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 * 1 JOB, COMPZ
  INTEGER N, ILO, IHI, LDH, LDZ, LWORK, INFO
  DOUBLE PRECISION H(LDH,*), WR(*), WI(*), Z(LDZ,*), WORK(*)

  SUBROUTINE DHSEQR_64( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, 
 *      LDZ, WORK, LWORK, INFO)
  CHARACTER * 1 JOB, COMPZ
  INTEGER*8 N, ILO, IHI, LDH, LDZ, LWORK, INFO
  DOUBLE PRECISION H(LDH,*), WR(*), WI(*), Z(LDZ,*), WORK(*)

F95 INTERFACE

  SUBROUTINE HSEQR( JOB, COMPZ, N, ILO, IHI, H, [LDH], WR, WI, Z, [LDZ], 
 *       [WORK], [LWORK], [INFO])
  CHARACTER(LEN=1) :: JOB, COMPZ
  INTEGER :: N, ILO, IHI, LDH, LDZ, LWORK, INFO
  REAL(8), DIMENSION(:) :: WR, WI, WORK
  REAL(8), DIMENSION(:,:) :: H, Z

  SUBROUTINE HSEQR_64( JOB, COMPZ, N, ILO, IHI, H, [LDH], WR, WI, Z, 
 *       [LDZ], [WORK], [LWORK], [INFO])
  CHARACTER(LEN=1) :: JOB, COMPZ
  INTEGER(8) :: N, ILO, IHI, LDH, LDZ, LWORK, INFO
  REAL(8), DIMENSION(:) :: WR, WI, WORK
  REAL(8), DIMENSION(:,:) :: H, Z

C INTERFACE

#include <sunperf.h>

void dhseqr(char job, char compz, int n, int ilo, int ihi, double *h, int ldh, double *wr, double *wi, double *z, int ldz, int *info);

void dhseqr_64(char job, char compz, long n, long ilo, long ihi, double *h, long ldh, double *wr, double *wi, double *z, long ldz, long *info);

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)

 = 'E':  compute eigenvalues only;

 = 'S':  compute eigenvalues and the Schur form T.

COMPZ (input)

 = '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)
The order of the matrix H. N > = 0.
ILO (input)
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 SGEBAL, and then passed to SGEHRD when the matrix output by SGEBAL 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.
IHI (input)
See the description of ILO.
H (input/output)
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)
The leading dimension of the array H. LDH > = max(1,N).
WR (output)
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).
WI (output)
See the description of WR.
Z (input/output)
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 SORGHR after the call to SGEHRD which formed the Hessenberg matrix H.
LDZ (input)
The leading dimension of the array Z. LDZ > = max(1,N) if COMPZ = 'I' or 'V'; LDZ > = 1 otherwise.
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input)
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)

 = 0:  successful exit

 < 0:  if INFO  = -i, the i-th argument had an illegal value

 > 0:  if INFO  = i, SHSEQR 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.