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LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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LAPACK 3.12.0
LAPACK: Linear Algebra PACKage
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| subroutine sgghrd | ( | character | compq, |
| character | compz, | ||
| integer | n, | ||
| integer | ilo, | ||
| integer | ihi, | ||
| real, dimension( lda, * ) | a, | ||
| integer | lda, | ||
| real, dimension( ldb, * ) | b, | ||
| integer | ldb, | ||
| real, dimension( ldq, * ) | q, | ||
| integer | ldq, | ||
| real, dimension( ldz, * ) | z, | ||
| integer | ldz, | ||
| integer | info | ||
| ) |
SGGHRD
Download SGGHRD + dependencies [TGZ] [ZIP] [TXT]
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. The form of the
generalized eigenvalue problem is
A*x = lambda*B*x,
and B is typically made upper triangular by computing its QR
factorization and moving the orthogonal matrix Q to the left side
of the equation.
This subroutine simultaneously reduces A to a Hessenberg matrix H:
Q**T*A*Z = H
and transforms B to another upper triangular matrix T:
Q**T*B*Z = T
in order to reduce the problem to its standard form
H*y = lambda*T*y
where y = Z**T*x.
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**T = (Q1*Q) * H * (Z1*Z)**T
Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
If Q1 is the orthogonal matrix from the QR factorization of B in the
original equation A*x = lambda*B*x, then SGGHRD reduces the original
problem to generalized Hessenberg form. | [in] | COMPQ | COMPQ is 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. |
| [in] | COMPZ | COMPZ is 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. |
| [in] | N | N is INTEGER
The order of the matrices A and B. N >= 0. |
| [in] | ILO | ILO is INTEGER |
| [in] | IHI | IHI is INTEGER
ILO and IHI mark the rows and columns of A which are to be
reduced. 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. |
| [in,out] | A | A is 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. |
| [in] | LDA | LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N). |
| [in,out] | B | B is REAL array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B.
On exit, the upper triangular matrix T = Q**T B Z. The
elements below the diagonal are set to zero. |
| [in] | LDB | LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N). |
| [in,out] | Q | Q is REAL array, dimension (LDQ, N)
On entry, if COMPQ = 'V', the orthogonal matrix Q1,
typically from the QR factorization of B.
On exit, if COMPQ='I', the orthogonal matrix Q, and if
COMPQ = 'V', the product Q1*Q.
Not referenced if COMPQ='N'. |
| [in] | LDQ | LDQ is INTEGER
The leading dimension of the array Q.
LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise. |
| [in,out] | Z | Z is REAL array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the orthogonal matrix Z1.
On exit, if COMPZ='I', the orthogonal matrix Z, and if
COMPZ = 'V', the product Z1*Z.
Not referenced if COMPZ='N'. |
| [in] | LDZ | LDZ is INTEGER
The leading dimension of the array Z.
LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise. |
| [out] | INFO | INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value. |
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.)
Definition at line 205 of file sgghrd.f.
1.9.7