claqr4.f man page

claqr4.f —

Synopsis

Functions/Subroutines

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.

Function/Subroutine Documentation

subroutine claqr4 (logicalWANTT, logicalWANTZ, integerN, integerILO, integerIHI, complex, dimension( ldh, * )H, integerLDH, complex, dimension( * )W, integerILOZ, integerIHIZ, complex, dimension( ldz, * )Z, integerLDZ, complex, dimension( * )WORK, integerLWORK, integerINFO)

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 .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 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.GT.0, then 1.LE.ILO.LE.IHI.LE.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.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).

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 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. 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.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 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 .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 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

verbatim
         INFO is INTEGER
            =  0:  successful exit
          .GT. 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 .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 unitary 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 unitary 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 249 of file claqr4.f.

Author

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Referenced By

claqr4(3) is an alias of claqr4.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK