dhseqr(3S)

NAME
     DHSEQR - compute the eigenvalues of a real upper 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

SYNOPSIS
     SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, LDZ, WORK,
                        LWORK, INFO )

         CHARACTER      COMPZ, JOB

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

         DOUBLE         PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ), 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
     DHSEQR computes the eigenvalues of a real upper 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.


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
             orthogonal 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 DGEBAL, and then passed to
             SGEHRD when the matrix output by DGEBAL 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) DOUBLE PRECISION array, dimension (LDH,N)
             On entry, the upper Hessenberg matrix H.  On exit, if JOB = 'S',
             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) < 0. 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).

     WR      (output) DOUBLE PRECISION array, dimension (N)
             WI      (output) DOUBLE PRECISION array, dimension (N) The real
             and imaginary parts, respectively, of the computed eigenvalues.
             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) > 0 and WI(i+1) < 0. If JOB = 'S', 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).

     Z       (input/output) DOUBLE PRECISION 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 orthogonal 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
             orthogonal matrix generated by DORGHR after the call to DGEHRD
             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) DOUBLE PRECISION 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, DHSEQR failed to compute all of the
             eigenvalues in a total of 30*(IHI-ILO+1) iterations; elements
             1:ilo-1 and i+1:n of WR and WI contain those eigenvalues which
             have been successfully computed.

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

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