?gghrd

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?gghrd

Reduces a pair of matrices to generalized upper Hessenberg form using orthogonal/unitary transformations.

Syntax

lapack_int LAPACKE_sgghrd (int matrix_layout, char compq, char compz, lapack_int n, lapack_int ilo, lapack_int ihi, float* a, lapack_int lda, float* b, lapack_int ldb, float* q, lapack_int ldq, float* z, lapack_int ldz);

lapack_int LAPACKE_dgghrd (int matrix_layout, char compq, char compz, lapack_int n, lapack_int ilo, lapack_int ihi, double* a, lapack_int lda, double* b, lapack_int ldb, double* q, lapack_int ldq, double* z, lapack_int ldz);

lapack_int LAPACKE_cgghrd (int matrix_layout, char compq, char compz, lapack_int n, lapack_int ilo, lapack_int ihi, lapack_complex_float* a, lapack_int lda, lapack_complex_float* b, lapack_int ldb, lapack_complex_float* q, lapack_int ldq, lapack_complex_float* z, lapack_int ldz);

lapack_int LAPACKE_zgghrd (int matrix_layout, char compq, char compz, lapack_int n, lapack_int ilo, lapack_int ihi, lapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int ldb, lapack_complex_double* q, lapack_int ldq, lapack_complex_double* z, lapack_int ldz);

Include Files

mkl.h

Description

The routine reduces a pair of real/complex matrices (A,B) to generalized upper Hessenberg form using orthogonal/unitary transformations, where A is a general matrix and B is upper triangular. The form of the generalized eigenvalue problem is A*x = λ*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 routine simultaneously reduces A to a Hessenberg matrix H:

Q^H*A*Z = H

and transforms B to another upper triangular matrix T:

Q^H*B*Z = T

in order to reduce the problem to its standard form H*y = λ*T*y, where y = Z^H*x.

The orthogonal/unitary 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^H = (Q₁*Q)*H*(Z₁*Z)^H

Q₁*B*Z₁^H = (Q₁*Q)*T*(Z₁*Z)^H

If Q₁ is the orthogonal/unitary matrix from the QR factorization of B in the original equation A*x = λ*B*x, then the routine ?gghrd reduces the original problem to generalized Hessenberg form.

Input Parameters

matrix_layout

Specifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).

compq

Must be 'N', 'I', or 'V'.

If compq = 'N', matrix Q is not computed.

If compq = 'I', Q is initialized to the unit matrix, and the orthogonal/unitary matrix Q is returned;

If compq = 'V', Q must contain an orthogonal/unitary matrix Q₁ on entry, and the product Q₁*Q is returned.

compz

Must be 'N', 'I', or 'V'.

If compz = 'N', matrix Z is not computed.

If compz = 'I', Z is initialized to the unit matrix, and the orthogonal/unitary matrix Z is returned;

If compz = 'V', Z must contain an orthogonal/unitary matrix Z₁ on entry, and the product Z₁*Z is returned.

n

The order of the matrices A and B (n≥ 0).

ilo, ihi

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. Values of ilo and ihi are normally set by a previous call to ggbal; otherwise they should be set to 1 and n respectively.

Constraint:

If n > 0, then 1 ≤ilo≤ihi≤n;

if n = 0, then ilo = 1 and ihi = 0.

a, b, q, z

Arrays:

a (size max(1, lda*n)) contains the n-by-n general matrix A.

b (size max(1, ldb*n)) contains the n-by-n upper triangular matrix B.

q (size max(1, ldq*n))

If compq = 'N', then q is not referenced.

If compq = 'V', then q must contain the orthogonal/unitary matrix Q₁, typically from the QR factorization of B.

z (size max(1, ldz*n))

If compz = 'N', then z is not referenced.

If compz = 'V', then z must contain the orthogonal/unitary matrix Z₁.

lda

The leading dimension of a; at least max(1, n).

ldb

The leading dimension of b; at least max(1, n).

ldq

The leading dimension of q;

If compq = 'N', then ldq≥ 1.

If compq = 'I'or 'V', then ldq≥ max(1, n).

ldz

The leading dimension of z;

If compz = 'N', then ldz≥ 1.

If compz = 'I'or 'V', then ldz≥ max(1, n).

Output Parameters

a

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.

b

On exit, overwritten by the upper triangular matrix T = Q^H*B*Z. The elements below the diagonal are set to zero.

q

If compq = 'I', then q contains the orthogonal/unitary matrix Q, ;

If compq = 'V', then q is overwritten by the product Q₁*Q.

z

If compz = 'I', then z contains the orthogonal/unitary matrix Z;

If compz = 'V', then z is overwritten by the product Z₁*Z.

Return Values

This function returns a value info.

If info=0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

Parent topic: Generalized Nonsymmetric Eigenvalue Problems: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?gghrd