shseqr(3P) Sun Performance Library shseqr(3P)NAMEshseqr - 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 SHSEQR(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
REAL H(LDH,*), WR(*), WI(*), Z(LDZ,*), WORK(*)
SUBROUTINE SHSEQR_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
REAL 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, DIMENSION(:) :: WR, WI, WORK
REAL, 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, DIMENSION(:) :: WR, WI, WORK
REAL, DIMENSION(:,:) :: H, Z
C INTERFACE
#include <sunperf.h>
void shseqr(char job, char compz, int n, int ilo, int ihi, float *h,
int ldh, float *wr, float *wi, float *z, int ldz, int *info);
void shseqr_64(char job, char compz, long n, long ilo, long ihi, float
*h, long ldh, float *wr, float *wi, float *z, long ldz, long
*info);
PURPOSEshseqr 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. Other‐
wise 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 diago‐
nal 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) 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 eigen‐
values 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.
6 Mar 2009 shseqr(3P)
