LAPACK version 3.0.
     It is (1) more robust against overflow and underflow and
     (2) adopts the more conservative Ahues & Tisseur stopping
     criterion (LAWN 122, 1997).

Definition at line 193 of file clahqr.f.

subroutine clahr2 (integer N, integer K, integer NB, complex, dimension( lda, * ) A, integer LDA, complex, dimension( nb ) TAU, complex, dimension( ldt, nb ) T, integer LDT, complex, dimension( ldy, nb ) Y, integer LDY)

CLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.  

Purpose:

 CLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1)
 matrix A so that elements below the k-th subdiagonal are zero. The
 reduction is performed by an unitary similarity transformation
 Q**H * A * Q. The routine returns the matrices V and T which determine
 Q as a block reflector I - V*T*v**H, and also the matrix Y = A * V * T.

 This is an auxiliary routine called by CGEHRD.
Parameters

N

          N is INTEGER
          The order of the matrix A.

K

          K is INTEGER
          The offset for the reduction. Elements below the k-th
          subdiagonal in the first NB columns are reduced to zero.
          K < N.

NB

          NB is INTEGER
          The number of columns to be reduced.

A

          A is COMPLEX array, dimension (LDA,N-K+1)
          On entry, the n-by-(n-k+1) general matrix A.
          On exit, the elements on and above the k-th subdiagonal in
          the first NB columns are overwritten with the corresponding
          elements of the reduced matrix; the elements below the k-th
          subdiagonal, with the array TAU, represent the matrix Q as a
          product of elementary reflectors. The other columns of A are
          unchanged. See Further Details.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

TAU

          TAU is COMPLEX array, dimension (NB)
          The scalar factors of the elementary reflectors. See Further
          Details.

T

          T is COMPLEX array, dimension (LDT,NB)
          The upper triangular matrix T.

LDT

          LDT is INTEGER
          The leading dimension of the array T.  LDT >= NB.

Y

          Y is COMPLEX array, dimension (LDY,NB)
          The n-by-nb matrix Y.

LDY

          LDY is INTEGER
          The leading dimension of the array Y. LDY >= N.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  The matrix Q is represented as a product of nb elementary reflectors

     Q = H(1) H(2) . . . H(nb).

  Each H(i) has the form

     H(i) = I - tau * v * v**H

  where tau is a complex scalar, and v is a complex vector with
  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
  A(i+k+1:n,i), and tau in TAU(i).

  The elements of the vectors v together form the (n-k+1)-by-nb matrix
  V which is needed, with T and Y, to apply the transformation to the
  unreduced part of the matrix, using an update of the form:
  A := (I - V*T*V**H) * (A - Y*V**H).

  The contents of A on exit are illustrated by the following example
  with n = 7, k = 3 and nb = 2:

     ( a   a   a   a   a )
     ( a   a   a   a   a )
     ( a   a   a   a   a )
     ( h   h   a   a   a )
     ( v1  h   a   a   a )
     ( v1  v2  a   a   a )
     ( v1  v2  a   a   a )

  where a denotes an element of the original matrix A, h denotes a
  modified element of the upper Hessenberg matrix H, and vi denotes an
  element of the vector defining H(i).

  This subroutine is a slight modification of LAPACK-3.0's DLAHRD
  incorporating improvements proposed by Quintana-Orti and Van de
  Gejin. Note that the entries of A(1:K,2:NB) differ from those
  returned by the original LAPACK-3.0's DLAHRD routine. (This
  subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
References:

Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the
 performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180-194, June 2006.

Definition at line 180 of file clahr2.f.

subroutine claic1 (integer JOB, integer J, complex, dimension( j ) X, real SEST, complex, dimension( j ) W, complex GAMMA, real SESTPR, complex S, complex C)

CLAIC1 applies one step of incremental condition estimation.  

Purpose:

 CLAIC1 applies one step of incremental condition estimation in
 its simplest version:

 Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
 lower triangular matrix L, such that
          twonorm(L*x) = sest
 Then CLAIC1 computes sestpr, s, c such that
 the vector
                 [ s*x ]
          xhat = [  c  ]
 is an approximate singular vector of
                 [ L      0  ]
          Lhat = [ w**H gamma ]
 in the sense that
          twonorm(Lhat*xhat) = sestpr.

 Depending on JOB, an estimate for the largest or smallest singular
 value is computed.

 Note that [s c]**H and sestpr**2 is an eigenpair of the system

     diag(sest*sest, 0) + [alpha  gamma] * [ conjg(alpha) ]
                                           [ conjg(gamma) ]

 where  alpha =  x**H*w.
Parameters

JOB

          JOB is INTEGER
          = 1: an estimate for the largest singular value is computed.
          = 2: an estimate for the smallest singular value is computed.

J

          J is INTEGER
          Length of X and W

X

          X is COMPLEX array, dimension (J)
          The j-vector x.

SEST

          SEST is REAL
          Estimated singular value of j by j matrix L

W

          W is COMPLEX array, dimension (J)
          The j-vector w.

GAMMA

          GAMMA is COMPLEX
          The diagonal element gamma.

SESTPR

          SESTPR is REAL
          Estimated singular value of (j+1) by (j+1) matrix Lhat.

S

          S is COMPLEX
          Sine needed in forming xhat.

C

          C is COMPLEX
          Cosine needed in forming xhat.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 134 of file claic1.f.

real function clangt (character NORM, integer N, complex, dimension( * ) DL, complex, dimension( * ) D, complex, dimension( * ) DU)

CLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of a general tridiagonal matrix.  

Purpose:

 CLANGT  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 complex tridiagonal matrix A.
Returns

CLANGT

    CLANGT = ( max(abs(A(i,j))), NORM = 'M' or 'm'
             (
             ( norm1(A),         NORM = '1', 'O' or 'o'
             (
             ( normI(A),         NORM = 'I' or 'i'
             (
             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

 where  norm1  denotes the  one norm of a matrix (maximum column sum),
 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
 normF  denotes the  Frobenius norm of a matrix (square root of sum of
 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
Parameters

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in CLANGT as described
          above.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, CLANGT is
          set to zero.

DL

          DL is COMPLEX array, dimension (N-1)
          The (n-1) sub-diagonal elements of A.

D

          D is COMPLEX array, dimension (N)
          The diagonal elements of A.

DU

          DU is COMPLEX array, dimension (N-1)
          The (n-1) super-diagonal elements of A.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 105 of file clangt.f.

real function clanhb (character NORM, character UPLO, integer N, integer K, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) WORK)

CLANHB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian band matrix.  

Purpose:

 CLANHB  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the element of  largest absolute value  of an
 n by n hermitian band matrix A,  with k super-diagonals.
Returns

CLANHB

    CLANHB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
             (
             ( norm1(A),         NORM = '1', 'O' or 'o'
             (
             ( normI(A),         NORM = 'I' or 'i'
             (
             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

 where  norm1  denotes the  one norm of a matrix (maximum column sum),
 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
 normF  denotes the  Frobenius norm of a matrix (square root of sum of
 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
Parameters

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in CLANHB as described
          above.

UPLO

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          band matrix A is supplied.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

N

          N is INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, CLANHB is
          set to zero.

K

          K is INTEGER
          The number of super-diagonals or sub-diagonals of the
          band matrix A.  K >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          The upper or lower triangle of the hermitian band matrix A,
          stored in the first K+1 rows of AB.  The j-th column of A is
          stored in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).
          Note that the imaginary parts of the diagonal elements need
          not be set and are assumed to be zero.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= K+1.

WORK

          WORK is REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
          WORK is not referenced.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 130 of file clanhb.f.

real function clanhp (character NORM, character UPLO, integer N, complex, dimension( * ) AP, real, dimension( * ) WORK)

CLANHP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian matrix supplied in packed form.  

Purpose:

 CLANHP  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 complex hermitian matrix A,  supplied in packed form.
Returns

CLANHP

    CLANHP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
             (
             ( norm1(A),         NORM = '1', 'O' or 'o'
             (
             ( normI(A),         NORM = 'I' or 'i'
             (
             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

 where  norm1  denotes the  one norm of a matrix (maximum column sum),
 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
 normF  denotes the  Frobenius norm of a matrix (square root of sum of
 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
Parameters

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in CLANHP as described
          above.

UPLO

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          hermitian matrix A is supplied.
          = 'U':  Upper triangular part of A is supplied
          = 'L':  Lower triangular part of A is supplied

N

          N is INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, CLANHP is
          set to zero.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The upper or lower triangle of the hermitian matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          Note that the  imaginary parts of the diagonal elements need
          not be set and are assumed to be zero.

WORK

          WORK is REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
          WORK is not referenced.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 116 of file clanhp.f.

real function clanhs (character NORM, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK)

CLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value of any element of an upper Hessenberg matrix.  

Purpose:

 CLANHS  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 Hessenberg matrix A.
Returns

CLANHS

    CLANHS = ( max(abs(A(i,j))), NORM = 'M' or 'm'
             (
             ( norm1(A),         NORM = '1', 'O' or 'o'
             (
             ( normI(A),         NORM = 'I' or 'i'
             (
             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

 where  norm1  denotes the  one norm of a matrix (maximum column sum),
 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
 normF  denotes the  Frobenius norm of a matrix (square root of sum of
 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
Parameters

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in CLANHS as described
          above.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, CLANHS is
          set to zero.

A

          A is COMPLEX array, dimension (LDA,N)
          The n by n upper Hessenberg matrix A; the part of A below the
          first sub-diagonal is not referenced.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(N,1).

WORK

          WORK is REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
          referenced.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 108 of file clanhs.f.

real function clanht (character NORM, integer N, real, dimension( * ) D, complex, dimension( * ) E)

CLANHT returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a complex Hermitian tridiagonal matrix.  

Purpose:

 CLANHT  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 complex Hermitian tridiagonal matrix A.
Returns

CLANHT

    CLANHT = ( max(abs(A(i,j))), NORM = 'M' or 'm'
             (
             ( norm1(A),         NORM = '1', 'O' or 'o'
             (
             ( normI(A),         NORM = 'I' or 'i'
             (
             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

 where  norm1  denotes the  one norm of a matrix (maximum column sum),
 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
 normF  denotes the  Frobenius norm of a matrix (square root of sum of
 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
Parameters

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in CLANHT as described
          above.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, CLANHT is
          set to zero.

D

          D is REAL array, dimension (N)
          The diagonal elements of A.

E

          E is COMPLEX array, dimension (N-1)
          The (n-1) sub-diagonal or super-diagonal elements of A.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 100 of file clanht.f.

real function clansb (character NORM, character UPLO, integer N, integer K, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) WORK)

CLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric band matrix.  

Purpose:

 CLANSB  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the element of  largest absolute value  of an
 n by n symmetric band matrix A,  with k super-diagonals.
Returns

CLANSB

    CLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
             (
             ( norm1(A),         NORM = '1', 'O' or 'o'
             (
             ( normI(A),         NORM = 'I' or 'i'
             (
             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

 where  norm1  denotes the  one norm of a matrix (maximum column sum),
 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
 normF  denotes the  Frobenius norm of a matrix (square root of sum of
 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
Parameters

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in CLANSB as described
          above.

UPLO

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          band matrix A is supplied.
          = 'U':  Upper triangular part is supplied
          = 'L':  Lower triangular part is supplied

N

          N is INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, CLANSB is
          set to zero.

K

          K is INTEGER
          The number of super-diagonals or sub-diagonals of the
          band matrix A.  K >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          The upper or lower triangle of the symmetric band matrix A,
          stored in the first K+1 rows of AB.  The j-th column of A is
          stored in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= K+1.

WORK

          WORK is REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
          WORK is not referenced.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 128 of file clansb.f.

real function clansp (character NORM, character UPLO, integer N, complex, dimension( * ) AP, real, dimension( * ) WORK)

CLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix supplied in packed form.  

Purpose:

 CLANSP  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 complex symmetric matrix A,  supplied in packed form.
Returns

CLANSP

    CLANSP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
             (
             ( norm1(A),         NORM = '1', 'O' or 'o'
             (
             ( normI(A),         NORM = 'I' or 'i'
             (
             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

 where  norm1  denotes the  one norm of a matrix (maximum column sum),
 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
 normF  denotes the  Frobenius norm of a matrix (square root of sum of
 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
Parameters

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in CLANSP as described
          above.

UPLO

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is supplied.
          = 'U':  Upper triangular part of A is supplied
          = 'L':  Lower triangular part of A is supplied

N

          N is INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, CLANSP is
          set to zero.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The upper or lower triangle of the symmetric matrix A, packed
          columnwise in a linear array.  The j-th column of A is stored
          in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

WORK

          WORK is REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
          WORK is not referenced.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 114 of file clansp.f.

real function clantb (character NORM, character UPLO, character DIAG, integer N, integer K, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) WORK)

CLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular band matrix.  

Purpose:

 CLANTB  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the element of  largest absolute value  of an
 n by n triangular band matrix A,  with ( k + 1 ) diagonals.
Returns

CLANTB

    CLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
             (
             ( norm1(A),         NORM = '1', 'O' or 'o'
             (
             ( normI(A),         NORM = 'I' or 'i'
             (
             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

 where  norm1  denotes the  one norm of a matrix (maximum column sum),
 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
 normF  denotes the  Frobenius norm of a matrix (square root of sum of
 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
Parameters

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in CLANTB as described
          above.

UPLO

          UPLO is CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

DIAG

          DIAG is CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

N

          N is INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, CLANTB is
          set to zero.

K

          K is INTEGER
          The number of super-diagonals of the matrix A if UPLO = 'U',
          or the number of sub-diagonals of the matrix A if UPLO = 'L'.
          K >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first k+1 rows of AB.  The j-th column of A is stored
          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)   = A(i,j) for j<=i<=min(n,j+k).
          Note that when DIAG = 'U', the elements of the array AB
          corresponding to the diagonal elements of the matrix A are
          not referenced, but are assumed to be one.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= K+1.

WORK

          WORK is REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
          referenced.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 139 of file clantb.f.

real function clantp (character NORM, character UPLO, character DIAG, integer N, complex, dimension( * ) AP, real, dimension( * ) WORK)

CLANTP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a triangular matrix supplied in packed form.  

Purpose:

 CLANTP  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 triangular matrix A, supplied in packed form.
Returns

CLANTP

    CLANTP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
             (
             ( norm1(A),         NORM = '1', 'O' or 'o'
             (
             ( normI(A),         NORM = 'I' or 'i'
             (
             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

 where  norm1  denotes the  one norm of a matrix (maximum column sum),
 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
 normF  denotes the  Frobenius norm of a matrix (square root of sum of
 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
Parameters

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in CLANTP as described
          above.

UPLO

          UPLO is CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

DIAG

          DIAG is CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

N

          N is INTEGER
          The order of the matrix A.  N >= 0.  When N = 0, CLANTP is
          set to zero.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in
          a linear array.  The j-th column of A is stored in the array
          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
          Note that when DIAG = 'U', the elements of the array AP
          corresponding to the diagonal elements of the matrix A are
          not referenced, but are assumed to be one.

WORK

          WORK is REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= N when NORM = 'I'; otherwise, WORK is not
          referenced.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 124 of file clantp.f.

real function clantr (character NORM, character UPLO, character DIAG, integer M, integer N, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) WORK)

CLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix.  

Purpose:

 CLANTR  returns the value of the one norm,  or the Frobenius norm, or
 the  infinity norm,  or the  element of  largest absolute value  of a
 trapezoidal or triangular matrix A.
Returns

CLANTR

    CLANTR = ( max(abs(A(i,j))), NORM = 'M' or 'm'
             (
             ( norm1(A),         NORM = '1', 'O' or 'o'
             (
             ( normI(A),         NORM = 'I' or 'i'
             (
             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'

 where  norm1  denotes the  one norm of a matrix (maximum column sum),
 normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
 normF  denotes the  Frobenius norm of a matrix (square root of sum of
 squares).  Note that  max(abs(A(i,j)))  is not a consistent matrix norm.
Parameters

NORM

          NORM is CHARACTER*1
          Specifies the value to be returned in CLANTR as described
          above.

UPLO

          UPLO is CHARACTER*1
          Specifies whether the matrix A is upper or lower trapezoidal.
          = 'U':  Upper trapezoidal
          = 'L':  Lower trapezoidal
          Note that A is triangular instead of trapezoidal if M = N.

DIAG

          DIAG is CHARACTER*1
          Specifies whether or not the matrix A has unit diagonal.
          = 'N':  Non-unit diagonal
          = 'U':  Unit diagonal

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0, and if
          UPLO = 'U', M <= N.  When M = 0, CLANTR is set to zero.

N

          N is INTEGER
          The number of columns of the matrix A.  N >= 0, and if
          UPLO = 'L', N <= M.  When N = 0, CLANTR is set to zero.

A

          A is COMPLEX array, dimension (LDA,N)
          The trapezoidal matrix A (A is triangular if M = N).
          If UPLO = 'U', the leading m by n upper trapezoidal part of
          the array A contains the upper trapezoidal matrix, and the
          strictly lower triangular part of A is not referenced.
          If UPLO = 'L', the leading m by n lower trapezoidal part of
          the array A contains the lower trapezoidal matrix, and the
          strictly upper triangular part of A is not referenced.  Note
          that when DIAG = 'U', the diagonal elements of A are not
          referenced and are assumed to be one.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(M,1).

WORK

          WORK is REAL array, dimension (MAX(1,LWORK)),
          where LWORK >= M when NORM = 'I'; otherwise, WORK is not
          referenced.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 140 of file clantr.f.

subroutine clapll (integer N, complex, dimension( * ) X, integer INCX, complex, dimension( * ) Y, integer INCY, real SSMIN)

CLAPLL measures the linear dependence of two vectors.  

Purpose:

 Given two column vectors X and Y, let

                      A = ( X Y ).

 The subroutine first computes the QR factorization of A = Q*R,
 and then computes the SVD of the 2-by-2 upper triangular matrix R.
 The smaller singular value of R is returned in SSMIN, which is used
 as the measurement of the linear dependency of the vectors X and Y.
Parameters

N

          N is INTEGER
          The length of the vectors X and Y.

X

          X is COMPLEX array, dimension (1+(N-1)*INCX)
          On entry, X contains the N-vector X.
          On exit, X is overwritten.

