dlaqr4.f man page

dlaqr4.f —

Synopsis

Functions/Subroutines

subroutine dlaqr4 (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)
DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Function/Subroutine Documentation

subroutine dlaqr4 (logicalWANTT, logicalWANTZ, integerN, integerILO, integerIHI, double precision, dimension( ldh, * )H, integerLDH, double precision, dimension( * )WR, double precision, dimension( * )WI, integerILOZ, integerIHIZ, double precision, dimension( ldz, * )Z, integerLDZ, double precision, dimension( * )WORK, integerLWORK, integerINFO)

DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Purpose:

DLAQR4 implements one level of recursion for DLAQR0.
It is a complete implementation of the small bulge multi-shift
QR algorithm.  It may be called by DLAQR0 and, for large enough
deflation window size, it may be called by DLAQR3.  This
subroutine is identical to DLAQR0 except that it calls DLAQR2
instead of DLAQR3.

DLAQR4 computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**T, where T is an upper quasi-triangular matrix (the
Schur form), and Z is the orthogonal matrix of Schur vectors.

Optionally Z may be postmultiplied into an input orthogonal
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 orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

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 .GE. 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.GT.1,
 H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
 previous call to DGEBAL, and then passed to DGEHRD when the
 matrix output by DGEBAL is reduced to Hessenberg form.
 Otherwise, ILO and IHI should be set to 1 and N,
 respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
 If N = 0, then ILO = 1 and IHI = 0.

H

H is DOUBLE PRECISION 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 quasi-triangular matrix T from the Schur
 decomposition (the Schur form); 2-by-2 diagonal blocks
 (corresponding to complex conjugate pairs of eigenvalues)
 are returned in standard form, with H(i,i) = H(i+1,i+1)
 and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
 .FALSE., then the contents of H are unspecified on exit.
 (The output value of H when INFO.GT.0 is given under the
 description of INFO below.)

 This subroutine may explicitly set H(i,j) = 0 for i.GT.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 .GE. max(1,N).

WR

WR is DOUBLE PRECISION array, dimension (IHI)

WI

WI is DOUBLE PRECISION array, dimension (IHI)
 The real and imaginary parts, respectively, of the computed
 eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
 and WI(ILO:IHI). If two eigenvalues are computed as a
 complex conjugate pair, they are stored in consecutive
 elements of WR and WI, say the i-th and (i+1)th, with
 WI(i) .GT. 0 and WI(i+1) .LT. 0. 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
 WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
 block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
 WI(i+1) = -WI(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 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.

Z

Z is DOUBLE PRECISION 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.GT.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.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.

WORK

WORK is DOUBLE PRECISION 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 .GE. 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 DLAQR4 does a workspace query.
 In this case, DLAQR4 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
 .GT. 0:  if INFO = i, DLAQR4 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 .GT. 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 .GT. 0 and WANTT is .TRUE., then on exit

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

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

      If INFO .GT. 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 orthogonal matrix in (*) (regard-
      less of the value of WANTT.)

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

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

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 263 of file dlaqr4.f.

Author

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Referenced By

dlaqr4(3) is an alias of dlaqr4.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK