# 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***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*

**Internal Parameters:**

**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**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**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**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**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).