zhseqr(3S)

NAME
     ZHSEQR - compute the eigenvalues of a complex upper 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

SYNOPSIS
     SUBROUTINE ZHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK,
                        LWORK, INFO )

         CHARACTER      COMPZ, JOB

         INTEGER        IHI, ILO, INFO, LDH, LDZ, LWORK, N

         COMPLEX*16     H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     ZHSEQR computes the eigenvalues of a complex upper 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)*T*(QZ)**H.


ARGUMENTS
     JOB     (input) CHARACTER*1
             = 'E': compute eigenvalues only;
             = 'S': compute eigenvalues and the Schur form T.

     COMPZ   (input) CHARACTER*1
             = 'N': no Schur vectors are computed;
             = 'I': Z is initialized to the unit matrix and the matrix Z of
             Schur vectors of H is returned; = 'V': Z must contain an unitary
             matrix Q on entry, and the product Q*Z is returned.

     N       (input) INTEGER
             The order of the matrix H.  N >= 0.

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

     H       (input/output) COMPLEX*16 array, dimension (LDH,N)
             On entry, the upper Hessenberg matrix H.  On exit, if JOB = 'S',
             H contains the upper triangular matrix T from the Schur
             decomposition (the Schur form). If JOB = 'E', the contents of H
             are unspecified on exit.

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

     W       (output) COMPLEX*16 array, dimension (N)
             The computed eigenvalues. If JOB = 'S', 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).

     Z       (input/output) COMPLEX*16 array, dimension (LDZ,N)
             If COMPZ = 'N': Z is not referenced.
             If COMPZ = 'I': on entry, Z need not be set, and on exit, Z
             contains the unitary matrix Z of the Schur vectors of H.  If
             COMPZ = 'V': on entry Z must contain an N-by-N matrix Q, which is
             assumed to be equal to the unit matrix except for the submatrix
             Z(ILO:IHI,ILO:IHI); on exit Z contains Q*Z.  Normally Q is the
             unitary matrix generated by ZUNGHR after the call to ZGEHRD which
             formed the Hessenberg matrix H.

     LDZ     (input) INTEGER
             The leading dimension of the array Z.  LDZ >= max(1,N) if COMPZ =
             'I' or 'V'; LDZ >= 1 otherwise.

     WORK    (workspace/output) COMPLEX*16 array, dimension (LWORK)
             On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

     LWORK   (input) INTEGER
             The dimension of the array WORK.  LWORK >= max(1,N).

             If LWORK = -1, then a workspace query is assumed; the routine
             only calculates the optimal size of the WORK array, returns this
             value as the first entry of the WORK array, and no error message
             related to LWORK is issued by XERBLA.
     INFO    (output) INTEGER
             = 0:  successful exit
             < 0:  if INFO = -i, the i-th argument had an illegal value
             > 0:  if INFO = i, ZHSEQR failed to compute all the eigenvalues
             in a total of 30*(IHI-ILO+1) iterations; elements 1:ilo-1 and
             i+1:n of W contain those eigenvalues which have been successfully
             computed.

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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