INCX

          INCX is INTEGER
          The increment between successive elements of X. INCX > 0.

Y

          Y is COMPLEX array, dimension (1+(N-1)*INCY)
          On entry, Y contains the N-vector Y.
          On exit, Y is overwritten.

INCY

          INCY is INTEGER
          The increment between successive elements of Y. INCY > 0.

SSMIN

          SSMIN is REAL
          The smallest singular value of the N-by-2 matrix A = ( X Y ).
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 99 of file clapll.f.

subroutine clapmr (logical FORWRD, integer M, integer N, complex, dimension( ldx, * ) X, integer LDX, integer, dimension( * ) K)

CLAPMR rearranges rows of a matrix as specified by a permutation vector.  

Purpose:

 CLAPMR rearranges the rows of the M by N matrix X as specified
 by the permutation K(1),K(2),...,K(M) of the integers 1,...,M.
 If FORWRD = .TRUE.,  forward permutation:

      X(K(I),*) is moved X(I,*) for I = 1,2,...,M.

 If FORWRD = .FALSE., backward permutation:

      X(I,*) is moved to X(K(I),*) for I = 1,2,...,M.
Parameters

FORWRD

          FORWRD is LOGICAL
          = .TRUE., forward permutation
          = .FALSE., backward permutation

M

          M is INTEGER
          The number of rows of the matrix X. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix X. N >= 0.

X

          X is COMPLEX array, dimension (LDX,N)
          On entry, the M by N matrix X.
          On exit, X contains the permuted matrix X.

LDX

          LDX is INTEGER
          The leading dimension of the array X, LDX >= MAX(1,M).

K

          K is INTEGER array, dimension (M)
          On entry, K contains the permutation vector. K is used as
          internal workspace, but reset to its original value on
          output.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 103 of file clapmr.f.

subroutine clapmt (logical FORWRD, integer M, integer N, complex, dimension( ldx, * ) X, integer LDX, integer, dimension( * ) K)

CLAPMT performs a forward or backward permutation of the columns of a matrix.  

Purpose:

 CLAPMT rearranges the columns of the M by N matrix X as specified
 by the permutation K(1),K(2),...,K(N) of the integers 1,...,N.
 If FORWRD = .TRUE.,  forward permutation:

      X(*,K(J)) is moved X(*,J) for J = 1,2,...,N.

 If FORWRD = .FALSE., backward permutation:

      X(*,J) is moved to X(*,K(J)) for J = 1,2,...,N.
Parameters

FORWRD

          FORWRD is LOGICAL
          = .TRUE., forward permutation
          = .FALSE., backward permutation

M

          M is INTEGER
          The number of rows of the matrix X. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix X. N >= 0.

X

          X is COMPLEX array, dimension (LDX,N)
          On entry, the M by N matrix X.
          On exit, X contains the permuted matrix X.

LDX

          LDX is INTEGER
          The leading dimension of the array X, LDX >= MAX(1,M).

K

          K is INTEGER array, dimension (N)
          On entry, K contains the permutation vector. K is used as
          internal workspace, but reset to its original value on
          output.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 103 of file clapmt.f.

subroutine claqhb (character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) S, real SCOND, real AMAX, character EQUED)

CLAQHB scales a Hermitian band matrix, using scaling factors computed by cpbequ.  

Purpose:

 CLAQHB equilibrates an Hermitian band matrix A using the scaling
 factors in the vector S.
Parameters

UPLO

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of super-diagonals of the matrix A if UPLO = 'U',
          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U**H *U or A = L*L**H of the band
          matrix A, in the same storage format as A.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

S

          S is REAL array, dimension (N)
          The scale factors for A.

SCOND

          SCOND is REAL
          Ratio of the smallest S(i) to the largest S(i).

AMAX

          AMAX is REAL
          Absolute value of largest matrix entry.

EQUED

          EQUED is CHARACTER*1
          Specifies whether or not equilibration was done.
          = 'N':  No equilibration.
          = 'Y':  Equilibration was done, i.e., A has been replaced by
                  diag(S) * A * diag(S).

Internal Parameters:

  THRESH is a threshold value used to decide if scaling should be done
  based on the ratio of the scaling factors.  If SCOND < THRESH,
  scaling is done.

  LARGE and SMALL are threshold values used to decide if scaling should
  be done based on the absolute size of the largest matrix element.
  If AMAX > LARGE or AMAX < SMALL, scaling is done.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 140 of file claqhb.f.

subroutine claqhp (character UPLO, integer N, complex, dimension( * ) AP, real, dimension( * ) S, real SCOND, real AMAX, character EQUED)

CLAQHP scales a Hermitian matrix stored in packed form.  

Purpose:

 CLAQHP equilibrates a Hermitian matrix A using the scaling factors
 in the vector S.
Parameters

UPLO

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the Hermitian matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

          On exit, the equilibrated matrix:  diag(S) * A * diag(S), in
          the same storage format as A.

S

          S is REAL array, dimension (N)
          The scale factors for A.

SCOND

          SCOND is REAL
          Ratio of the smallest S(i) to the largest S(i).

AMAX

          AMAX is REAL
          Absolute value of largest matrix entry.

EQUED

          EQUED is CHARACTER*1
          Specifies whether or not equilibration was done.
          = 'N':  No equilibration.
          = 'Y':  Equilibration was done, i.e., A has been replaced by
                  diag(S) * A * diag(S).

Internal Parameters:

  THRESH is a threshold value used to decide if scaling should be done
  based on the ratio of the scaling factors.  If SCOND < THRESH,
  scaling is done.

  LARGE and SMALL are threshold values used to decide if scaling should
  be done based on the absolute size of the largest matrix element.
  If AMAX > LARGE or AMAX < SMALL, scaling is done.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 125 of file claqhp.f.

subroutine claqp2 (integer M, integer N, integer OFFSET, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, complex, dimension( * ) TAU, real, dimension( * ) VN1, real, dimension( * ) VN2, complex, dimension( * ) WORK)

CLAQP2 computes a QR factorization with column pivoting of the matrix block.  

Purpose:

 CLAQP2 computes a QR factorization with column pivoting of
 the block A(OFFSET+1:M,1:N).
 The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
Parameters

M

          M is INTEGER
          The number of rows of the matrix A. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix A. N >= 0.

OFFSET

          OFFSET is INTEGER
          The number of rows of the matrix A that must be pivoted
          but no factorized. OFFSET >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, the upper triangle of block A(OFFSET+1:M,1:N) is
          the triangular factor obtained; the elements in block
          A(OFFSET+1:M,1:N) below the diagonal, together with the
          array TAU, represent the orthogonal matrix Q as a product of
          elementary reflectors. Block A(1:OFFSET,1:N) has been
          accordingly pivoted, but no factorized.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

JPVT

          JPVT is INTEGER array, dimension (N)
          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
          to the front of A*P (a leading column); if JPVT(i) = 0,
          the i-th column of A is a free column.
          On exit, if JPVT(i) = k, then the i-th column of A*P
          was the k-th column of A.

TAU

          TAU is COMPLEX array, dimension (min(M,N))
          The scalar factors of the elementary reflectors.

VN1

          VN1 is REAL array, dimension (N)
          The vector with the partial column norms.

VN2

          VN2 is REAL array, dimension (N)
          The vector with the exact column norms.

WORK

          WORK is COMPLEX array, dimension (N)
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA
Partial column norm updating strategy modified on April 2011 Z. Drmac and Z. Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia.

References:

LAPACK Working Note 176  

Definition at line 147 of file claqp2.f.

subroutine claqps (integer M, integer N, integer OFFSET, integer NB, integer KB, complex, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, complex, dimension( * ) TAU, real, dimension( * ) VN1, real, dimension( * ) VN2, complex, dimension( * ) AUXV, complex, dimension( ldf, * ) F, integer LDF)

CLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A by using BLAS level 3.  

Purpose:

 CLAQPS computes a step of QR factorization with column pivoting
 of a complex M-by-N matrix A by using Blas-3.  It tries to factorize
 NB columns from A starting from the row OFFSET+1, and updates all
 of the matrix with Blas-3 xGEMM.

 In some cases, due to catastrophic cancellations, it cannot
 factorize NB columns.  Hence, the actual number of factorized
 columns is returned in KB.

 Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
Parameters

M

          M is INTEGER
          The number of rows of the matrix A. M >= 0.

N

          N is INTEGER
          The number of columns of the matrix A. N >= 0

OFFSET

          OFFSET is INTEGER
          The number of rows of A that have been factorized in
          previous steps.

NB

          NB is INTEGER
          The number of columns to factorize.

KB

          KB is INTEGER
          The number of columns actually factorized.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the M-by-N matrix A.
          On exit, block A(OFFSET+1:M,1:KB) is the triangular
          factor obtained and block A(1:OFFSET,1:N) has been
          accordingly pivoted, but no factorized.
          The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
          been updated.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,M).

JPVT

          JPVT is INTEGER array, dimension (N)
          JPVT(I) = K <==> Column K of the full matrix A has been
          permuted into position I in AP.

TAU

          TAU is COMPLEX array, dimension (KB)
          The scalar factors of the elementary reflectors.

VN1

          VN1 is REAL array, dimension (N)
          The vector with the partial column norms.

VN2

          VN2 is REAL array, dimension (N)
          The vector with the exact column norms.

AUXV

          AUXV is COMPLEX array, dimension (NB)
          Auxiliary vector.

F

          F is COMPLEX array, dimension (LDF,NB)
          Matrix  F**H = L * Y**H * A.

LDF

          LDF is INTEGER
          The leading dimension of the array F. LDF >= max(1,N).
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA

Partial column norm updating strategy modified on April 2011 Z. Drmac and Z. Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia.

References:

LAPACK Working Note 176  

Definition at line 176 of file claqps.f.

subroutine claqr0 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, complex, dimension( ldh, * ) H, integer LDH, complex, dimension( * ) W, integer ILOZ, integer IHIZ, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.  

Purpose:

    CLAQR0 computes the eigenvalues of a Hessenberg matrix H
    and, optionally, the matrices T and Z from the Schur decomposition
    H = Z T Z**H, where T is an upper triangular matrix (the
    Schur form), and Z is the unitary matrix of Schur vectors.

    Optionally Z may be postmultiplied into an input unitary
    matrix Q so that this routine can give the Schur factorization
    of a matrix A which has been reduced to the Hessenberg form H
    by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.
Parameters

WANTT

          WANTT is LOGICAL
          = .TRUE. : the full Schur form T is required;
          = .FALSE.: only eigenvalues are required.

WANTZ

          WANTZ is LOGICAL
          = .TRUE. : the matrix of Schur vectors Z is required;
          = .FALSE.: Schur vectors are not required.

N

          N is INTEGER
           The order of the matrix H.  N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
           It is assumed that H is already upper triangular in rows
           and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
           previous call to CGEBAL, and then passed to CGEHRD when the
           matrix output by CGEBAL is reduced to Hessenberg form.
           Otherwise, ILO and IHI should be set to 1 and N,
           respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
           If N = 0, then ILO = 1 and IHI = 0.

H

          H is COMPLEX array, dimension (LDH,N)
           On entry, the upper Hessenberg matrix H.
           On exit, if INFO = 0 and WANTT is .TRUE., then H
           contains the upper triangular matrix T from the Schur
           decomposition (the Schur form). If INFO = 0 and WANT is
           .FALSE., then the contents of H are unspecified on exit.
           (The output value of H when INFO > 0 is given under the
           description of INFO below.)

           This subroutine may explicitly set H(i,j) = 0 for i > j and
           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

LDH

          LDH is INTEGER
           The leading dimension of the array H. LDH >= max(1,N).

W

          W is COMPLEX array, dimension (N)
           The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
           in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
           stored in the same order as on the diagonal of the Schur
           form returned in H, with W(i) = H(i,i).

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER
           Specify the rows of Z to which transformations must be
           applied if WANTZ is .TRUE..
           1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

Z

          Z is COMPLEX array, dimension (LDZ,IHI)
           If WANTZ is .FALSE., then Z is not referenced.
           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
           (The output value of Z when INFO > 0 is given under
           the description of INFO below.)

LDZ

          LDZ is INTEGER
           The leading dimension of the array Z.  if WANTZ is .TRUE.
           then LDZ >= MAX(1,IHIZ).  Otherwise, LDZ >= 1.

WORK

          WORK is COMPLEX array, dimension LWORK
           On exit, if LWORK = -1, WORK(1) returns an estimate of
           the optimal value for LWORK.

LWORK

          LWORK is INTEGER
           The dimension of the array WORK.  LWORK >= max(1,N)
           is sufficient, but LWORK typically as large as 6*N may
           be required for optimal performance.  A workspace query
           to determine the optimal workspace size is recommended.

           If LWORK = -1, then CLAQR0 does a workspace query.
           In this case, CLAQR0 checks the input parameters and
           estimates the optimal workspace size for the given
           values of N, ILO and IHI.  The estimate is returned
           in WORK(1).  No error message related to LWORK is
           issued by XERBLA.  Neither H nor Z are accessed.

INFO

          INFO is INTEGER
             = 0:  successful exit
             > 0:  if INFO = i, CLAQR0 failed to compute all of
                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
                and WI contain those eigenvalues which have been
                successfully computed.  (Failures are rare.)

                If INFO > 0 and WANT is .FALSE., then on exit,
                the remaining unconverged eigenvalues are the eigen-
                values of the upper Hessenberg matrix rows and
                columns ILO through INFO of the final, output
                value of H.

                If INFO > 0 and WANTT is .TRUE., then on exit

           (*)  (initial value of H)*U  = U*(final value of H)

                where U is a unitary matrix.  The final
                value of  H is upper Hessenberg and triangular in
                rows and columns INFO+1 through IHI.

                If INFO > 0 and WANTZ is .TRUE., then on exit

                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U

                where U is the unitary matrix in (*) (regard-
                less of the value of WANTT.)

                If INFO > 0 and WANTZ is .FALSE., then Z is not
                accessed.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

References:

  K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
  Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
  Performance, SIAM Journal of Matrix Analysis, volume 23, pages
  929--947, 2002.

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

Definition at line 238 of file claqr0.f.

subroutine claqr1 (integer N, complex, dimension( ldh, * ) H, integer LDH, complex S1, complex S2, complex, dimension( * ) V)

CLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3 matrix H and specified shifts.  

Purpose:

      Given a 2-by-2 or 3-by-3 matrix H, CLAQR1 sets v to a
      scalar multiple of the first column of the product

      (*)  K = (H - s1*I)*(H - s2*I)

      scaling to avoid overflows and most underflows.

      This is useful for starting double implicit shift bulges
      in the QR algorithm.
Parameters

N

          N is INTEGER
              Order of the matrix H. N must be either 2 or 3.

H

          H is COMPLEX array, dimension (LDH,N)
              The 2-by-2 or 3-by-3 matrix H in (*).

LDH

          LDH is INTEGER
              The leading dimension of H as declared in
              the calling procedure.  LDH >= N

S1

          S1 is COMPLEX

S2

          S2 is COMPLEX

          S1 and S2 are the shifts defining K in (*) above.

V

          V is COMPLEX array, dimension (N)
              A scalar multiple of the first column of the
              matrix K in (*).
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

Definition at line 106 of file claqr1.f.

subroutine claqr2 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT, integer NW, complex, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, complex, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, complex, dimension( * ) SH, complex, dimension( ldv, * ) V, integer LDV, integer NH, complex, dimension( ldt, * ) T, integer LDT, integer NV, complex, dimension( ldwv, * ) WV, integer LDWV, complex, dimension( * ) WORK, integer LWORK)

CLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).  

Purpose:

    CLAQR2 is identical to CLAQR3 except that it avoids
    recursion by calling CLAHQR instead of CLAQR4.

    Aggressive early deflation:

    This subroutine accepts as input an upper Hessenberg matrix
    H and performs an unitary similarity transformation
    designed to detect and deflate fully converged eigenvalues from
    a trailing principal submatrix.  On output H has been over-
    written by a new Hessenberg matrix that is a perturbation of
    an unitary similarity transformation of H.  It is to be
    hoped that the final version of H has many zero subdiagonal
    entries.
Parameters

WANTT

          WANTT is LOGICAL
          If .TRUE., then the Hessenberg matrix H is fully updated
          so that the triangular Schur factor may be
          computed (in cooperation with the calling subroutine).
          If .FALSE., then only enough of H is updated to preserve
          the eigenvalues.

WANTZ

          WANTZ is LOGICAL
          If .TRUE., then the unitary matrix Z is updated so
          so that the unitary Schur factor may be computed
          (in cooperation with the calling subroutine).
          If .FALSE., then Z is not referenced.

N

          N is INTEGER
          The order of the matrix H and (if WANTZ is .TRUE.) the
          order of the unitary matrix Z.

KTOP

          KTOP is INTEGER
          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
          KBOT and KTOP together determine an isolated block
          along the diagonal of the Hessenberg matrix.

KBOT

          KBOT is INTEGER
          It is assumed without a check that either
          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
          determine an isolated block along the diagonal of the
          Hessenberg matrix.

NW

          NW is INTEGER
          Deflation window size.  1 <= NW <= (KBOT-KTOP+1).

H

          H is COMPLEX array, dimension (LDH,N)
          On input the initial N-by-N section of H stores the
          Hessenberg matrix undergoing aggressive early deflation.
          On output H has been transformed by a unitary
          similarity transformation, perturbed, and the returned
          to Hessenberg form that (it is to be hoped) has some
          zero subdiagonal entries.

LDH

          LDH is INTEGER
          Leading dimension of H just as declared in the calling
          subroutine.  N <= LDH

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER
          Specify the rows of Z to which transformations must be
          applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.

Z

          Z is COMPLEX array, dimension (LDZ,N)
          IF WANTZ is .TRUE., then on output, the unitary
          similarity transformation mentioned above has been
          accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
          If WANTZ is .FALSE., then Z is unreferenced.

LDZ

          LDZ is INTEGER
          The leading dimension of Z just as declared in the
          calling subroutine.  1 <= LDZ.

NS

          NS is INTEGER
          The number of unconverged (ie approximate) eigenvalues
          returned in SR and SI that may be used as shifts by the
          calling subroutine.

ND

          ND is INTEGER
          The number of converged eigenvalues uncovered by this
          subroutine.

SH

          SH is COMPLEX array, dimension (KBOT)
          On output, approximate eigenvalues that may
          be used for shifts are stored in SH(KBOT-ND-NS+1)
          through SR(KBOT-ND).  Converged eigenvalues are
          stored in SH(KBOT-ND+1) through SH(KBOT).

V

          V is COMPLEX array, dimension (LDV,NW)
          An NW-by-NW work array.

LDV

          LDV is INTEGER
          The leading dimension of V just as declared in the
          calling subroutine.  NW <= LDV

NH

          NH is INTEGER
          The number of columns of T.  NH >= NW.

T

          T is COMPLEX array, dimension (LDT,NW)

LDT

          LDT is INTEGER
          The leading dimension of T just as declared in the
          calling subroutine.  NW <= LDT

NV

          NV is INTEGER
          The number of rows of work array WV available for
          workspace.  NV >= NW.

WV

          WV is COMPLEX array, dimension (LDWV,NW)

LDWV

          LDWV is INTEGER
          The leading dimension of W just as declared in the
          calling subroutine.  NW <= LDV

WORK

          WORK is COMPLEX array, dimension (LWORK)
          On exit, WORK(1) is set to an estimate of the optimal value
          of LWORK for the given values of N, NW, KTOP and KBOT.

LWORK

          LWORK is INTEGER
          The dimension of the work array WORK.  LWORK = 2*NW
          suffices, but greater efficiency may result from larger
          values of LWORK.

          If LWORK = -1, then a workspace query is assumed; CLAQR2
          only estimates the optimal workspace size for the given
          values of N, NW, KTOP and KBOT.  The estimate is returned
          in WORK(1).  No error message related to LWORK is issued
          by XERBLA.  Neither H nor Z are accessed.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

Definition at line 266 of file claqr2.f.

subroutine claqr3 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT, integer NW, complex, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, complex, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, complex, dimension( * ) SH, complex, dimension( ldv, * ) V, integer LDV, integer NH, complex, dimension( ldt, * ) T, integer LDT, integer NV, complex, dimension( ldwv, * ) WV, integer LDWV, complex, dimension( * ) WORK, integer LWORK)

CLAQR3 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).  

Purpose:

    Aggressive early deflation:

    CLAQR3 accepts as input an upper Hessenberg matrix
    H and performs an unitary similarity transformation
    designed to detect and deflate fully converged eigenvalues from
    a trailing principal submatrix.  On output H has been over-
    written by a new Hessenberg matrix that is a perturbation of
    an unitary similarity transformation of H.  It is to be
    hoped that the final version of H has many zero subdiagonal
    entries.
Parameters

WANTT

          WANTT is LOGICAL
          If .TRUE., then the Hessenberg matrix H is fully updated
          so that the triangular Schur factor may be
          computed (in cooperation with the calling subroutine).
          If .FALSE., then only enough of H is updated to preserve
          the eigenvalues.

WANTZ

          WANTZ is LOGICAL
          If .TRUE., then the unitary matrix Z is updated so
          so that the unitary Schur factor may be computed
          (in cooperation with the calling subroutine).
          If .FALSE., then Z is not referenced.

N

          N is INTEGER
          The order of the matrix H and (if WANTZ is .TRUE.) the
          order of the unitary matrix Z.

KTOP

          KTOP is INTEGER
          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
          KBOT and KTOP together determine an isolated block
          along the diagonal of the Hessenberg matrix.

KBOT

          KBOT is INTEGER
          It is assumed without a check that either
          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
          determine an isolated block along the diagonal of the
          Hessenberg matrix.

NW

          NW is INTEGER
          Deflation window size.  1 <= NW <= (KBOT-KTOP+1).

H

          H is COMPLEX array, dimension (LDH,N)
          On input the initial N-by-N section of H stores the
          Hessenberg matrix undergoing aggressive early deflation.
          On output H has been transformed by a unitary
          similarity transformation, perturbed, and the returned
          to Hessenberg form that (it is to be hoped) has some
          zero subdiagonal entries.

LDH

          LDH is INTEGER
          Leading dimension of H just as declared in the calling
          subroutine.  N <= LDH

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER
          Specify the rows of Z to which transformations must be
          applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.

Z

          Z is COMPLEX array, dimension (LDZ,N)
          IF WANTZ is .TRUE., then on output, the unitary
          similarity transformation mentioned above has been
          accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
          If WANTZ is .FALSE., then Z is unreferenced.

LDZ

          LDZ is INTEGER
          The leading dimension of Z just as declared in the
          calling subroutine.  1 <= LDZ.

NS

          NS is INTEGER
          The number of unconverged (ie approximate) eigenvalues
          returned in SR and SI that may be used as shifts by the
          calling subroutine.

ND

          ND is INTEGER
          The number of converged eigenvalues uncovered by this
          subroutine.

SH

          SH is COMPLEX array, dimension (KBOT)
          On output, approximate eigenvalues that may
          be used for shifts are stored in SH(KBOT-ND-NS+1)
          through SR(KBOT-ND).  Converged eigenvalues are
          stored in SH(KBOT-ND+1) through SH(KBOT).

V

          V is COMPLEX array, dimension (LDV,NW)
          An NW-by-NW work array.

LDV

          LDV is INTEGER
          The leading dimension of V just as declared in the
          calling subroutine.  NW <= LDV

NH

          NH is INTEGER
          The number of columns of T.  NH >= NW.

T

          T is COMPLEX array, dimension (LDT,NW)

LDT

          LDT is INTEGER
          The leading dimension of T just as declared in the
          calling subroutine.  NW <= LDT

NV

          NV is INTEGER
          The number of rows of work array WV available for
          workspace.  NV >= NW.

WV

          WV is COMPLEX array, dimension (LDWV,NW)

LDWV

          LDWV is INTEGER
          The leading dimension of W just as declared in the
          calling subroutine.  NW <= LDV

WORK

          WORK is COMPLEX array, dimension (LWORK)
          On exit, WORK(1) is set to an estimate of the optimal value
          of LWORK for the given values of N, NW, KTOP and KBOT.

LWORK

          LWORK is INTEGER
          The dimension of the work array WORK.  LWORK = 2*NW
          suffices, but greater efficiency may result from larger
          values of LWORK.

          If LWORK = -1, then a workspace query is assumed; CLAQR3
          only estimates the optimal workspace size for the given
          values of N, NW, KTOP and KBOT.  The estimate is returned
          in WORK(1).  No error message related to LWORK is issued
          by XERBLA.  Neither H nor Z are accessed.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

Definition at line 263 of file claqr3.f.

subroutine claqr4 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, complex, dimension( ldh, * ) H, integer LDH, complex, dimension( * ) W, integer ILOZ, integer IHIZ, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, integer INFO)

CLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.  

Purpose:

    CLAQR4 implements one level of recursion for CLAQR0.
    It is a complete implementation of the small bulge multi-shift
    QR algorithm.  It may be called by CLAQR0 and, for large enough
    deflation window size, it may be called by CLAQR3.  This
    subroutine is identical to CLAQR0 except that it calls CLAQR2
    instead of CLAQR3.

    CLAQR4 computes the eigenvalues of a Hessenberg matrix H
    and, optionally, the matrices T and Z from the Schur decomposition
    H = Z T Z**H, where T is an upper triangular matrix (the
    Schur form), and Z is the unitary matrix of Schur vectors.

    Optionally Z may be postmultiplied into an input unitary
    matrix Q so that this routine can give the Schur factorization
    of a matrix A which has been reduced to the Hessenberg form H
    by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.
Parameters

WANTT

          WANTT is LOGICAL
          = .TRUE. : the full Schur form T is required;
          = .FALSE.: only eigenvalues are required.

WANTZ

          WANTZ is LOGICAL
          = .TRUE. : the matrix of Schur vectors Z is required;
          = .FALSE.: Schur vectors are not required.

N

          N is INTEGER
           The order of the matrix H.  N >= 0.

ILO

          ILO is INTEGER

IHI

          IHI is INTEGER
           It is assumed that H is already upper triangular in rows
           and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
           previous call to CGEBAL, and then passed to CGEHRD when the
           matrix output by CGEBAL is reduced to Hessenberg form.
           Otherwise, ILO and IHI should be set to 1 and N,
           respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
           If N = 0, then ILO = 1 and IHI = 0.

H

          H is COMPLEX array, dimension (LDH,N)
           On entry, the upper Hessenberg matrix H.
           On exit, if INFO = 0 and WANTT is .TRUE., then H
           contains the upper triangular matrix T from the Schur
           decomposition (the Schur form). If INFO = 0 and WANT is
           .FALSE., then the contents of H are unspecified on exit.
           (The output value of H when INFO > 0 is given under the
           description of INFO below.)

           This subroutine may explicitly set H(i,j) = 0 for i > j and
           j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

LDH

          LDH is INTEGER
           The leading dimension of the array H. LDH >= max(1,N).

W

          W is COMPLEX array, dimension (N)
           The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
           in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
           stored in the same order as on the diagonal of the Schur
           form returned in H, with W(i) = H(i,i).

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER
           Specify the rows of Z to which transformations must be
           applied if WANTZ is .TRUE..
           1 <= ILOZ <= ILO; IHI <= IHIZ <= N.

Z

          Z is COMPLEX array, dimension (LDZ,IHI)
           If WANTZ is .FALSE., then Z is not referenced.
           If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
           replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
           orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
           (The output value of Z when INFO > 0 is given under
           the description of INFO below.)

LDZ

          LDZ is INTEGER
           The leading dimension of the array Z.  if WANTZ is .TRUE.
           then LDZ >= MAX(1,IHIZ).  Otherwise, LDZ >= 1.

WORK

          WORK is COMPLEX array, dimension LWORK
           On exit, if LWORK = -1, WORK(1) returns an estimate of
           the optimal value for LWORK.

LWORK

          LWORK is INTEGER
           The dimension of the array WORK.  LWORK >= max(1,N)
           is sufficient, but LWORK typically as large as 6*N may
           be required for optimal performance.  A workspace query
           to determine the optimal workspace size is recommended.

           If LWORK = -1, then CLAQR4 does a workspace query.
           In this case, CLAQR4 checks the input parameters and
           estimates the optimal workspace size for the given
           values of N, ILO and IHI.  The estimate is returned
           in WORK(1).  No error message related to LWORK is
           issued by XERBLA.  Neither H nor Z are accessed.

INFO

          INFO is INTEGER
             = 0:  successful exit
             > 0:  if INFO = i, CLAQR4 failed to compute all of
                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
                and WI contain those eigenvalues which have been
                successfully computed.  (Failures are rare.)

                If INFO > 0 and WANT is .FALSE., then on exit,
                the remaining unconverged eigenvalues are the eigen-
                values of the upper Hessenberg matrix rows and
                columns ILO through INFO of the final, output
                value of H.

                If INFO > 0 and WANTT is .TRUE., then on exit

           (*)  (initial value of H)*U  = U*(final value of H)

                where U is a unitary matrix.  The final
                value of  H is upper Hessenberg and triangular in
                rows and columns INFO+1 through IHI.

                If INFO > 0 and WANTZ is .TRUE., then on exit

                  (final value of Z(ILO:IHI,ILOZ:IHIZ)
                   =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U

                where U is the unitary matrix in (*) (regard-
                less of the value of WANTT.)

                If INFO > 0 and WANTZ is .FALSE., then Z is not
                accessed.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

References:

  K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
  Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
  Performance, SIAM Journal of Matrix Analysis, volume 23, pages
  929--947, 2002.

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

Definition at line 246 of file claqr4.f.

subroutine claqr5 (logical WANTT, logical WANTZ, integer KACC22, integer N, integer KTOP, integer KBOT, integer NSHFTS, complex, dimension( * ) S, complex, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( ldu, * ) U, integer LDU, integer NV, complex, dimension( ldwv, * ) WV, integer LDWV, integer NH, complex, dimension( ldwh, * ) WH, integer LDWH)

CLAQR5 performs a single small-bulge multi-shift QR sweep.  

Purpose:

    CLAQR5 called by CLAQR0 performs a
    single small-bulge multi-shift QR sweep.
Parameters

WANTT

          WANTT is LOGICAL
             WANTT = .true. if the triangular Schur factor
             is being computed.  WANTT is set to .false. otherwise.

WANTZ

          WANTZ is LOGICAL
             WANTZ = .true. if the unitary Schur factor is being
             computed.  WANTZ is set to .false. otherwise.

KACC22

          KACC22 is INTEGER with value 0, 1, or 2.
             Specifies the computation mode of far-from-diagonal
             orthogonal updates.
        = 0: CLAQR5 does not accumulate reflections and does not
             use matrix-matrix multiply to update far-from-diagonal
             matrix entries.
        = 1: CLAQR5 accumulates reflections and uses matrix-matrix
             multiply to update the far-from-diagonal matrix entries.
        = 2: Same as KACC22 = 1. This option used to enable exploiting
             the 2-by-2 structure during matrix multiplications, but
             this is no longer supported.

N

          N is INTEGER
             N is the order of the Hessenberg matrix H upon which this
             subroutine operates.

KTOP

          KTOP is INTEGER

KBOT

          KBOT is INTEGER
             These are the first and last rows and columns of an
             isolated diagonal block upon which the QR sweep is to be
             applied. It is assumed without a check that
                       either KTOP = 1  or   H(KTOP,KTOP-1) = 0
             and
                       either KBOT = N  or   H(KBOT+1,KBOT) = 0.

NSHFTS

          NSHFTS is INTEGER
             NSHFTS gives the number of simultaneous shifts.  NSHFTS
             must be positive and even.

S

          S is COMPLEX array, dimension (NSHFTS)
             S contains the shifts of origin that define the multi-
             shift QR sweep.  On output S may be reordered.

H

          H is COMPLEX array, dimension (LDH,N)
             On input H contains a Hessenberg matrix.  On output a
             multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied
             to the isolated diagonal block in rows and columns KTOP
             through KBOT.

LDH

          LDH is INTEGER
             LDH is the leading dimension of H just as declared in the
             calling procedure.  LDH >= MAX(1,N).

ILOZ

          ILOZ is INTEGER

IHIZ

          IHIZ is INTEGER
             Specify the rows of Z to which transformations must be
             applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N

Z

          Z is COMPLEX array, dimension (LDZ,IHIZ)
             If WANTZ = .TRUE., then the QR Sweep unitary
             similarity transformation is accumulated into
             Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
             If WANTZ = .FALSE., then Z is unreferenced.

LDZ

          LDZ is INTEGER
             LDA is the leading dimension of Z just as declared in
             the calling procedure. LDZ >= N.

V

          V is COMPLEX array, dimension (LDV,NSHFTS/2)

LDV

          LDV is INTEGER
             LDV is the leading dimension of V as declared in the
             calling procedure.  LDV >= 3.

U

          U is COMPLEX array, dimension (LDU,2*NSHFTS)

LDU

          LDU is INTEGER
             LDU is the leading dimension of U just as declared in the
             in the calling subroutine.  LDU >= 2*NSHFTS.

NV

          NV is INTEGER
             NV is the number of rows in WV agailable for workspace.
             NV >= 1.

WV

          WV is COMPLEX array, dimension (LDWV,2*NSHFTS)

LDWV

          LDWV is INTEGER
             LDWV is the leading dimension of WV as declared in the
             in the calling subroutine.  LDWV >= NV.

NH

          NH is INTEGER
             NH is the number of columns in array WH available for
             workspace. NH >= 1.

WH

          WH is COMPLEX array, dimension (LDWH,NH)

LDWH

          LDWH is INTEGER
             Leading dimension of WH just as declared in the
             calling procedure.  LDWH >= 2*NSHFTS.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

Lars Karlsson, Daniel Kressner, and Bruno Lang

Thijs Steel, Department of Computer science, KU Leuven, Belgium

References:

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume 23, pages 929--947, 2002.

Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed chains of bulges in multishift QR algorithms. ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014).

Definition at line 254 of file claqr5.f.

subroutine claqsb (character UPLO, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, real, dimension( * ) S, real SCOND, real AMAX, character EQUED)

CLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ.  

Purpose:

 CLAQSB equilibrates a symmetric band matrix A using the scaling
 factors in the vector S.
Parameters

UPLO

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of super-diagonals of the matrix A if UPLO = 'U',
          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band
          matrix A, stored in the first KD+1 rows of the array.  The
          j-th column of A is stored in the j-th column of the array AB
          as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

          On exit, if INFO = 0, the triangular factor U or L from the
          Cholesky factorization A = U**H *U or A = L*L**H of the band
          matrix A, in the same storage format as A.

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

S

          S is REAL array, dimension (N)
          The scale factors for A.

SCOND

          SCOND is REAL
          Ratio of the smallest S(i) to the largest S(i).

AMAX

          AMAX is REAL
          Absolute value of largest matrix entry.

EQUED

          EQUED is CHARACTER*1
          Specifies whether or not equilibration was done.
          = 'N':  No equilibration.
          = 'Y':  Equilibration was done, i.e., A has been replaced by
                  diag(S) * A * diag(S).

Internal Parameters:

  THRESH is a threshold value used to decide if scaling should be done
  based on the ratio of the scaling factors.  If SCOND < THRESH,
  scaling is done.

  LARGE and SMALL are threshold values used to decide if scaling should
  be done based on the absolute size of the largest matrix element.
  If AMAX > LARGE or AMAX < SMALL, scaling is done.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 140 of file claqsb.f.

subroutine claqsp (character UPLO, integer N, complex, dimension( * ) AP, real, dimension( * ) S, real SCOND, real AMAX, character EQUED)

CLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors computed by sppequ.  

Purpose:

 CLAQSP equilibrates a symmetric matrix A using the scaling factors
 in the vector S.
Parameters

UPLO

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          symmetric matrix A is stored.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          On entry, the upper or lower triangle of the symmetric matrix
          A, packed columnwise in a linear array.  The j-th column of A
          is stored in the array AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

          On exit, the equilibrated matrix:  diag(S) * A * diag(S), in
          the same storage format as A.

S

          S is REAL array, dimension (N)
          The scale factors for A.

SCOND

          SCOND is REAL
          Ratio of the smallest S(i) to the largest S(i).

AMAX

          AMAX is REAL
          Absolute value of largest matrix entry.

EQUED

          EQUED is CHARACTER*1
          Specifies whether or not equilibration was done.
          = 'N':  No equilibration.
          = 'Y':  Equilibration was done, i.e., A has been replaced by
                  diag(S) * A * diag(S).

