SGGHRD(3F) SGGHRD(3F)
SGGHRD - reduce a pair of real matrices (A,B) to generalized upper
Hessenberg form using orthogonal transformations, where A is a general
matrix and B is upper triangular
SUBROUTINE SGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z,
LDZ, INFO )
CHARACTER COMPQ, COMPZ
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
REAL A( LDA, * ), B( LDB, * ), Q( LDQ, * ), Z( LDZ, * )
SGGHRD reduces a pair of real matrices (A,B) to generalized upper
Hessenberg form using orthogonal transformations, where A is a general
matrix and B is upper triangular: Q' * A * Z = H and Q' * B * Z = T,
where H is upper Hessenberg, T is upper triangular, and Q and Z are
orthogonal, and ' means transpose.
The orthogonal matrices Q and Z are determined as products of Givens
rotations. They may either be formed explicitly, or they may be
postmultiplied into input matrices Q1 and Z1, so that
Q1 * A * Z1' = (Q1*Q) * H * (Z1*Z)'
Q1 * B * Z1' = (Q1*Q) * T * (Z1*Z)'
COMPQ (input) CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the orthogonal
matrix Q is returned; = 'V': Q must contain an orthogonal matrix
Q1 on entry, and the product Q1*Q is returned.
COMPZ (input) CHARACTER*1
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the orthogonal
matrix Z is returned; = 'V': Z must contain an orthogonal matrix
Z1 on entry, and the product Z1*Z is returned.
N (input) INTEGER
The order of the matrices A and B. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER It is assumed that A 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 SGGBAL; otherwise they
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
SGGHRD(3F) SGGHRD(3F)
A (input/output) REAL array, dimension (LDA, N)
On entry, the N-by-N general matrix to be reduced. On exit, the
upper triangle and the first subdiagonal of A are overwritten
with the upper Hessenberg matrix H, and the rest is set to zero.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) REAL array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B. On exit, the
upper triangular matrix T = Q' B Z. The elements below the
diagonal are set to zero.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Q (input/output) REAL array, dimension (LDQ, N)
If COMPQ='N': Q is not referenced.
If COMPQ='I': on entry, Q need not be set, and on exit it
contains the orthogonal matrix Q, where Q' is the product of the
Givens transformations which are applied to A and B on the left.
If COMPQ='V': on entry, Q must contain an orthogonal matrix Q1,
and on exit this is overwritten by Q1*Q.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= N if COMPQ='V' or
'I'; LDQ >= 1 otherwise.
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 it
contains the orthogonal matrix Z, which is the product of the
Givens transformations which are applied to A and B on the right.
If COMPZ='V': on entry, Z must contain an orthogonal matrix Z1,
and on exit this is overwritten by Z1*Z.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= N if COMPZ='V' or
'I'; LDZ >= 1 otherwise.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
FURTHER DETAILS
This routine reduces A to Hessenberg and B to triangular form by an
unblocked reduction, as described in _Matrix_Computations_, by Golub and
Van Loan (Johns Hopkins Press.)
SGGHRD(3F) SGGHRD(3F)
SGGHRD - reduce a pair of real matrices (A,B) to generalized upper
Hessenberg form using orthogonal transformations, where A is a general
matrix and B is upper triangular
SUBROUTINE SGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z,
LDZ, INFO )
CHARACTER COMPQ, COMPZ
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
REAL A( LDA, * ), B( LDB, * ), Q( LDQ, * ), Z( LDZ, * )
SGGHRD reduces a pair of real matrices (A,B) to generalized upper
Hessenberg form using orthogonal transformations, where A is a general
matrix and B is upper triangular: Q' * A * Z = H and Q' * B * Z = T,
where H is upper Hessenberg, T is upper triangular, and Q and Z are
orthogonal, and ' means transpose.
The orthogonal matrices Q and Z are determined as products of Givens
rotations. They may either be formed explicitly, or they may be
postmultiplied into input matrices Q1 and Z1, so that
Q1 * A * Z1' = (Q1*Q) * H * (Z1*Z)'
Q1 * B * Z1' = (Q1*Q) * T * (Z1*Z)'
COMPQ (input) CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the orthogonal
matrix Q is returned; = 'V': Q must contain an orthogonal matrix
Q1 on entry, and the product Q1*Q is returned.
COMPZ (input) CHARACTER*1
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the orthogonal
matrix Z is returned; = 'V': Z must contain an orthogonal matrix
Z1 on entry, and the product Z1*Z is returned.
N (input) INTEGER
The order of the matrices A and B. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER It is assumed that A 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 SGGBAL; otherwise they
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
SGGHRD(3F) SGGHRD(3F)
A (input/output) REAL array, dimension (LDA, N)
On entry, the N-by-N general matrix to be reduced. On exit, the
upper triangle and the first subdiagonal of A are overwritten
with the upper Hessenberg matrix H, and the rest is set to zero.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) REAL array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B. On exit, the
upper triangular matrix T = Q' B Z. The elements below the
diagonal are set to zero.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Q (input/output) REAL array, dimension (LDQ, N)
If COMPQ='N': Q is not referenced.
If COMPQ='I': on entry, Q need not be set, and on exit it
contains the orthogonal matrix Q, where Q' is the product of the
Givens transformations which are applied to A and B on the left.
If COMPQ='V': on entry, Q must contain an orthogonal matrix Q1,
and on exit this is overwritten by Q1*Q.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= N if COMPQ='V' or
'I'; LDQ >= 1 otherwise.
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 it
contains the orthogonal matrix Z, which is the product of the
Givens transformations which are applied to A and B on the right.
If COMPZ='V': on entry, Z must contain an orthogonal matrix Z1,
and on exit this is overwritten by Z1*Z.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= N if COMPZ='V' or
'I'; LDZ >= 1 otherwise.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
FURTHER DETAILS
This routine reduces A to Hessenberg and B to triangular form by an
unblocked reduction, as described in _Matrix_Computations_, by Golub and
Van Loan (Johns Hopkins Press.)
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