?ggev

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 12/16/2022

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?ggev

Computes the generalized eigenvalues, and the left and/or right generalized eigenvectors for a pair of nonsymmetric matrices.

Syntax

call sggev(jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info)

call dggev(jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info)

call cggev(jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info)

call zggev(jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info)

call ggev(a, b, alphar, alphai, beta [,vl] [,vr] [,info])

call ggev(a, b, alpha, beta [, vl] [,vr] [,info])

Include Files

mkl.fi, lapack.f90

Description

The ?ggev routine computes the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors for a pair of n-by-n real/complex nonsymmetric matrices (A,B).

A generalized eigenvalue for a pair of matrices (A,B) is a scalar λ or a ratio alpha / beta = λ, such that A - λ*B is singular. It is usually represented as the pair (alpha, beta), as there is a reasonable interpretation for beta =0 and even for both being zero.

The right generalized eigenvector v(j) corresponding to the generalized eigenvalue λ(j) of (A,B) satisfies

A*v(j) = λ(j)*B*v(j).

The left generalized eigenvector u(j) corresponding to the generalized eigenvalue λ(j) of (A,B) satisfies

u(j)^H*A = λ(j)*u(j)^H*B

where u(j)^H denotes the conjugate transpose of u(j).

The ?ggev routine replaces the deprecated ?gegv routine.

Input Parameters

jobvl

CHARACTER*1. Must be 'N' or 'V'.

If jobvl = 'N', the left generalized eigenvectors are not computed;

If jobvl = 'V', the left generalized eigenvectors are computed.

jobvr

CHARACTER*1. Must be 'N' or 'V'.

If jobvr = 'N', the right generalized eigenvectors are not computed;

If jobvr = 'V', the right generalized eigenvectors are computed.

n

INTEGER. The order of the matrices A, B, vl, and vr (n≥ 0).

a, b, work

REAL for sggev

DOUBLE PRECISION for dggev

COMPLEX for cggev

DOUBLE COMPLEX for zggev.

Arrays:

a(lda,*) is an array containing the n-by-n matrix A (first of the pair of matrices).

The second dimension of a must be at least max(1, n).

b(ldb,*) is an array containing the n-by-n matrix B (second of the pair of matrices).

The second dimension of b must be at least max(1, n).

work is a workspace array, its dimension max(1, lwork).

lda

INTEGER. The leading dimension of the array a. Must be at least max(1, n).

ldb

INTEGER. The leading dimension of the array b. Must be at least max(1, n).

ldvl, ldvr

INTEGER. The leading dimensions of the output matrices vl and vr, respectively.

Constraints:

ldvl≥ 1. If jobvl = 'V', ldvl≥ max(1, n).

ldvr≥ 1. If jobvr = 'V', ldvr≥ max(1, n).

lwork

INTEGER.

The dimension of the array work.

lwork≥ max(1, 8n+16) for real flavors;

lwork≥ max(1, 2n) for complex flavors.

For good performance, lwork must generally be larger.

If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued by xerbla.

rwork

REAL for cggev

DOUBLE PRECISION for zggev

Workspace array, size at least max(1, 8n).

This array is used in complex flavors only.

Output Parameters

a, b

On exit, these arrays have been overwritten.

alphar, alphai

REAL for sggev;

DOUBLE PRECISION for dggev.

Arrays, size at least max(1, n) each. Contain values that form generalized eigenvalues in real flavors.

See beta.

alpha

COMPLEX for cggev;

DOUBLE COMPLEX for zggev.

Array, size at least max(1, n). Contain values that form generalized eigenvalues in complex flavors. See beta.

beta

REAL for sggev

DOUBLE PRECISION for dggev

COMPLEX for cggev

DOUBLE COMPLEX for zggev.

Array, size at least max(1, n).

For real flavors:

On exit, (alphar(j) + alphai(j)*i)/beta(j), j=1,..., n, are the generalized eigenvalues.

If alphai(j) is zero, then the j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with alphai(j+1) negative.

For complex flavors:

On exit, alpha(j)/beta(j), j=1,..., n, are the generalized eigenvalues.

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?ggev