Internal Parameters:

  THRESH is a threshold value used to decide if scaling should be done
  based on the ratio of the scaling factors.  If SCOND < THRESH,
  scaling is done.

  LARGE and SMALL are threshold values used to decide if scaling should
  be done based on the absolute size of the largest matrix element.
  If AMAX > LARGE or AMAX < SMALL, scaling is done.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 125 of file claqsp.f.

subroutine clar1v (integer N, integer B1, integer BN, real LAMBDA, real, dimension( * ) D, real, dimension( * ) L, real, dimension( * ) LD, real, dimension( * ) LLD, real PIVMIN, real GAPTOL, complex, dimension( * ) Z, logical WANTNC, integer NEGCNT, real ZTZ, real MINGMA, integer R, integer, dimension( * ) ISUPPZ, real NRMINV, real RESID, real RQCORR, real, dimension( * ) WORK)

CLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.  

Purpose:

 CLAR1V computes the (scaled) r-th column of the inverse of
 the sumbmatrix in rows B1 through BN of the tridiagonal matrix
 L D L**T - sigma I. When sigma is close to an eigenvalue, the
 computed vector is an accurate eigenvector. Usually, r corresponds
 to the index where the eigenvector is largest in magnitude.
 The following steps accomplish this computation :
 (a) Stationary qd transform,  L D L**T - sigma I = L(+) D(+) L(+)**T,
 (b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T,
 (c) Computation of the diagonal elements of the inverse of
     L D L**T - sigma I by combining the above transforms, and choosing
     r as the index where the diagonal of the inverse is (one of the)
     largest in magnitude.
 (d) Computation of the (scaled) r-th column of the inverse using the
     twisted factorization obtained by combining the top part of the
     the stationary and the bottom part of the progressive transform.
Parameters

N

          N is INTEGER
           The order of the matrix L D L**T.

B1

          B1 is INTEGER
           First index of the submatrix of L D L**T.

BN

          BN is INTEGER
           Last index of the submatrix of L D L**T.

LAMBDA

          LAMBDA is REAL
           The shift. In order to compute an accurate eigenvector,
           LAMBDA should be a good approximation to an eigenvalue
           of L D L**T.

L

          L is REAL array, dimension (N-1)
           The (n-1) subdiagonal elements of the unit bidiagonal matrix
           L, in elements 1 to N-1.

D

          D is REAL array, dimension (N)
           The n diagonal elements of the diagonal matrix D.

LD

          LD is REAL array, dimension (N-1)
           The n-1 elements L(i)*D(i).

LLD

          LLD is REAL array, dimension (N-1)
           The n-1 elements L(i)*L(i)*D(i).

PIVMIN

          PIVMIN is REAL
           The minimum pivot in the Sturm sequence.

GAPTOL

          GAPTOL is REAL
           Tolerance that indicates when eigenvector entries are negligible
           w.r.t. their contribution to the residual.

Z

          Z is COMPLEX array, dimension (N)
           On input, all entries of Z must be set to 0.
           On output, Z contains the (scaled) r-th column of the
           inverse. The scaling is such that Z(R) equals 1.

WANTNC

          WANTNC is LOGICAL
           Specifies whether NEGCNT has to be computed.

NEGCNT

          NEGCNT is INTEGER
           If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin
           in the  matrix factorization L D L**T, and NEGCNT = -1 otherwise.

ZTZ

          ZTZ is REAL
           The square of the 2-norm of Z.

MINGMA

          MINGMA is REAL
           The reciprocal of the largest (in magnitude) diagonal
           element of the inverse of L D L**T - sigma I.

R

          R is INTEGER
           The twist index for the twisted factorization used to
           compute Z.
           On input, 0 <= R <= N. If R is input as 0, R is set to
           the index where (L D L**T - sigma I)^{-1} is largest
           in magnitude. If 1 <= R <= N, R is unchanged.
           On output, R contains the twist index used to compute Z.
           Ideally, R designates the position of the maximum entry in the
           eigenvector.

ISUPPZ

          ISUPPZ is INTEGER array, dimension (2)
           The support of the vector in Z, i.e., the vector Z is
           nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).

NRMINV

          NRMINV is REAL
           NRMINV = 1/SQRT( ZTZ )

RESID

          RESID is REAL
           The residual of the FP vector.
           RESID = ABS( MINGMA )/SQRT( ZTZ )

RQCORR

          RQCORR is REAL
           The Rayleigh Quotient correction to LAMBDA.
           RQCORR = MINGMA*TMP

WORK

          WORK is REAL array, dimension (4*N)
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA

Definition at line 227 of file clar1v.f.

subroutine clar2v (integer N, complex, dimension( * ) X, complex, dimension( * ) Y, complex, dimension( * ) Z, integer INCX, real, dimension( * ) C, complex, dimension( * ) S, integer INCC)

CLAR2V applies a vector of plane rotations with real cosines and complex sines from both sides to a sequence of 2-by-2 symmetric/Hermitian matrices.  

Purpose:

 CLAR2V applies a vector of complex plane rotations with real cosines
 from both sides to a sequence of 2-by-2 complex Hermitian matrices,
 defined by the elements of the vectors x, y and z. For i = 1,2,...,n

    (       x(i)  z(i) ) :=
    ( conjg(z(i)) y(i) )

      (  c(i) conjg(s(i)) ) (       x(i)  z(i) ) ( c(i) -conjg(s(i)) )
      ( -s(i)       c(i)  ) ( conjg(z(i)) y(i) ) ( s(i)        c(i)  )
Parameters

N

          N is INTEGER
          The number of plane rotations to be applied.

X

          X is COMPLEX array, dimension (1+(N-1)*INCX)
          The vector x; the elements of x are assumed to be real.

Y

          Y is COMPLEX array, dimension (1+(N-1)*INCX)
          The vector y; the elements of y are assumed to be real.

Z

          Z is COMPLEX array, dimension (1+(N-1)*INCX)
          The vector z.

INCX

          INCX is INTEGER
          The increment between elements of X, Y and Z. INCX > 0.

C

          C is REAL array, dimension (1+(N-1)*INCC)
          The cosines of the plane rotations.

S

          S is COMPLEX array, dimension (1+(N-1)*INCC)
          The sines of the plane rotations.

INCC

          INCC is INTEGER
          The increment between elements of C and S. INCC > 0.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 110 of file clar2v.f.

subroutine clarcm (integer M, integer N, real, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldc, * ) C, integer LDC, real, dimension( * ) RWORK)

CLARCM copies all or part of a real two-dimensional array to a complex array.  

Purpose:

 CLARCM performs a very simple matrix-matrix multiplication:
          C := A * B,
 where A is M by M and real; B is M by N and complex;
 C is M by N and complex.
Parameters

M

          M is INTEGER
          The number of rows of the matrix A and of the matrix C.
          M >= 0.

N

          N is INTEGER
          The number of columns and rows of the matrix B and
          the number of columns of the matrix C.
          N >= 0.

A

          A is REAL array, dimension (LDA, M)
          On entry, A contains the M by M matrix A.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >=max(1,M).

B

          B is COMPLEX array, dimension (LDB, N)
          On entry, B contains the M by N matrix B.

LDB

          LDB is INTEGER
          The leading dimension of the array B. LDB >=max(1,M).

C

          C is COMPLEX array, dimension (LDC, N)
          On exit, C contains the M by N matrix C.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >=max(1,M).

RWORK

          RWORK is REAL array, dimension (2*M*N)
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 113 of file clarcm.f.

subroutine clarf (character SIDE, integer M, integer N, complex, dimension( * ) V, integer INCV, complex TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK)

CLARF applies an elementary reflector to a general rectangular matrix.  

Purpose:

 CLARF applies a complex elementary reflector H to a complex M-by-N
 matrix C, from either the left or the right. H is represented in the
 form

       H = I - tau * v * v**H

 where tau is a complex scalar and v is a complex vector.

 If tau = 0, then H is taken to be the unit matrix.

 To apply H**H (the conjugate transpose of H), supply conjg(tau) instead
 tau.
Parameters

SIDE

          SIDE is CHARACTER*1
          = 'L': form  H * C
          = 'R': form  C * H

M

          M is INTEGER
          The number of rows of the matrix C.

N

          N is INTEGER
          The number of columns of the matrix C.

V

          V is COMPLEX array, dimension
                     (1 + (M-1)*abs(INCV)) if SIDE = 'L'
                  or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
          The vector v in the representation of H. V is not used if
          TAU = 0.

INCV

          INCV is INTEGER
          The increment between elements of v. INCV <> 0.

TAU

          TAU is COMPLEX
          The value tau in the representation of H.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by the matrix H * C if SIDE = 'L',
          or C * H if SIDE = 'R'.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension
                         (N) if SIDE = 'L'
                      or (M) if SIDE = 'R'
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 127 of file clarf.f.

subroutine clarfb (character SIDE, character TRANS, character DIRECT, character STOREV, integer M, integer N, integer K, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( ldwork, * ) WORK, integer LDWORK)

CLARFB applies a block reflector or its conjugate-transpose to a general rectangular matrix.  

Purpose:

 CLARFB applies a complex block reflector H or its transpose H**H to a
 complex M-by-N matrix C, from either the left or the right.
Parameters

SIDE

          SIDE is CHARACTER*1
          = 'L': apply H or H**H from the Left
          = 'R': apply H or H**H from the Right

TRANS

          TRANS is CHARACTER*1
          = 'N': apply H (No transpose)
          = 'C': apply H**H (Conjugate transpose)

DIRECT

          DIRECT is CHARACTER*1
          Indicates how H is formed from a product of elementary
          reflectors
          = 'F': H = H(1) H(2) . . . H(k) (Forward)
          = 'B': H = H(k) . . . H(2) H(1) (Backward)

STOREV

          STOREV is CHARACTER*1
          Indicates how the vectors which define the elementary
          reflectors are stored:
          = 'C': Columnwise
          = 'R': Rowwise

M

          M is INTEGER
          The number of rows of the matrix C.

N

          N is INTEGER
          The number of columns of the matrix C.

K

          K is INTEGER
          The order of the matrix T (= the number of elementary
          reflectors whose product defines the block reflector).
          If SIDE = 'L', M >= K >= 0;
          if SIDE = 'R', N >= K >= 0.

V

          V is COMPLEX array, dimension
                                (LDV,K) if STOREV = 'C'
                                (LDV,M) if STOREV = 'R' and SIDE = 'L'
                                (LDV,N) if STOREV = 'R' and SIDE = 'R'
          The matrix V. See Further Details.

LDV

          LDV is INTEGER
          The leading dimension of the array V.
          If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
          if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
          if STOREV = 'R', LDV >= K.

T

          T is COMPLEX array, dimension (LDT,K)
          The triangular K-by-K matrix T in the representation of the
          block reflector.

LDT

          LDT is INTEGER
          The leading dimension of the array T. LDT >= K.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the M-by-N matrix C.
          On exit, C is overwritten by H*C or H**H*C or C*H or C*H**H.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension (LDWORK,K)

LDWORK

          LDWORK is INTEGER
          The leading dimension of the array WORK.
          If SIDE = 'L', LDWORK >= max(1,N);
          if SIDE = 'R', LDWORK >= max(1,M).
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  The shape of the matrix V and the storage of the vectors which define
  the H(i) is best illustrated by the following example with n = 5 and
  k = 3. The elements equal to 1 are not stored; the corresponding
  array elements are modified but restored on exit. The rest of the
  array is not used.

  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':

               V = (  1       )                 V = (  1 v1 v1 v1 v1 )
                   ( v1  1    )                     (     1 v2 v2 v2 )
                   ( v1 v2  1 )                     (        1 v3 v3 )
                   ( v1 v2 v3 )
                   ( v1 v2 v3 )

  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':

               V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )
                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
                   (     1 v3 )
                   (        1 )

Definition at line 195 of file clarfb.f.

subroutine clarfb_gett (character IDENT, integer M, integer N, integer K, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldwork, * ) WORK, integer LDWORK)

CLARFB_GETT  

Purpose:

 CLARFB_GETT applies a complex Householder block reflector H from the
 left to a complex (K+M)-by-N  "triangular-pentagonal" matrix
 composed of two block matrices: an upper trapezoidal K-by-N matrix A
 stored in the array A, and a rectangular M-by-(N-K) matrix B, stored
 in the array B. The block reflector H is stored in a compact
 WY-representation, where the elementary reflectors are in the
 arrays A, B and T. See Further Details section.
Parameters

IDENT

          IDENT is CHARACTER*1
          If IDENT = not 'I', or not 'i', then V1 is unit
             lower-triangular and stored in the left K-by-K block of
             the input matrix A,
          If IDENT = 'I' or 'i', then  V1 is an identity matrix and
             not stored.
          See Further Details section.

M

          M is INTEGER
          The number of rows of the matrix B.
          M >= 0.

N

          N is INTEGER
          The number of columns of the matrices A and B.
          N >= 0.

K

          K is INTEGER
          The number or rows of the matrix A.
          K is also order of the matrix T, i.e. the number of
          elementary reflectors whose product defines the block
          reflector. 0 <= K <= N.

T

          T is COMPLEX array, dimension (LDT,K)
          The upper-triangular K-by-K matrix T in the representation
          of the block reflector.

LDT

          LDT is INTEGER
          The leading dimension of the array T. LDT >= K.

A

          A is COMPLEX array, dimension (LDA,N)

          On entry:
           a) In the K-by-N upper-trapezoidal part A: input matrix A.
           b) In the columns below the diagonal: columns of V1
              (ones are not stored on the diagonal).

          On exit:
            A is overwritten by rectangular K-by-N product H*A.

          See Further Details section.

LDA

          LDB is INTEGER
          The leading dimension of the array A. LDA >= max(1,K).

B

          B is COMPLEX array, dimension (LDB,N)

          On entry:
            a) In the M-by-(N-K) right block: input matrix B.
            b) In the M-by-N left block: columns of V2.

          On exit:
            B is overwritten by rectangular M-by-N product H*B.

          See Further Details section.

LDB

          LDB is INTEGER
          The leading dimension of the array B. LDB >= max(1,M).

WORK

          WORK is COMPLEX array,
          dimension (LDWORK,max(K,N-K))

LDWORK

          LDWORK is INTEGER
          The leading dimension of the array WORK. LDWORK>=max(1,K).
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

 November 2020, Igor Kozachenko,
                Computer Science Division,
                University of California, Berkeley

Further Details:

    (1) Description of the Algebraic Operation.

    The matrix A is a K-by-N matrix composed of two column block
    matrices, A1, which is K-by-K, and A2, which is K-by-(N-K):
    A = ( A1, A2 ).
    The matrix B is an M-by-N matrix composed of two column block
    matrices, B1, which is M-by-K, and B2, which is M-by-(N-K):
    B = ( B1, B2 ).

    Perform the operation:

       ( A_out ) := H * ( A_in ) = ( I - V * T * V**H ) * ( A_in ) =
       ( B_out )        ( B_in )                          ( B_in )
                  = ( I - ( V1 ) * T * ( V1**H, V2**H ) ) * ( A_in )
                          ( V2 )                            ( B_in )
     On input:

    a) ( A_in )  consists of two block columns:
       ( B_in )

       ( A_in ) = (( A1_in ) ( A2_in )) = (( A1_in ) ( A2_in ))
       ( B_in )   (( B1_in ) ( B2_in ))   ((     0 ) ( B2_in )),

       where the column blocks are:

       (  A1_in )  is a K-by-K upper-triangular matrix stored in the
                   upper triangular part of the array A(1:K,1:K).
       (  B1_in )  is an M-by-K rectangular ZERO matrix and not stored.

       ( A2_in )  is a K-by-(N-K) rectangular matrix stored
                  in the array A(1:K,K+1:N).
       ( B2_in )  is an M-by-(N-K) rectangular matrix stored
                  in the array B(1:M,K+1:N).

    b) V = ( V1 )
           ( V2 )

       where:
       1) if IDENT == 'I',V1 is a K-by-K identity matrix, not stored;
       2) if IDENT != 'I',V1 is a K-by-K unit lower-triangular matrix,
          stored in the lower-triangular part of the array
          A(1:K,1:K) (ones are not stored),
       and V2 is an M-by-K rectangular stored the array B(1:M,1:K),
                 (because on input B1_in is a rectangular zero
                  matrix that is not stored and the space is
                  used to store V2).

    c) T is a K-by-K upper-triangular matrix stored
       in the array T(1:K,1:K).

    On output:

    a) ( A_out ) consists of two  block columns:
       ( B_out )

       ( A_out ) = (( A1_out ) ( A2_out ))
       ( B_out )   (( B1_out ) ( B2_out )),

       where the column blocks are:

       ( A1_out )  is a K-by-K square matrix, or a K-by-K
                   upper-triangular matrix, if V1 is an
                   identity matrix. AiOut is stored in
                   the array A(1:K,1:K).
       ( B1_out )  is an M-by-K rectangular matrix stored
                   in the array B(1:M,K:N).

       ( A2_out )  is a K-by-(N-K) rectangular matrix stored
                   in the array A(1:K,K+1:N).
       ( B2_out )  is an M-by-(N-K) rectangular matrix stored
                   in the array B(1:M,K+1:N).


    The operation above can be represented as the same operation
    on each block column:

       ( A1_out ) := H * ( A1_in ) = ( I - V * T * V**H ) * ( A1_in )
       ( B1_out )        (     0 )                          (     0 )

       ( A2_out ) := H * ( A2_in ) = ( I - V * T * V**H ) * ( A2_in )
       ( B2_out )        ( B2_in )                          ( B2_in )

    If IDENT != 'I':

       The computation for column block 1:

       A1_out: = A1_in - V1*T*(V1**H)*A1_in

       B1_out: = - V2*T*(V1**H)*A1_in

       The computation for column block 2, which exists if N > K:

       A2_out: = A2_in - V1*T*( (V1**H)*A2_in + (V2**H)*B2_in )

       B2_out: = B2_in - V2*T*( (V1**H)*A2_in + (V2**H)*B2_in )

    If IDENT == 'I':

       The operation for column block 1:

       A1_out: = A1_in - V1*T*A1_in

       B1_out: = - V2*T*A1_in

       The computation for column block 2, which exists if N > K:

       A2_out: = A2_in - T*( A2_in + (V2**H)*B2_in )

       B2_out: = B2_in - V2*T*( A2_in + (V2**H)*B2_in )

    (2) Description of the Algorithmic Computation.

    In the first step, we compute column block 2, i.e. A2 and B2.
    Here, we need to use the K-by-(N-K) rectangular workspace
    matrix W2 that is of the same size as the matrix A2.
    W2 is stored in the array WORK(1:K,1:(N-K)).

    In the second step, we compute column block 1, i.e. A1 and B1.
    Here, we need to use the K-by-K square workspace matrix W1
    that is of the same size as the as the matrix A1.
    W1 is stored in the array WORK(1:K,1:K).

    NOTE: Hence, in this routine, we need the workspace array WORK
    only of size WORK(1:K,1:max(K,N-K)) so it can hold both W2 from
    the first step and W1 from the second step.

    Case (A), when V1 is unit lower-triangular, i.e. IDENT != 'I',
    more computations than in the Case (B).

    if( IDENT != 'I' ) then
     if ( N > K ) then
       (First Step - column block 2)
       col2_(1) W2: = A2
       col2_(2) W2: = (V1**H) * W2 = (unit_lower_tr_of_(A1)**H) * W2
       col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2
       col2_(4) W2: = T * W2
       col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2
       col2_(6) W2: = V1 * W2 = unit_lower_tr_of_(A1) * W2
       col2_(7) A2: = A2 - W2
     else
       (Second Step - column block 1)
       col1_(1) W1: = A1
       col1_(2) W1: = (V1**H) * W1 = (unit_lower_tr_of_(A1)**H) * W1
       col1_(3) W1: = T * W1
       col1_(4) B1: = - V2 * W1 = - B1 * W1
       col1_(5) square W1: = V1 * W1 = unit_lower_tr_of_(A1) * W1
       col1_(6) square A1: = A1 - W1
     end if
    end if

    Case (B), when V1 is an identity matrix, i.e. IDENT == 'I',
    less computations than in the Case (A)

    if( IDENT == 'I' ) then
     if ( N > K ) then
       (First Step - column block 2)
       col2_(1) W2: = A2
       col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2
       col2_(4) W2: = T * W2
       col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2
       col2_(7) A2: = A2 - W2
     else
       (Second Step - column block 1)
       col1_(1) W1: = A1
       col1_(3) W1: = T * W1
       col1_(4) B1: = - V2 * W1 = - B1 * W1
       col1_(6) upper-triangular_of_(A1): = A1 - W1
     end if
    end if

    Combine these cases (A) and (B) together, this is the resulting
    algorithm:

    if ( N > K ) then

      (First Step - column block 2)

      col2_(1)  W2: = A2
      if( IDENT != 'I' ) then
        col2_(2)  W2: = (V1**H) * W2
                      = (unit_lower_tr_of_(A1)**H) * W2
      end if
      col2_(3)  W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2]
      col2_(4)  W2: = T * W2
      col2_(5)  B2: = B2 - V2 * W2 = B2 - B1 * W2
      if( IDENT != 'I' ) then
        col2_(6)    W2: = V1 * W2 = unit_lower_tr_of_(A1) * W2
      end if
      col2_(7) A2: = A2 - W2

    else

    (Second Step - column block 1)

      col1_(1) W1: = A1
      if( IDENT != 'I' ) then
        col1_(2) W1: = (V1**H) * W1
                    = (unit_lower_tr_of_(A1)**H) * W1
      end if
      col1_(3) W1: = T * W1
      col1_(4) B1: = - V2 * W1 = - B1 * W1
      if( IDENT != 'I' ) then
        col1_(5) square W1: = V1 * W1 = unit_lower_tr_of_(A1) * W1
        col1_(6_a) below_diag_of_(A1): =  - below_diag_of_(W1)
      end if
      col1_(6_b) up_tr_of_(A1): = up_tr_of_(A1) - up_tr_of_(W1)

    end if

Definition at line 390 of file clarfb_gett.f.

subroutine clarfg (integer N, complex ALPHA, complex, dimension( * ) X, integer INCX, complex TAU)

CLARFG generates an elementary reflector (Householder matrix).  

Purpose:

 CLARFG generates a complex elementary reflector H of order n, such
 that

       H**H * ( alpha ) = ( beta ),   H**H * H = I.
              (   x   )   (   0  )

 where alpha and beta are scalars, with beta real, and x is an
 (n-1)-element complex vector. H is represented in the form

       H = I - tau * ( 1 ) * ( 1 v**H ) ,
                     ( v )

 where tau is a complex scalar and v is a complex (n-1)-element
 vector. Note that H is not hermitian.

 If the elements of x are all zero and alpha is real, then tau = 0
 and H is taken to be the unit matrix.

 Otherwise  1 <= real(tau) <= 2  and  abs(tau-1) <= 1 .
Parameters

N

          N is INTEGER
          The order of the elementary reflector.

ALPHA

          ALPHA is COMPLEX
          On entry, the value alpha.
          On exit, it is overwritten with the value beta.

X

          X is COMPLEX array, dimension
                         (1+(N-2)*abs(INCX))
          On entry, the vector x.
          On exit, it is overwritten with the vector v.

INCX

          INCX is INTEGER
          The increment between elements of X. INCX > 0.

TAU

          TAU is COMPLEX
          The value tau.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 105 of file clarfg.f.

subroutine clarfgp (integer N, complex ALPHA, complex, dimension( * ) X, integer INCX, complex TAU)

CLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.  

Purpose:

 CLARFGP generates a complex elementary reflector H of order n, such
 that

       H**H * ( alpha ) = ( beta ),   H**H * H = I.
              (   x   )   (   0  )

 where alpha and beta are scalars, beta is real and non-negative, and
 x is an (n-1)-element complex vector.  H is represented in the form

       H = I - tau * ( 1 ) * ( 1 v**H ) ,
                     ( v )

 where tau is a complex scalar and v is a complex (n-1)-element
 vector. Note that H is not hermitian.

 If the elements of x are all zero and alpha is real, then tau = 0
 and H is taken to be the unit matrix.
Parameters

N

          N is INTEGER
          The order of the elementary reflector.

ALPHA

          ALPHA is COMPLEX
          On entry, the value alpha.
          On exit, it is overwritten with the value beta.

X

          X is COMPLEX array, dimension
                         (1+(N-2)*abs(INCX))
          On entry, the vector x.
          On exit, it is overwritten with the vector v.

INCX

          INCX is INTEGER
          The increment between elements of X. INCX > 0.

TAU

          TAU is COMPLEX
          The value tau.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 103 of file clarfgp.f.

subroutine clarft (character DIRECT, character STOREV, integer N, integer K, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( * ) TAU, complex, dimension( ldt, * ) T, integer LDT)

CLARFT forms the triangular factor T of a block reflector H = I - vtvH  

Purpose:

 CLARFT forms the triangular factor T of a complex block reflector H
 of order n, which is defined as a product of k elementary reflectors.

 If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;

 If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.

 If STOREV = 'C', the vector which defines the elementary reflector
 H(i) is stored in the i-th column of the array V, and

    H  =  I - V * T * V**H

 If STOREV = 'R', the vector which defines the elementary reflector
 H(i) is stored in the i-th row of the array V, and

    H  =  I - V**H * T * V
Parameters

DIRECT

          DIRECT is CHARACTER*1
          Specifies the order in which the elementary reflectors are
          multiplied to form the block reflector:
          = 'F': H = H(1) H(2) . . . H(k) (Forward)
          = 'B': H = H(k) . . . H(2) H(1) (Backward)

STOREV

          STOREV is CHARACTER*1
          Specifies how the vectors which define the elementary
          reflectors are stored (see also Further Details):
          = 'C': columnwise
          = 'R': rowwise

N

          N is INTEGER
          The order of the block reflector H. N >= 0.

K

          K is INTEGER
          The order of the triangular factor T (= the number of
          elementary reflectors). K >= 1.

V

          V is COMPLEX array, dimension
                               (LDV,K) if STOREV = 'C'
                               (LDV,N) if STOREV = 'R'
          The matrix V. See further details.

LDV

          LDV is INTEGER
          The leading dimension of the array V.
          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

TAU

          TAU is COMPLEX array, dimension (K)
          TAU(i) must contain the scalar factor of the elementary
          reflector H(i).

T

          T is COMPLEX array, dimension (LDT,K)
          The k by k triangular factor T of the block reflector.
          If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
          lower triangular. The rest of the array is not used.

LDT

          LDT is INTEGER
          The leading dimension of the array T. LDT >= K.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  The shape of the matrix V and the storage of the vectors which define
  the H(i) is best illustrated by the following example with n = 5 and
  k = 3. The elements equal to 1 are not stored.

  DIRECT = 'F' and STOREV = 'C':         DIRECT = 'F' and STOREV = 'R':

               V = (  1       )                 V = (  1 v1 v1 v1 v1 )
                   ( v1  1    )                     (     1 v2 v2 v2 )
                   ( v1 v2  1 )                     (        1 v3 v3 )
                   ( v1 v2 v3 )
                   ( v1 v2 v3 )

  DIRECT = 'B' and STOREV = 'C':         DIRECT = 'B' and STOREV = 'R':

               V = ( v1 v2 v3 )                 V = ( v1 v1  1       )
                   ( v1 v2 v3 )                     ( v2 v2 v2  1    )
                   (  1 v2 v3 )                     ( v3 v3 v3 v3  1 )
                   (     1 v3 )
                   (        1 )

Definition at line 162 of file clarft.f.

subroutine clarfx (character SIDE, integer M, integer N, complex, dimension( * ) V, complex TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK)

CLARFX applies an elementary reflector to a general rectangular matrix, with loop unrolling when the reflector has order ≤ 10.  

Purpose:

 CLARFX applies a complex elementary reflector H to a complex m by n
 matrix C, from either the left or the right. H is represented in the
 form

       H = I - tau * v * v**H

 where tau is a complex scalar and v is a complex vector.

 If tau = 0, then H is taken to be the unit matrix

 This version uses inline code if H has order < 11.
Parameters

SIDE

          SIDE is CHARACTER*1
          = 'L': form  H * C
          = 'R': form  C * H

M

          M is INTEGER
          The number of rows of the matrix C.

N

          N is INTEGER
          The number of columns of the matrix C.

V

          V is COMPLEX array, dimension (M) if SIDE = 'L'
                                        or (N) if SIDE = 'R'
          The vector v in the representation of H.

TAU

          TAU is COMPLEX
          The value tau in the representation of H.

C

          C is COMPLEX array, dimension (LDC,N)
          On entry, the m by n matrix C.
          On exit, C is overwritten by the matrix H * C if SIDE = 'L',
          or C * H if SIDE = 'R'.

LDC

          LDC is INTEGER
          The leading dimension of the array C. LDC >= max(1,M).

WORK

          WORK is COMPLEX array, dimension (N) if SIDE = 'L'
                                            or (M) if SIDE = 'R'
          WORK is not referenced if H has order < 11.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 118 of file clarfx.f.

subroutine clarfy (character UPLO, integer N, complex, dimension( * ) V, integer INCV, complex TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK)

CLARFY

Purpose:

 CLARFY applies an elementary reflector, or Householder matrix, H,
 to an n x n Hermitian matrix C, from both the left and the right.

 H is represented in the form

    H = I - tau * v * v'

 where  tau  is a scalar and  v  is a vector.

 If  tau  is  zero, then  H  is taken to be the unit matrix.
Parameters

UPLO

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian matrix C is stored.
          = 'U':  Upper triangle
          = 'L':  Lower triangle

N

          N is INTEGER
          The number of rows and columns of the matrix C.  N >= 0.

V

          V is COMPLEX array, dimension
                  (1 + (N-1)*abs(INCV))
          The vector v as described above.

INCV

          INCV is INTEGER
          The increment between successive elements of v.  INCV must
          not be zero.

TAU

          TAU is COMPLEX
          The value tau as described above.

C

          C is COMPLEX array, dimension (LDC, N)
          On entry, the matrix C.
          On exit, C is overwritten by H * C * H'.

LDC

          LDC is INTEGER
          The leading dimension of the array C.  LDC >= max( 1, N ).

WORK

          WORK is COMPLEX array, dimension (N)
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 107 of file clarfy.f.

subroutine clargv (integer N, complex, dimension( * ) X, integer INCX, complex, dimension( * ) Y, integer INCY, real, dimension( * ) C, integer INCC)

CLARGV generates a vector of plane rotations with real cosines and complex sines.  

Purpose:

 CLARGV generates a vector of complex plane rotations with real
 cosines, determined by elements of the complex vectors x and y.
 For i = 1,2,...,n

    (        c(i)   s(i) ) ( x(i) ) = ( r(i) )
    ( -conjg(s(i))  c(i) ) ( y(i) ) = (   0  )

    where c(i)**2 + ABS(s(i))**2 = 1

 The following conventions are used (these are the same as in CLARTG,
 but differ from the BLAS1 routine CROTG):
    If y(i)=0, then c(i)=1 and s(i)=0.
    If x(i)=0, then c(i)=0 and s(i) is chosen so that r(i) is real.
Parameters

N

          N is INTEGER
          The number of plane rotations to be generated.

X

          X is COMPLEX array, dimension (1+(N-1)*INCX)
          On entry, the vector x.
          On exit, x(i) is overwritten by r(i), for i = 1,...,n.

INCX

          INCX is INTEGER
          The increment between elements of X. INCX > 0.

Y

          Y is COMPLEX array, dimension (1+(N-1)*INCY)
          On entry, the vector y.
          On exit, the sines of the plane rotations.

INCY

          INCY is INTEGER
          The increment between elements of Y. INCY > 0.

C

          C is REAL array, dimension (1+(N-1)*INCC)
          The cosines of the plane rotations.

INCC

          INCC is INTEGER
          The increment between elements of C. INCC > 0.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  6-6-96 - Modified with a new algorithm by W. Kahan and J. Demmel

  This version has a few statements commented out for thread safety
  (machine parameters are computed on each entry). 10 feb 03, SJH.

Definition at line 121 of file clargv.f.

subroutine clarnv (integer IDIST, integer, dimension( 4 ) ISEED, integer N, complex, dimension( * ) X)

CLARNV returns a vector of random numbers from a uniform or normal distribution.  

Purpose:

 CLARNV returns a vector of n random complex numbers from a uniform or
 normal distribution.
Parameters

IDIST

          IDIST is INTEGER
          Specifies the distribution of the random numbers:
          = 1:  real and imaginary parts each uniform (0,1)
          = 2:  real and imaginary parts each uniform (-1,1)
          = 3:  real and imaginary parts each normal (0,1)
          = 4:  uniformly distributed on the disc abs(z) < 1
          = 5:  uniformly distributed on the circle abs(z) = 1

ISEED

          ISEED is INTEGER array, dimension (4)
          On entry, the seed of the random number generator; the array
          elements must be between 0 and 4095, and ISEED(4) must be
          odd.
          On exit, the seed is updated.

N

          N is INTEGER
          The number of random numbers to be generated.

X

          X is COMPLEX array, dimension (N)
          The generated random numbers.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  This routine calls the auxiliary routine SLARUV to generate random
  real numbers from a uniform (0,1) distribution, in batches of up to
  128 using vectorisable code. The Box-Muller method is used to
  transform numbers from a uniform to a normal distribution.

Definition at line 98 of file clarnv.f.

subroutine clarrv (integer N, real VL, real VU, real, dimension( * ) D, real, dimension( * ) L, real PIVMIN, integer, dimension( * ) ISPLIT, integer M, integer DOL, integer DOU, real MINRGP, real RTOL1, real RTOL2, real, dimension( * ) W, real, dimension( * ) WERR, real, dimension( * ) WGAP, integer, dimension( * ) IBLOCK, integer, dimension( * ) INDEXW, real, dimension( * ) GERS, complex, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)

CLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the eigenvalues of L D LT.  

Purpose:

 CLARRV computes the eigenvectors of the tridiagonal matrix
 T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T.
 The input eigenvalues should have been computed by SLARRE.
Parameters

N

          N is INTEGER
          The order of the matrix.  N >= 0.

VL

          VL is REAL
          Lower bound of the interval that contains the desired
          eigenvalues. VL < VU. Needed to compute gaps on the left or right
          end of the extremal eigenvalues in the desired RANGE.

VU

          VU is REAL
          Upper bound of the interval that contains the desired
          eigenvalues. VL < VU. Needed to compute gaps on the left or right
          end of the extremal eigenvalues in the desired RANGE.

D

          D is REAL array, dimension (N)
          On entry, the N diagonal elements of the diagonal matrix D.
          On exit, D may be overwritten.

L

          L is REAL array, dimension (N)
          On entry, the (N-1) subdiagonal elements of the unit
          bidiagonal matrix L are in elements 1 to N-1 of L
          (if the matrix is not split.) At the end of each block
          is stored the corresponding shift as given by SLARRE.
          On exit, L is overwritten.

PIVMIN

          PIVMIN is REAL
          The minimum pivot allowed in the Sturm sequence.

ISPLIT

          ISPLIT is INTEGER array, dimension (N)
          The splitting points, at which T breaks up into blocks.
          The first block consists of rows/columns 1 to
          ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
          through ISPLIT( 2 ), etc.

M

          M is INTEGER
          The total number of input eigenvalues.  0 <= M <= N.

DOL

          DOL is INTEGER

DOU

          DOU is INTEGER
          If the user wants to compute only selected eigenvectors from all
          the eigenvalues supplied, he can specify an index range DOL:DOU.
          Or else the setting DOL=1, DOU=M should be applied.
          Note that DOL and DOU refer to the order in which the eigenvalues
          are stored in W.
          If the user wants to compute only selected eigenpairs, then
          the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
          computed eigenvectors. All other columns of Z are set to zero.

MINRGP

          MINRGP is REAL

RTOL1

          RTOL1 is REAL

RTOL2

          RTOL2 is REAL
           Parameters for bisection.
           An interval [LEFT,RIGHT] has converged if
           RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

W

          W is REAL array, dimension (N)
          The first M elements of W contain the APPROXIMATE eigenvalues for
          which eigenvectors are to be computed.  The eigenvalues
          should be grouped by split-off block and ordered from
          smallest to largest within the block ( The output array
          W from SLARRE is expected here ). Furthermore, they are with
          respect to the shift of the corresponding root representation
          for their block. On exit, W holds the eigenvalues of the
          UNshifted matrix.

