SHSEQR(3F) SHSEQR(3F)
SHSEQR - 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
SUBROUTINE SHSEQR( 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
REAL H( LDH, * ), WI( * ), WORK( * ), WR( * ), Z( LDZ, * )
SHSEQR 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.
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 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.
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SHSEQR(3F) SHSEQR(3F)
H (input/output) REAL 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) REAL array, dimension (N)
WI (output) REAL 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) REAL 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 SORGHR after the call to SGEHRD
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) REAL array, dimension (N)
LWORK (input) INTEGER
This argument is currently redundant.
INFO (output) INTEGER
= 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.
SHSEQR(3F) SHSEQR(3F)
SHSEQR - 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
SUBROUTINE SHSEQR( 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
REAL H( LDH, * ), WI( * ), WORK( * ), WR( * ), Z( LDZ, * )
SHSEQR 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.
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 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.
Page 1
SHSEQR(3F) SHSEQR(3F)
H (input/output) REAL 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) REAL array, dimension (N)
WI (output) REAL 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) REAL 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 SORGHR after the call to SGEHRD
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) REAL array, dimension (N)
LWORK (input) INTEGER
This argument is currently redundant.
INFO (output) INTEGER
= 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.
PPPPaaaaggggeeee 2222 [ Back ]
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