?gghd3

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 11/07/2023

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?gghd3

Reduces a pair of matrices to generalized upper Hessenberg form.

Syntax

lapack_int LAPACKE_sgghd3 (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_dgghd3 (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_cgghd3 (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_zgghd3 (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

?gghd3 reduces a pair of real or 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/unitary matrix Q to the left side of the equation.

This subroutine simultaneously reduces A to a Hessenberg matrix H:

Q^T*A*Z = H for real flavors

Q^T*A*Z = H for complex flavors

and transforms B to another upper triangular matrix T:

Q^T*B*Z = T for real flavors

Q^T*B*Z = T for complex flavors

in order to reduce the problem to its standard form

H*y = λ*T*y

where y = Z^T*x for real flavors

y = Z^T*x for complex flavors.

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

for real flavors:

Q₁ * A * Z₁^T = (Q₁*Q) * H * (Z₁*Z)^T

Q₁ * B * Z₁^T = (Q₁*Q) * T * (Z₁*Z)^T

for complex flavors:

Q₁ * A * Z₁^H = (Q₁*Q) * H * (Z₁*Z)^T

Q₁ * B * Z₁^T = (Q₁*Q) * T * (Z₁*Z)^T

If Q₁ is the orthogonal/unitary matrix from the QR factorization of B in the original equation A*x = λ*B*x, then ?gghd3 reduces the original problem to generalized Hessenberg form.

This is a blocked variant of ?gghrd, using matrix-matrix multiplications for parts of the computation to enhance performance.

Input Parameters

matrix_layout

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

compq

= 'N': do not compute q;

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

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

compz

= 'N': do not compute z;

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

= '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. ilo and ihi are normally set by a previous call to ?ggbal; 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.

a

Array, size (lda*n).

On entry, the n-by-n general matrix to be reduced.

lda

The leading dimension of the array a.

lda≥ max(1,n).

b

Array, (ldb*n).

On entry, then-by-n upper triangular matrix B.

ldb

The leading dimension of the array b.

ldb≥ max(1,n).

q

Array, size (ldq*n).

On entry, if compq = 'V', the orthogonal/unitary matrix Q₁, typically from the QR factorization of b.

ldq

The leading dimension of the array q.

ldq≥n if compq='V' or 'I'; ldq≥ 1 otherwise.

z

Array, size (ldz*n).

On entry, if compz = 'V', the orthogonal/unitary matrix Z₁.

Not referenced if compz='N'.

ldz

The leading dimension of the array z. ldz≥n if compz='V' or 'I'; ldz≥ 1 otherwise.

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, the upper triangular matrix `T` = `Q`^T`BZ` for real flavors or `T` = `Q`^H`BZ` for complex flavors. The elements below the diagonal are set to zero.
q	On exit, if compq='I', the orthogonal/unitary matrix `Q`, and if compq = 'V', the product `Q`₁*`Q`. Not referenced if compq='N'.
z	On exit, if compz='I', the orthogonal/unitary matrix `Z`, and if compz = 'V', the product `Z`₁*`Z`. Not referenced if compz='N'.

Return Values

This function returns a value info.

= 0: successful exit.

< 0: if info = -i, the i-th argument had an illegal value.

Application Notes

This routine reduces A to Hessenberg form and maintains B in using a blocked variant of Moler and Stewart's original algorithm, as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti (BIT 2008).

Parent topic: Generalized Nonsymmetric Eigenvalue Problems: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?gghd3

Syntax

Include Files

Description

Input Parameters

Output Parameters

Return Values

Application Notes