WERR

          WERR is REAL array, dimension (N)
          The first M elements contain the semiwidth of the uncertainty
          interval of the corresponding eigenvalue in W

WGAP

          WGAP is REAL array, dimension (N)
          The separation from the right neighbor eigenvalue in W.

IBLOCK

          IBLOCK is INTEGER array, dimension (N)
          The indices of the blocks (submatrices) associated with the
          corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
          W(i) belongs to the first block from the top, =2 if W(i)
          belongs to the second block, etc.

INDEXW

          INDEXW is INTEGER array, dimension (N)
          The indices of the eigenvalues within each block (submatrix);
          for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
          i-th eigenvalue W(i) is the 10-th eigenvalue in the second block.

GERS

          GERS is REAL array, dimension (2*N)
          The N Gerschgorin intervals (the i-th Gerschgorin interval
          is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should
          be computed from the original UNshifted matrix.

Z

          Z is COMPLEX array, dimension (LDZ, max(1,M) )
          If INFO = 0, the first M columns of Z contain the
          orthonormal eigenvectors of the matrix T
          corresponding to the input eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z.

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDZ >= 1, and if
          JOBZ = 'V', LDZ >= max(1,N).

ISUPPZ

          ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
          The support of the eigenvectors in Z, i.e., the indices
          indicating the nonzero elements in Z. The I-th eigenvector
          is nonzero only in elements ISUPPZ( 2*I-1 ) through
          ISUPPZ( 2*I ).

WORK

          WORK is REAL array, dimension (12*N)

IWORK

          IWORK is INTEGER array, dimension (7*N)

INFO

          INFO is INTEGER
          = 0:  successful exit

          > 0:  A problem occurred in CLARRV.
          < 0:  One of the called subroutines signaled an internal problem.
                Needs inspection of the corresponding parameter IINFO
                for further information.

          =-1:  Problem in SLARRB when refining a child's eigenvalues.
          =-2:  Problem in SLARRF when computing the RRR of a child.
                When a child is inside a tight cluster, it can be difficult
                to find an RRR. A partial remedy from the user's point of
                view is to make the parameter MINRGP smaller and recompile.
                However, as the orthogonality of the computed vectors is
                proportional to 1/MINRGP, the user should be aware that
                he might be trading in precision when he decreases MINRGP.
          =-3:  Problem in SLARRB when refining a single eigenvalue
                after the Rayleigh correction was rejected.
          = 5:  The Rayleigh Quotient Iteration failed to converge to
                full accuracy in MAXITR steps.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA

Definition at line 281 of file clarrv.f.

subroutine clartg (complex F, complex G, real CS, complex SN, complex R)

CLARTG generates a plane rotation with real cosine and complex sine.  

Purpose:

 CLARTG generates a plane rotation so that

    [  CS  SN  ]     [ F ]     [ R ]
    [  __      ]  .  [   ]  =  [   ]   where CS**2 + |SN|**2 = 1.
    [ -SN  CS  ]     [ G ]     [ 0 ]

 This is a faster version of the BLAS1 routine CROTG, except for
 the following differences:
    F and G are unchanged on return.
    If G=0, then CS=1 and SN=0.
    If F=0, then CS=0 and SN is chosen so that R is real.
Parameters

F

          F is COMPLEX
          The first component of vector to be rotated.

G

          G is COMPLEX
          The second component of vector to be rotated.

CS

          CS is REAL
          The cosine of the rotation.

SN

          SN is COMPLEX
          The sine of the rotation.

R

          R is COMPLEX
          The nonzero component of the rotated vector.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  3-5-96 - Modified with a new algorithm by W. Kahan and J. Demmel

  This version has a few statements commented out for thread safety
  (machine parameters are computed on each entry). 10 feb 03, SJH.

Definition at line 102 of file clartg.f.

subroutine clartv (integer N, complex, dimension( * ) X, integer INCX, complex, dimension( * ) Y, integer INCY, real, dimension( * ) C, complex, dimension( * ) S, integer INCC)

CLARTV applies a vector of plane rotations with real cosines and complex sines to the elements of a pair of vectors.  

Purpose:

 CLARTV applies a vector of complex plane rotations with real cosines
 to elements of the complex vectors x and y. For i = 1,2,...,n

    ( x(i) ) := (        c(i)   s(i) ) ( x(i) )
    ( y(i) )    ( -conjg(s(i))  c(i) ) ( y(i) )
Parameters

N

          N is INTEGER
          The number of plane rotations to be applied.

X

          X is COMPLEX array, dimension (1+(N-1)*INCX)
          The vector x.

INCX

          INCX is INTEGER
          The increment between elements of X. INCX > 0.

Y

          Y is COMPLEX array, dimension (1+(N-1)*INCY)
          The vector y.

INCY

          INCY is INTEGER
          The increment between elements of Y. INCY > 0.

C

          C is REAL array, dimension (1+(N-1)*INCC)
          The cosines of the plane rotations.

S

          S is COMPLEX array, dimension (1+(N-1)*INCC)
          The sines of the plane rotations.

INCC

          INCC is INTEGER
          The increment between elements of C and S. INCC > 0.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 106 of file clartv.f.

subroutine clascl (character TYPE, integer KL, integer KU, real CFROM, real CTO, integer M, integer N, complex, dimension( lda, * ) A, integer LDA, integer INFO)

CLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.  

Purpose:

 CLASCL multiplies the M by N complex matrix A by the real scalar
 CTO/CFROM.  This is done without over/underflow as long as the final
 result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that
 A may be full, upper triangular, lower triangular, upper Hessenberg,
 or banded.
Parameters

TYPE

          TYPE is CHARACTER*1
          TYPE indices the storage type of the input matrix.
          = 'G':  A is a full matrix.
          = 'L':  A is a lower triangular matrix.
          = 'U':  A is an upper triangular matrix.
          = 'H':  A is an upper Hessenberg matrix.
          = 'B':  A is a symmetric band matrix with lower bandwidth KL
                  and upper bandwidth KU and with the only the lower
                  half stored.
          = 'Q':  A is a symmetric band matrix with lower bandwidth KL
                  and upper bandwidth KU and with the only the upper
                  half stored.
          = 'Z':  A is a band matrix with lower bandwidth KL and upper
                  bandwidth KU. See CGBTRF for storage details.

KL

          KL is INTEGER
          The lower bandwidth of A.  Referenced only if TYPE = 'B',
          'Q' or 'Z'.

KU

          KU is INTEGER
          The upper bandwidth of A.  Referenced only if TYPE = 'B',
          'Q' or 'Z'.

CFROM

          CFROM is REAL

CTO

          CTO is REAL

          The matrix A is multiplied by CTO/CFROM. A(I,J) is computed
          without over/underflow if the final result CTO*A(I,J)/CFROM
          can be represented without over/underflow.  CFROM must be
          nonzero.

M

          M is INTEGER
          The number of rows of the matrix A.  M >= 0.

N

          N is INTEGER
          The number of columns of the matrix A.  N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          The matrix to be multiplied by CTO/CFROM.  See TYPE for the
          storage type.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          If TYPE = 'G', 'L', 'U', 'H', LDA >= max(1,M);
             TYPE = 'B', LDA >= KL+1;
             TYPE = 'Q', LDA >= KU+1;
             TYPE = 'Z', LDA >= 2*KL+KU+1.

INFO

          INFO is INTEGER
          0  - successful exit
          <0 - if INFO = -i, the i-th argument had an illegal value.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 142 of file clascl.f.

subroutine claset (character UPLO, integer M, integer N, complex ALPHA, complex BETA, complex, dimension( lda, * ) A, integer LDA)

CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.  

Purpose:

 CLASET initializes a 2-D array A to BETA on the diagonal and
 ALPHA on the offdiagonals.
Parameters

UPLO

          UPLO is CHARACTER*1
          Specifies the part of the matrix A to be set.
          = 'U':      Upper triangular part is set. The lower triangle
                      is unchanged.
          = 'L':      Lower triangular part is set. The upper triangle
                      is unchanged.
          Otherwise:  All of the matrix A is set.

M

          M is INTEGER
          On entry, M specifies the number of rows of A.

N

          N is INTEGER
          On entry, N specifies the number of columns of A.

ALPHA

          ALPHA is COMPLEX
          All the offdiagonal array elements are set to ALPHA.

BETA

          BETA is COMPLEX
          All the diagonal array elements are set to BETA.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the m by n matrix A.
          On exit, A(i,j) = ALPHA, 1 <= i <= m, 1 <= j <= n, i.ne.j;
                   A(i,i) = BETA , 1 <= i <= min(m,n)

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 105 of file claset.f.

subroutine clasr (character SIDE, character PIVOT, character DIRECT, integer M, integer N, real, dimension( * ) C, real, dimension( * ) S, complex, dimension( lda, * ) A, integer LDA)

CLASR applies a sequence of plane rotations to a general rectangular matrix.  

Purpose:

 CLASR applies a sequence of real plane rotations to a complex matrix
 A, from either the left or the right.

 When SIDE = 'L', the transformation takes the form

    A := P*A

 and when SIDE = 'R', the transformation takes the form

    A := A*P**T

 where P is an orthogonal matrix consisting of a sequence of z plane
 rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R',
 and P**T is the transpose of P.

 When DIRECT = 'F' (Forward sequence), then

    P = P(z-1) * ... * P(2) * P(1)

 and when DIRECT = 'B' (Backward sequence), then

    P = P(1) * P(2) * ... * P(z-1)

 where P(k) is a plane rotation matrix defined by the 2-by-2 rotation

    R(k) = (  c(k)  s(k) )
         = ( -s(k)  c(k) ).

 When PIVOT = 'V' (Variable pivot), the rotation is performed
 for the plane (k,k+1), i.e., P(k) has the form

    P(k) = (  1                                            )
           (       ...                                     )
           (              1                                )
           (                   c(k)  s(k)                  )
           (                  -s(k)  c(k)                  )
           (                                1              )
           (                                     ...       )
           (                                            1  )

 where R(k) appears as a rank-2 modification to the identity matrix in
 rows and columns k and k+1.

 When PIVOT = 'T' (Top pivot), the rotation is performed for the
 plane (1,k+1), so P(k) has the form

    P(k) = (  c(k)                    s(k)                 )
           (         1                                     )
           (              ...                              )
           (                     1                         )
           ( -s(k)                    c(k)                 )
           (                                 1             )
           (                                      ...      )
           (                                             1 )

 where R(k) appears in rows and columns 1 and k+1.

 Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is
 performed for the plane (k,z), giving P(k) the form

    P(k) = ( 1                                             )
           (      ...                                      )
           (             1                                 )
           (                  c(k)                    s(k) )
           (                         1                     )
           (                              ...              )
           (                                     1         )
           (                 -s(k)                    c(k) )

 where R(k) appears in rows and columns k and z.  The rotations are
 performed without ever forming P(k) explicitly.
Parameters

SIDE

          SIDE is CHARACTER*1
          Specifies whether the plane rotation matrix P is applied to
          A on the left or the right.
          = 'L':  Left, compute A := P*A
          = 'R':  Right, compute A:= A*P**T

PIVOT

          PIVOT is CHARACTER*1
          Specifies the plane for which P(k) is a plane rotation
          matrix.
          = 'V':  Variable pivot, the plane (k,k+1)
          = 'T':  Top pivot, the plane (1,k+1)
          = 'B':  Bottom pivot, the plane (k,z)

DIRECT

          DIRECT is CHARACTER*1
          Specifies whether P is a forward or backward sequence of
          plane rotations.
          = 'F':  Forward, P = P(z-1)*...*P(2)*P(1)
          = 'B':  Backward, P = P(1)*P(2)*...*P(z-1)

M

          M is INTEGER
          The number of rows of the matrix A.  If m <= 1, an immediate
          return is effected.

N

          N is INTEGER
          The number of columns of the matrix A.  If n <= 1, an
          immediate return is effected.

C

          C is REAL array, dimension
                  (M-1) if SIDE = 'L'
                  (N-1) if SIDE = 'R'
          The cosines c(k) of the plane rotations.

S

          S is REAL array, dimension
                  (M-1) if SIDE = 'L'
                  (N-1) if SIDE = 'R'
          The sines s(k) of the plane rotations.  The 2-by-2 plane
          rotation part of the matrix P(k), R(k), has the form
          R(k) = (  c(k)  s(k) )
                 ( -s(k)  c(k) ).

A

          A is COMPLEX array, dimension (LDA,N)
          The M-by-N matrix A.  On exit, A is overwritten by P*A if
          SIDE = 'R' or by A*P**T if SIDE = 'L'.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 199 of file clasr.f.

subroutine classq (integer N, complex, dimension( * ) X, integer INCX, real SCALE, real SUMSQ)

CLASSQ updates a sum of squares represented in scaled form.  

Purpose:

 CLASSQ returns the values scl and ssq such that

    ( scl**2 )*ssq = x( 1 )**2 +...+ x( n )**2 + ( scale**2 )*sumsq,

 where x( i ) = abs( X( 1 + ( i - 1 )*INCX ) ). The value of sumsq is
 assumed to be at least unity and the value of ssq will then satisfy

    1.0 <= ssq <= ( sumsq + 2*n ).

 scale is assumed to be non-negative and scl returns the value

    scl = max( scale, abs( real( x( i ) ) ), abs( aimag( x( i ) ) ) ),
           i

 scale and sumsq must be supplied in SCALE and SUMSQ respectively.
 SCALE and SUMSQ are overwritten by scl and ssq respectively.

 The routine makes only one pass through the vector X.
Parameters

N

          N is INTEGER
          The number of elements to be used from the vector X.

X

          X is COMPLEX array, dimension (1+(N-1)*INCX)
          The vector x as described above.
             x( i )  = X( 1 + ( i - 1 )*INCX ), 1 <= i <= n.

INCX

          INCX is INTEGER
          The increment between successive values of the vector X.
          INCX > 0.

SCALE

          SCALE is REAL
          On entry, the value  scale  in the equation above.
          On exit, SCALE is overwritten with the value  scl .

SUMSQ

          SUMSQ is REAL
          On entry, the value  sumsq  in the equation above.
          On exit, SUMSQ is overwritten with the value  ssq .
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 105 of file classq.f.

subroutine claswp (integer N, complex, dimension( lda, * ) A, integer LDA, integer K1, integer K2, integer, dimension( * ) IPIV, integer INCX)

CLASWP performs a series of row interchanges on a general rectangular matrix.  

Purpose:

 CLASWP performs a series of row interchanges on the matrix A.
 One row interchange is initiated for each of rows K1 through K2 of A.
Parameters

N

          N is INTEGER
          The number of columns of the matrix A.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the matrix of column dimension N to which the row
          interchanges will be applied.
          On exit, the permuted matrix.

LDA

          LDA is INTEGER
          The leading dimension of the array A.

K1

          K1 is INTEGER
          The first element of IPIV for which a row interchange will
          be done.

K2

          K2 is INTEGER
          (K2-K1+1) is the number of elements of IPIV for which a row
          interchange will be done.

IPIV

          IPIV is INTEGER array, dimension (K1+(K2-K1)*abs(INCX))
          The vector of pivot indices. Only the elements in positions
          K1 through K1+(K2-K1)*abs(INCX) of IPIV are accessed.
          IPIV(K1+(K-K1)*abs(INCX)) = L implies rows K and L are to be
          interchanged.

INCX

          INCX is INTEGER
          The increment between successive values of IPIV. If INCX
          is negative, the pivots are applied in reverse order.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  Modified by
   R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA

Definition at line 114 of file claswp.f.

subroutine clatbs (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, integer KD, complex, dimension( ldab, * ) AB, integer LDAB, complex, dimension( * ) X, real SCALE, real, dimension( * ) CNORM, integer INFO)

CLATBS solves a triangular banded system of equations.  

Purpose:

 CLATBS solves one of the triangular systems

    A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b,

 with scaling to prevent overflow, where A is an upper or lower
 triangular band matrix.  Here A**T denotes the transpose of A, x and b
 are n-element vectors, and s is a scaling factor, usually less than
 or equal to 1, chosen so that the components of x will be less than
 the overflow threshold.  If the unscaled problem will not cause
 overflow, the Level 2 BLAS routine CTBSV is called.  If the matrix A
 is singular (A(j,j) = 0 for some j), then s is set to 0 and a
 non-trivial solution to A*x = 0 is returned.
Parameters

UPLO

          UPLO is CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

TRANS

          TRANS is CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  Solve A * x = s*b     (No transpose)
          = 'T':  Solve A**T * x = s*b  (Transpose)
          = 'C':  Solve A**H * x = s*b  (Conjugate transpose)

DIAG

          DIAG is CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

NORMIN

          NORMIN is CHARACTER*1
          Specifies whether CNORM has been set or not.
          = 'Y':  CNORM contains the column norms on entry
          = 'N':  CNORM is not set on entry.  On exit, the norms will
                  be computed and stored in CNORM.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

KD

          KD is INTEGER
          The number of subdiagonals or superdiagonals in the
          triangular matrix A.  KD >= 0.

AB

          AB is COMPLEX array, dimension (LDAB,N)
          The upper or lower triangular band matrix A, stored in the
          first KD+1 rows of the array. The j-th column of A is stored
          in the j-th column of the array AB as follows:
          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).

LDAB

          LDAB is INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

X

          X is COMPLEX array, dimension (N)
          On entry, the right hand side b of the triangular system.
          On exit, X is overwritten by the solution vector x.

SCALE

          SCALE is REAL
          The scaling factor s for the triangular system
             A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b.
          If SCALE = 0, the matrix A is singular or badly scaled, and
          the vector x is an exact or approximate solution to A*x = 0.

CNORM

          CNORM is REAL array, dimension (N)

          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
          contains the norm of the off-diagonal part of the j-th column
          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
          must be greater than or equal to the 1-norm.

          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
          returns the 1-norm of the offdiagonal part of the j-th column
          of A.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -k, the k-th argument had an illegal value
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  A rough bound on x is computed; if that is less than overflow, CTBSV
  is called, otherwise, specific code is used which checks for possible
  overflow or divide-by-zero at every operation.

  A columnwise scheme is used for solving A*x = b.  The basic algorithm
  if A is lower triangular is

       x[1:n] := b[1:n]
       for j = 1, ..., n
            x(j) := x(j) / A(j,j)
            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
       end

  Define bounds on the components of x after j iterations of the loop:
     M(j) = bound on x[1:j]
     G(j) = bound on x[j+1:n]
  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

  Then for iteration j+1 we have
     M(j+1) <= G(j) / | A(j+1,j+1) |
     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )

  where CNORM(j+1) is greater than or equal to the infinity-norm of
  column j+1 of A, not counting the diagonal.  Hence

     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                  1<=i<=j
  and

     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                   1<=i< j

  Since |x(j)| <= M(j), we use the Level 2 BLAS routine CTBSV if the
  reciprocal of the largest M(j), j=1,..,n, is larger than
  max(underflow, 1/overflow).

  The bound on x(j) is also used to determine when a step in the
  columnwise method can be performed without fear of overflow.  If
  the computed bound is greater than a large constant, x is scaled to
  prevent overflow, but if the bound overflows, x is set to 0, x(j) to
  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.

  Similarly, a row-wise scheme is used to solve A**T *x = b  or
  A**H *x = b.  The basic algorithm for A upper triangular is

       for j = 1, ..., n
            x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
       end

  We simultaneously compute two bounds
       G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
       M(j) = bound on x(i), 1<=i<=j

  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
  Then the bound on x(j) is

       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |

            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                      1<=i<=j

  and we can safely call CTBSV if 1/M(n) and 1/G(n) are both greater
  than max(underflow, 1/overflow).

Definition at line 241 of file clatbs.f.

subroutine clatdf (integer IJOB, integer N, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) RHS, real RDSUM, real RDSCAL, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV)

CLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a contribution to the reciprocal Dif-estimate.  

Purpose:

 CLATDF computes the contribution to the reciprocal Dif-estimate
 by solving for x in Z * x = b, where b is chosen such that the norm
 of x is as large as possible. It is assumed that LU decomposition
 of Z has been computed by CGETC2. On entry RHS = f holds the
 contribution from earlier solved sub-systems, and on return RHS = x.

 The factorization of Z returned by CGETC2 has the form
 Z = P * L * U * Q, where P and Q are permutation matrices. L is lower
 triangular with unit diagonal elements and U is upper triangular.
Parameters

IJOB

          IJOB is INTEGER
          IJOB = 2: First compute an approximative null-vector e
              of Z using CGECON, e is normalized and solve for
              Zx = +-e - f with the sign giving the greater value of
              2-norm(x).  About 5 times as expensive as Default.
          IJOB .ne. 2: Local look ahead strategy where
              all entries of the r.h.s. b is chosen as either +1 or
              -1.  Default.

N

          N is INTEGER
          The number of columns of the matrix Z.

Z

          Z is COMPLEX array, dimension (LDZ, N)
          On entry, the LU part of the factorization of the n-by-n
          matrix Z computed by CGETC2:  Z = P * L * U * Q

LDZ

          LDZ is INTEGER
          The leading dimension of the array Z.  LDA >= max(1, N).

RHS

          RHS is COMPLEX array, dimension (N).
          On entry, RHS contains contributions from other subsystems.
          On exit, RHS contains the solution of the subsystem with
          entries according to the value of IJOB (see above).

RDSUM

          RDSUM is REAL
          On entry, the sum of squares of computed contributions to
          the Dif-estimate under computation by CTGSYL, where the
          scaling factor RDSCAL (see below) has been factored out.
          On exit, the corresponding sum of squares updated with the
          contributions from the current sub-system.
          If TRANS = 'T' RDSUM is not touched.
          NOTE: RDSUM only makes sense when CTGSY2 is called by CTGSYL.

RDSCAL

          RDSCAL is REAL
          On entry, scaling factor used to prevent overflow in RDSUM.
          On exit, RDSCAL is updated w.r.t. the current contributions
          in RDSUM.
          If TRANS = 'T', RDSCAL is not touched.
          NOTE: RDSCAL only makes sense when CTGSY2 is called by
          CTGSYL.

IPIV

          IPIV is INTEGER array, dimension (N).
          The pivot indices; for 1 <= i <= N, row i of the
          matrix has been interchanged with row IPIV(i).

JPIV

          JPIV is INTEGER array, dimension (N).
          The pivot indices; for 1 <= j <= N, column j of the
          matrix has been interchanged with column JPIV(j).
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

This routine is a further developed implementation of algorithm BSOLVE in [1] using complete pivoting in the LU factorization.

Contributors:

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

References:

[1] Bo Kagstrom and Lars Westin, Generalized Schur Methods with Condition Estimators for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic Control, Vol. 34, No. 7, July 1989, pp 745-751.

[2] Peter Poromaa, On Efficient and Robust Estimators for the Separation between two Regular Matrix Pairs with Applications in Condition Estimation. Report UMINF-95.05, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.

Definition at line 167 of file clatdf.f.

subroutine clatps (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, complex, dimension( * ) AP, complex, dimension( * ) X, real SCALE, real, dimension( * ) CNORM, integer INFO)

CLATPS solves a triangular system of equations with the matrix held in packed storage.  

Purpose:

 CLATPS solves one of the triangular systems

    A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b,

 with scaling to prevent overflow, where A is an upper or lower
 triangular matrix stored in packed form.  Here A**T denotes the
 transpose of A, A**H denotes the conjugate transpose of A, x and b
 are n-element vectors, and s is a scaling factor, usually less than
 or equal to 1, chosen so that the components of x will be less than
 the overflow threshold.  If the unscaled problem will not cause
 overflow, the Level 2 BLAS routine CTPSV is called. If the matrix A
 is singular (A(j,j) = 0 for some j), then s is set to 0 and a
 non-trivial solution to A*x = 0 is returned.
Parameters

UPLO

          UPLO is CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

TRANS

          TRANS is CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  Solve A * x = s*b     (No transpose)
          = 'T':  Solve A**T * x = s*b  (Transpose)
          = 'C':  Solve A**H * x = s*b  (Conjugate transpose)

DIAG

          DIAG is CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

NORMIN

          NORMIN is CHARACTER*1
          Specifies whether CNORM has been set or not.
          = 'Y':  CNORM contains the column norms on entry
          = 'N':  CNORM is not set on entry.  On exit, the norms will
                  be computed and stored in CNORM.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

AP

          AP is COMPLEX array, dimension (N*(N+1)/2)
          The upper or lower triangular matrix A, packed columnwise in
          a linear array.  The j-th column of A is stored in the array
          AP as follows:
          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.

X

          X is COMPLEX array, dimension (N)
          On entry, the right hand side b of the triangular system.
          On exit, X is overwritten by the solution vector x.

SCALE

          SCALE is REAL
          The scaling factor s for the triangular system
             A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b.
          If SCALE = 0, the matrix A is singular or badly scaled, and
          the vector x is an exact or approximate solution to A*x = 0.

CNORM

          CNORM is REAL array, dimension (N)

          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
          contains the norm of the off-diagonal part of the j-th column
          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
          must be greater than or equal to the 1-norm.

          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
          returns the 1-norm of the offdiagonal part of the j-th column
          of A.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -k, the k-th argument had an illegal value
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  A rough bound on x is computed; if that is less than overflow, CTPSV
  is called, otherwise, specific code is used which checks for possible
  overflow or divide-by-zero at every operation.

  A columnwise scheme is used for solving A*x = b.  The basic algorithm
  if A is lower triangular is

       x[1:n] := b[1:n]
       for j = 1, ..., n
            x(j) := x(j) / A(j,j)
            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
       end

  Define bounds on the components of x after j iterations of the loop:
     M(j) = bound on x[1:j]
     G(j) = bound on x[j+1:n]
  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

  Then for iteration j+1 we have
     M(j+1) <= G(j) / | A(j+1,j+1) |
     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )

  where CNORM(j+1) is greater than or equal to the infinity-norm of
  column j+1 of A, not counting the diagonal.  Hence

     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                  1<=i<=j
  and

     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                   1<=i< j

  Since |x(j)| <= M(j), we use the Level 2 BLAS routine CTPSV if the
  reciprocal of the largest M(j), j=1,..,n, is larger than
  max(underflow, 1/overflow).

  The bound on x(j) is also used to determine when a step in the
  columnwise method can be performed without fear of overflow.  If
  the computed bound is greater than a large constant, x is scaled to
  prevent overflow, but if the bound overflows, x is set to 0, x(j) to
  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.

  Similarly, a row-wise scheme is used to solve A**T *x = b  or
  A**H *x = b.  The basic algorithm for A upper triangular is

       for j = 1, ..., n
            x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
       end

  We simultaneously compute two bounds
       G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
       M(j) = bound on x(i), 1<=i<=j

  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
  Then the bound on x(j) is

       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |

            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                      1<=i<=j

  and we can safely call CTPSV if 1/M(n) and 1/G(n) are both greater
  than max(underflow, 1/overflow).

Definition at line 229 of file clatps.f.

subroutine clatrd (character UPLO, integer N, integer NB, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) E, complex, dimension( * ) TAU, complex, dimension( ldw, * ) W, integer LDW)

CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.  

Purpose:

 CLATRD reduces NB rows and columns of a complex Hermitian matrix A to
 Hermitian tridiagonal form by a unitary similarity
 transformation Q**H * A * Q, and returns the matrices V and W which are
 needed to apply the transformation to the unreduced part of A.

 If UPLO = 'U', CLATRD reduces the last NB rows and columns of a
 matrix, of which the upper triangle is supplied;
 if UPLO = 'L', CLATRD reduces the first NB rows and columns of a
 matrix, of which the lower triangle is supplied.

 This is an auxiliary routine called by CHETRD.
Parameters

UPLO

          UPLO is CHARACTER*1
          Specifies whether the upper or lower triangular part of the
          Hermitian matrix A is stored:
          = 'U': Upper triangular
          = 'L': Lower triangular

N

          N is INTEGER
          The order of the matrix A.

NB

          NB is INTEGER
          The number of rows and columns to be reduced.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
          n-by-n upper triangular part of A contains the upper
          triangular part of the matrix A, and the strictly lower
          triangular part of A is not referenced.  If UPLO = 'L', the
          leading n-by-n lower triangular part of A contains the lower
          triangular part of the matrix A, and the strictly upper
          triangular part of A is not referenced.
          On exit:
          if UPLO = 'U', the last NB columns have been reduced to
            tridiagonal form, with the diagonal elements overwriting
            the diagonal elements of A; the elements above the diagonal
            with the array TAU, represent the unitary matrix Q as a
            product of elementary reflectors;
          if UPLO = 'L', the first NB columns have been reduced to
            tridiagonal form, with the diagonal elements overwriting
            the diagonal elements of A; the elements below the diagonal
            with the array TAU, represent the  unitary matrix Q as a
            product of elementary reflectors.
          See Further Details.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

E

          E is REAL array, dimension (N-1)
          If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
          elements of the last NB columns of the reduced matrix;
          if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
          the first NB columns of the reduced matrix.

TAU

          TAU is COMPLEX array, dimension (N-1)
          The scalar factors of the elementary reflectors, stored in
          TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
          See Further Details.

W

          W is COMPLEX array, dimension (LDW,NB)
          The n-by-nb matrix W required to update the unreduced part
          of A.

LDW

          LDW is INTEGER
          The leading dimension of the array W. LDW >= max(1,N).
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  If UPLO = 'U', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(n) H(n-1) . . . H(n-nb+1).

  Each H(i) has the form

     H(i) = I - tau * v * v**H

  where tau is a complex scalar, and v is a complex vector with
  v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
  and tau in TAU(i-1).

  If UPLO = 'L', the matrix Q is represented as a product of elementary
  reflectors

     Q = H(1) H(2) . . . H(nb).

  Each H(i) has the form

     H(i) = I - tau * v * v**H

  where tau is a complex scalar, and v is a complex vector with
  v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
  and tau in TAU(i).

  The elements of the vectors v together form the n-by-nb matrix V
  which is needed, with W, to apply the transformation to the unreduced
  part of the matrix, using a Hermitian rank-2k update of the form:
  A := A - V*W**H - W*V**H.

  The contents of A on exit are illustrated by the following examples
  with n = 5 and nb = 2:

  if UPLO = 'U':                       if UPLO = 'L':

    (  a   a   a   v4  v5 )              (  d                  )
    (      a   a   v4  v5 )              (  1   d              )
    (          a   1   v5 )              (  v1  1   a          )
    (              d   1  )              (  v1  v2  a   a      )
    (                  d  )              (  v1  v2  a   a   a  )

  where d denotes a diagonal element of the reduced matrix, a denotes
  an element of the original matrix that is unchanged, and vi denotes
  an element of the vector defining H(i).

Definition at line 198 of file clatrd.f.

subroutine clatrs (character UPLO, character TRANS, character DIAG, character NORMIN, integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) X, real SCALE, real, dimension( * ) CNORM, integer INFO)

CLATRS solves a triangular system of equations with the scale factor set to prevent overflow.  

Purpose:

 CLATRS solves one of the triangular systems

    A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b,

 with scaling to prevent overflow.  Here A is an upper or lower
 triangular matrix, A**T denotes the transpose of A, A**H denotes the
 conjugate transpose of A, x and b are n-element vectors, and s is a
 scaling factor, usually less than or equal to 1, chosen so that the
 components of x will be less than the overflow threshold.  If the
 unscaled problem will not cause overflow, the Level 2 BLAS routine
 CTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
 then s is set to 0 and a non-trivial solution to A*x = 0 is returned.
Parameters

UPLO

          UPLO is CHARACTER*1
          Specifies whether the matrix A is upper or lower triangular.
          = 'U':  Upper triangular
          = 'L':  Lower triangular

TRANS

          TRANS is CHARACTER*1
          Specifies the operation applied to A.
          = 'N':  Solve A * x = s*b     (No transpose)
          = 'T':  Solve A**T * x = s*b  (Transpose)
          = 'C':  Solve A**H * x = s*b  (Conjugate transpose)

DIAG

          DIAG is CHARACTER*1
          Specifies whether or not the matrix A is unit triangular.
          = 'N':  Non-unit triangular
          = 'U':  Unit triangular

NORMIN

          NORMIN is CHARACTER*1
          Specifies whether CNORM has been set or not.
          = 'Y':  CNORM contains the column norms on entry
          = 'N':  CNORM is not set on entry.  On exit, the norms will
                  be computed and stored in CNORM.

N

          N is INTEGER
          The order of the matrix A.  N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          The triangular matrix A.  If UPLO = 'U', the leading n by n
          upper triangular part of the array A contains the upper
          triangular matrix, and the strictly lower triangular part of
          A is not referenced.  If UPLO = 'L', the leading n by n lower
          triangular part of the array A contains the lower triangular
          matrix, and the strictly upper triangular part of A is not
          referenced.  If DIAG = 'U', the diagonal elements of A are
          also not referenced and are assumed to be 1.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max (1,N).

X

          X is COMPLEX array, dimension (N)
          On entry, the right hand side b of the triangular system.
          On exit, X is overwritten by the solution vector x.

SCALE

          SCALE is REAL
          The scaling factor s for the triangular system
             A * x = s*b,  A**T * x = s*b,  or  A**H * x = s*b.
          If SCALE = 0, the matrix A is singular or badly scaled, and
          the vector x is an exact or approximate solution to A*x = 0.

CNORM

          CNORM is REAL array, dimension (N)

          If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
          contains the norm of the off-diagonal part of the j-th column
          of A.  If TRANS = 'N', CNORM(j) must be greater than or equal
          to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
          must be greater than or equal to the 1-norm.

          If NORMIN = 'N', CNORM is an output argument and CNORM(j)
          returns the 1-norm of the offdiagonal part of the j-th column
          of A.

INFO

          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -k, the k-th argument had an illegal value
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  A rough bound on x is computed; if that is less than overflow, CTRSV
  is called, otherwise, specific code is used which checks for possible
  overflow or divide-by-zero at every operation.

  A columnwise scheme is used for solving A*x = b.  The basic algorithm
  if A is lower triangular is

       x[1:n] := b[1:n]
       for j = 1, ..., n
            x(j) := x(j) / A(j,j)
            x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
       end

  Define bounds on the components of x after j iterations of the loop:
     M(j) = bound on x[1:j]
     G(j) = bound on x[j+1:n]
  Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

  Then for iteration j+1 we have
     M(j+1) <= G(j) / | A(j+1,j+1) |
     G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
            <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )

  where CNORM(j+1) is greater than or equal to the infinity-norm of
  column j+1 of A, not counting the diagonal.  Hence

     G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                  1<=i<=j
  and

     |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                   1<=i< j

  Since |x(j)| <= M(j), we use the Level 2 BLAS routine CTRSV if the
  reciprocal of the largest M(j), j=1,..,n, is larger than
  max(underflow, 1/overflow).

  The bound on x(j) is also used to determine when a step in the
  columnwise method can be performed without fear of overflow.  If
  the computed bound is greater than a large constant, x is scaled to
  prevent overflow, but if the bound overflows, x is set to 0, x(j) to
  1, and scale to 0, and a non-trivial solution to A*x = 0 is found.

  Similarly, a row-wise scheme is used to solve A**T *x = b  or
  A**H *x = b.  The basic algorithm for A upper triangular is

       for j = 1, ..., n
            x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
       end

  We simultaneously compute two bounds
       G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
       M(j) = bound on x(i), 1<=i<=j

  The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
  add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
  Then the bound on x(j) is

       M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |

            <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                      1<=i<=j

  and we can safely call CTRSV if 1/M(n) and 1/G(n) are both greater
  than max(underflow, 1/overflow).

Definition at line 237 of file clatrs.f.

subroutine clauu2 (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer INFO)

CLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular matrices (unblocked algorithm).  

Purpose:

 CLAUU2 computes the product U * U**H or L**H * L, where the triangular
 factor U or L is stored in the upper or lower triangular part of
 the array A.

 If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
 overwriting the factor U in A.
 If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
 overwriting the factor L in A.

 This is the unblocked form of the algorithm, calling Level 2 BLAS.
Parameters

UPLO

          UPLO is CHARACTER*1
          Specifies whether the triangular factor stored in the array A
          is upper or lower triangular:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

N

          N is INTEGER
          The order of the triangular factor U or L.  N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the triangular factor U or L.
          On exit, if UPLO = 'U', the upper triangle of A is
          overwritten with the upper triangle of the product U * U**H;
          if UPLO = 'L', the lower triangle of A is overwritten with
          the lower triangle of the product L**H * L.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 101 of file clauu2.f.

subroutine clauum (character UPLO, integer N, complex, dimension( lda, * ) A, integer LDA, integer INFO)

CLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular matrices (blocked algorithm).  

Purpose:

 CLAUUM computes the product U * U**H or L**H * L, where the triangular
 factor U or L is stored in the upper or lower triangular part of
 the array A.

 If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
 overwriting the factor U in A.
 If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
 overwriting the factor L in A.

 This is the blocked form of the algorithm, calling Level 3 BLAS.
Parameters

UPLO

          UPLO is CHARACTER*1
          Specifies whether the triangular factor stored in the array A
          is upper or lower triangular:
          = 'U':  Upper triangular
          = 'L':  Lower triangular

N

          N is INTEGER
          The order of the triangular factor U or L.  N >= 0.

A

          A is COMPLEX array, dimension (LDA,N)
          On entry, the triangular factor U or L.
          On exit, if UPLO = 'U', the upper triangle of A is
          overwritten with the upper triangle of the product U * U**H;
          if UPLO = 'L', the lower triangle of A is overwritten with
          the lower triangle of the product L**H * L.

LDA

          LDA is INTEGER
          The leading dimension of the array A.  LDA >= max(1,N).

INFO

          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -k, the k-th argument had an illegal value
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 101 of file clauum.f.

subroutine crot (integer N, complex, dimension( * ) CX, integer INCX, complex, dimension( * ) CY, integer INCY, real C, complex S)

CROT applies a plane rotation with real cosine and complex sine to a pair of complex vectors.  

Purpose:

 CROT   applies a plane rotation, where the cos (C) is real and the
 sin (S) is complex, and the vectors CX and CY are complex.
Parameters

N

          N is INTEGER
          The number of elements in the vectors CX and CY.

CX

          CX is COMPLEX array, dimension (N)
          On input, the vector X.
          On output, CX is overwritten with C*X + S*Y.

INCX

          INCX is INTEGER
          The increment between successive values of CY.  INCX <> 0.

CY

          CY is COMPLEX array, dimension (N)
          On input, the vector Y.
          On output, CY is overwritten with -CONJG(S)*X + C*Y.

INCY

          INCY is INTEGER
          The increment between successive values of CY.  INCX <> 0.

C

          C is REAL

S

          S is COMPLEX
          C and S define a rotation
             [  C          S  ]
             [ -conjg(S)   C  ]
          where C*C + S*CONJG(S) = 1.0.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 102 of file crot.f.

subroutine cspmv (character UPLO, integer N, complex ALPHA, complex, dimension( * ) AP, complex, dimension( * ) X, integer INCX, complex BETA, complex, dimension( * ) Y, integer INCY)

CSPMV computes a matrix-vector product for complex vectors using a complex symmetric packed matrix  

Purpose:

 CSPMV  performs the matrix-vector operation

    y := alpha*A*x + beta*y,

 where alpha and beta are scalars, x and y are n element vectors and
 A is an n by n symmetric matrix, supplied in packed form.
Parameters

UPLO

          UPLO is CHARACTER*1
           On entry, UPLO specifies whether the upper or lower
           triangular part of the matrix A is supplied in the packed
           array AP as follows:

              UPLO = 'U' or 'u'   The upper triangular part of A is
                                  supplied in AP.

              UPLO = 'L' or 'l'   The lower triangular part of A is
                                  supplied in AP.

           Unchanged on exit.

N

          N is INTEGER
           On entry, N specifies the order of the matrix A.
           N must be at least zero.
           Unchanged on exit.

ALPHA

          ALPHA is COMPLEX
           On entry, ALPHA specifies the scalar alpha.
           Unchanged on exit.

AP

          AP is COMPLEX array, dimension at least
           ( ( N*( N + 1 ) )/2 ).
           Before entry, with UPLO = 'U' or 'u', the array AP must
           contain the upper triangular part of the symmetric matrix
           packed sequentially, column by column, so that AP( 1 )
           contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
           and a( 2, 2 ) respectively, and so on.
           Before entry, with UPLO = 'L' or 'l', the array AP must
           contain the lower triangular part of the symmetric matrix
           packed sequentially, column by column, so that AP( 1 )
           contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
           and a( 3, 1 ) respectively, and so on.
           Unchanged on exit.

X

          X is COMPLEX array, dimension at least
           ( 1 + ( N - 1 )*abs( INCX ) ).
           Before entry, the incremented array X must contain the N-
           element vector x.
           Unchanged on exit.

INCX

          INCX is INTEGER
           On entry, INCX specifies the increment for the elements of
           X. INCX must not be zero.
           Unchanged on exit.

BETA

          BETA is COMPLEX
           On entry, BETA specifies the scalar beta. When BETA is
           supplied as zero then Y need not be set on input.
           Unchanged on exit.

Y

          Y is COMPLEX array, dimension at least
           ( 1 + ( N - 1 )*abs( INCY ) ).
           Before entry, the incremented array Y must contain the n
           element vector y. On exit, Y is overwritten by the updated
           vector y.

INCY

          INCY is INTEGER
           On entry, INCY specifies the increment for the elements of
           Y. INCY must not be zero.
           Unchanged on exit.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 150 of file cspmv.f.

subroutine cspr (character UPLO, integer N, complex ALPHA, complex, dimension( * ) X, integer INCX, complex, dimension( * ) AP)

CSPR performs the symmetrical rank-1 update of a complex symmetric packed matrix.  

Purpose:

 CSPR    performs the symmetric rank 1 operation

    A := alpha*x*x**H + A,

 where alpha is a complex scalar, x is an n element vector and A is an
 n by n symmetric matrix, supplied in packed form.
Parameters

UPLO

          UPLO is CHARACTER*1
           On entry, UPLO specifies whether the upper or lower
           triangular part of the matrix A is supplied in the packed
           array AP as follows:

              UPLO = 'U' or 'u'   The upper triangular part of A is
                                  supplied in AP.

              UPLO = 'L' or 'l'   The lower triangular part of A is
                                  supplied in AP.

           Unchanged on exit.

N

          N is INTEGER
           On entry, N specifies the order of the matrix A.
           N must be at least zero.
           Unchanged on exit.

ALPHA

          ALPHA is COMPLEX
           On entry, ALPHA specifies the scalar alpha.
           Unchanged on exit.

X

          X is COMPLEX array, dimension at least
           ( 1 + ( N - 1 )*abs( INCX ) ).
           Before entry, the incremented array X must contain the N-
           element vector x.
           Unchanged on exit.

INCX

          INCX is INTEGER
           On entry, INCX specifies the increment for the elements of
           X. INCX must not be zero.
           Unchanged on exit.

AP

          AP is COMPLEX array, dimension at least
           ( ( N*( N + 1 ) )/2 ).
           Before entry, with  UPLO = 'U' or 'u', the array AP must
           contain the upper triangular part of the symmetric matrix
           packed sequentially, column by column, so that AP( 1 )
           contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
           and a( 2, 2 ) respectively, and so on. On exit, the array
           AP is overwritten by the upper triangular part of the
           updated matrix.
           Before entry, with UPLO = 'L' or 'l', the array AP must
           contain the lower triangular part of the symmetric matrix
           packed sequentially, column by column, so that AP( 1 )
           contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
           and a( 3, 1 ) respectively, and so on. On exit, the array
           AP is overwritten by the lower triangular part of the
           updated matrix.
           Note that the imaginary parts of the diagonal elements need
           not be set, they are assumed to be zero, and on exit they
           are set to zero.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 131 of file cspr.f.

subroutine csrscl (integer N, real SA, complex, dimension( * ) SX, integer INCX)

CSRSCL multiplies a vector by the reciprocal of a real scalar.  

Purpose:

 CSRSCL multiplies an n-element complex vector x by the real scalar
 1/a.  This is done without overflow or underflow as long as
 the final result x/a does not overflow or underflow.
Parameters

N

          N is INTEGER
          The number of components of the vector x.

SA

          SA is REAL
          The scalar a which is used to divide each component of x.
          SA must be >= 0, or the subroutine will divide by zero.

SX

          SX is COMPLEX array, dimension
                         (1+(N-1)*abs(INCX))
          The n-element vector x.

INCX

          INCX is INTEGER
          The increment between successive values of the vector SX.
          > 0:  SX(1) = X(1) and SX(1+(i-1)*INCX) = x(i),     1< i<= n
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 83 of file csrscl.f.

subroutine ctprfb (character SIDE, character TRANS, character DIRECT, character STOREV, integer M, integer N, integer K, integer L, complex, dimension( ldv, * ) V, integer LDV, complex, dimension( ldt, * ) T, integer LDT, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldwork, * ) WORK, integer LDWORK)

CTPRFB applies a real or complex 'triangular-pentagonal' blocked reflector to a real or complex matrix, which is composed of two blocks.  

Purpose:

 CTPRFB applies a complex "triangular-pentagonal" block reflector H or its
 conjugate transpose H**H to a complex matrix C, which is composed of two
 blocks A and B, either from the left or right.
Parameters

SIDE

          SIDE is CHARACTER*1
          = 'L': apply H or H**H from the Left
          = 'R': apply H or H**H from the Right

TRANS

          TRANS is CHARACTER*1
          = 'N': apply H (No transpose)
          = 'C': apply H**H (Conjugate transpose)

DIRECT

          DIRECT is CHARACTER*1
          Indicates how H is formed from a product of elementary
          reflectors
          = 'F': H = H(1) H(2) . . . H(k) (Forward)
          = 'B': H = H(k) . . . H(2) H(1) (Backward)

STOREV

          STOREV is CHARACTER*1
          Indicates how the vectors which define the elementary
          reflectors are stored:
          = 'C': Columns
          = 'R': Rows

M

          M is INTEGER
          The number of rows of the matrix B.
          M >= 0.

N

          N is INTEGER
          The number of columns of the matrix B.
          N >= 0.

K

          K is INTEGER
          The order of the matrix T, i.e. the number of elementary
          reflectors whose product defines the block reflector.
          K >= 0.

L

          L is INTEGER
          The order of the trapezoidal part of V.
          K >= L >= 0.  See Further Details.

V

          V is COMPLEX array, dimension
                                (LDV,K) if STOREV = 'C'
                                (LDV,M) if STOREV = 'R' and SIDE = 'L'
                                (LDV,N) if STOREV = 'R' and SIDE = 'R'
          The pentagonal matrix V, which contains the elementary reflectors
          H(1), H(2), ..., H(K).  See Further Details.

LDV

          LDV is INTEGER
          The leading dimension of the array V.
          If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
          if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
          if STOREV = 'R', LDV >= K.

T

          T is COMPLEX array, dimension (LDT,K)
          The triangular K-by-K matrix T in the representation of the
          block reflector.

LDT

          LDT is INTEGER
          The leading dimension of the array T.
          LDT >= K.

A

          A is COMPLEX array, dimension
          (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R'
          On entry, the K-by-N or M-by-K matrix A.
          On exit, A is overwritten by the corresponding block of
          H*C or H**H*C or C*H or C*H**H.  See Further Details.

LDA

          LDA is INTEGER
          The leading dimension of the array A.
          If SIDE = 'L', LDA >= max(1,K);
          If SIDE = 'R', LDA >= max(1,M).

B

          B is COMPLEX array, dimension (LDB,N)
          On entry, the M-by-N matrix B.
          On exit, B is overwritten by the corresponding block of
          H*C or H**H*C or C*H or C*H**H.  See Further Details.

LDB

          LDB is INTEGER
          The leading dimension of the array B.
          LDB >= max(1,M).

WORK

          WORK is COMPLEX array, dimension
          (LDWORK,N) if SIDE = 'L',
          (LDWORK,K) if SIDE = 'R'.

LDWORK

          LDWORK is INTEGER
          The leading dimension of the array WORK.
          If SIDE = 'L', LDWORK >= K;
          if SIDE = 'R', LDWORK >= M.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

  The matrix C is a composite matrix formed from blocks A and B.
  The block B is of size M-by-N; if SIDE = 'R', A is of size M-by-K,
  and if SIDE = 'L', A is of size K-by-N.

  If SIDE = 'R' and DIRECT = 'F', C = [A B].

  If SIDE = 'L' and DIRECT = 'F', C = [A]
                                      [B].

  If SIDE = 'R' and DIRECT = 'B', C = [B A].

  If SIDE = 'L' and DIRECT = 'B', C = [B]
                                      [A].

  The pentagonal matrix V is composed of a rectangular block V1 and a
  trapezoidal block V2.  The size of the trapezoidal block is determined by
  the parameter L, where 0<=L<=K.  If L=K, the V2 block of V is triangular;
  if L=0, there is no trapezoidal block, thus V = V1 is rectangular.

  If DIRECT = 'F' and STOREV = 'C':  V = [V1]
                                         [V2]
     - V2 is upper trapezoidal (first L rows of K-by-K upper triangular)

  If DIRECT = 'F' and STOREV = 'R':  V = [V1 V2]

     - V2 is lower trapezoidal (first L columns of K-by-K lower triangular)

  If DIRECT = 'B' and STOREV = 'C':  V = [V2]
                                         [V1]
     - V2 is lower trapezoidal (last L rows of K-by-K lower triangular)

  If DIRECT = 'B' and STOREV = 'R':  V = [V2 V1]

     - V2 is upper trapezoidal (last L columns of K-by-K upper triangular)

  If STOREV = 'C' and SIDE = 'L', V is M-by-K with V2 L-by-K.

  If STOREV = 'C' and SIDE = 'R', V is N-by-K with V2 L-by-K.

  If STOREV = 'R' and SIDE = 'L', V is K-by-M with V2 K-by-L.

  If STOREV = 'R' and SIDE = 'R', V is K-by-N with V2 K-by-L.

Definition at line 249 of file ctprfb.f.

integer function icmax1 (integer N, complex, dimension(*) CX, integer INCX)

ICMAX1 finds the index of the first vector element of maximum absolute value.  

Purpose:

 ICMAX1 finds the index of the first vector element of maximum absolute value.

 Based on ICAMAX from Level 1 BLAS.
 The change is to use the 'genuine' absolute value.
Parameters

N

          N is INTEGER
          The number of elements in the vector CX.

CX

          CX is COMPLEX array, dimension (N)
          The vector CX. The ICMAX1 function returns the index of its first
          element of maximum absolute value.

INCX

          INCX is INTEGER
          The spacing between successive values of CX.  INCX >= 1.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Nick Higham for use with CLACON.

Definition at line 80 of file icmax1.f.

integer function ilaclc (integer M, integer N, complex, dimension( lda, * ) A, integer LDA)

ILACLC scans a matrix for its last non-zero column.  

Purpose:

 ILACLC scans A for its last non-zero column.
Parameters

M

          M is INTEGER
          The number of rows of the matrix A.

N

          N is INTEGER
          The number of columns of the matrix A.

A

          A is COMPLEX array, dimension (LDA,N)
          The m by n matrix A.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,M).
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 77 of file ilaclc.f.

integer function ilaclr (integer M, integer N, complex, dimension( lda, * ) A, integer LDA)

ILACLR scans a matrix for its last non-zero row.  

Purpose:

 ILACLR scans A for its last non-zero row.
Parameters

M

          M is INTEGER
          The number of rows of the matrix A.

N

          N is INTEGER
          The number of columns of the matrix A.

A

          A is COMPLEX array, dimension (LDA,N)
          The m by n matrix A.

LDA

          LDA is INTEGER
          The leading dimension of the array A. LDA >= max(1,M).
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Definition at line 77 of file ilaclr.f.

integer function izmax1 (integer N, complex*16, dimension(*) ZX, integer INCX)

IZMAX1 finds the index of the first vector element of maximum absolute value.  

Purpose:

 IZMAX1 finds the index of the first vector element of maximum absolute value.

 Based on IZAMAX from Level 1 BLAS.
 The change is to use the 'genuine' absolute value.
Parameters

N

          N is INTEGER
          The number of elements in the vector ZX.

ZX

          ZX is COMPLEX*16 array, dimension (N)
          The vector ZX. The IZMAX1 function returns the index of its first
          element of maximum absolute value.

INCX

          INCX is INTEGER
          The spacing between successive values of ZX.  INCX >= 1.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Nick Higham for use with ZLACON.

Definition at line 80 of file izmax1.f.

real function scsum1 (integer N, complex, dimension( * ) CX, integer INCX)

SCSUM1 forms the 1-norm of the complex vector using the true absolute value.  

Purpose:

 SCSUM1 takes the sum of the absolute values of a complex
 vector and returns a single precision result.

 Based on SCASUM from the Level 1 BLAS.
 The change is to use the 'genuine' absolute value.
Parameters

N

          N is INTEGER
          The number of elements in the vector CX.

CX

          CX is COMPLEX array, dimension (N)
          The vector whose elements will be summed.

INCX

          INCX is INTEGER
          The spacing between successive values of CX.  INCX > 0.
Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Nick Higham for use with CLACON.

Definition at line 80 of file scsum1.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.

Referenced By

The man pages clabrd(3), clacgv(3), clacn2(3), clacon(3), clacp2(3), clacpy(3), clacrm(3), clacrt(3), cladiv(3), claein(3), claev2(3), clags2(3), clagtm(3), clahqr(3), clahr2(3), claic1(3), clangt(3), clanhb(3), clanhp(3), clanhs(3), clanht(3), clansb(3), clansp(3), clantb(3), clantp(3), clantr(3), clapll(3), clapmr(3), clapmt(3), claqhb(3), claqhp(3), claqp2(3), claqps(3), claqr0(3), claqr1(3), claqr2(3), claqr3(3), claqr4(3), claqr5(3), claqsb(3), claqsp(3), clar1v(3), clar2v(3), clarcm(3), clarf(3), clarfb(3), clarfb_gett(3), clarfg(3), clarfgp(3), clarft(3), clarfx(3), clarfy(3), clargv(3), clarnv(3), clarrv(3), clartg(3), clartv(3), clascl(3), claset(3), clasr(3), classq(3), claswp(3), clatbs(3), clatdf(3), clatps(3), clatrd(3), clatrs(3), clauu2(3), clauum(3), crot(3), cspmv(3), cspr(3), csrscl(3), ctprfb(3), icmax1(3), ilaclc(3), ilaclr(3), izmax1(3) and scsum1(3) are aliases of complexOTHERauxiliary(3).

Thu Apr 1 2021 Version 3.9.1 LAPACK