?ggev3

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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?ggev3

Computes the generalized eigenvalues and the left and right generalized eigenvectors for a pair of matrices.

Syntax

lapack_int LAPACKE_sggev3 (int matrix_layout, char jobvl, char jobvr, lapack_int n, float * a, lapack_int lda, float * b, lapack_int ldb, float * alphar, float * alphai, float * beta, float * vl, lapack_int ldvl, float * vr, lapack_int ldvr);

lapack_int LAPACKE_dggev3 (int matrix_layout, char jobvl, char jobvr, lapack_int n, double * a, lapack_int lda, double * b, lapack_int ldb, double * alphar, double * alphai, double * beta, double * vl, lapack_int ldvl, double * vr, lapack_int ldvr);

lapack_int LAPACKE_cggev3 (int matrix_layout, char jobvl, char jobvr, lapack_int n, lapack_complex_float * a, lapack_int lda, lapack_complex_float * b, lapack_int ldb, lapack_complex_float * alpha, lapack_complex_float * beta, lapack_complex_float * vl, lapack_int ldvl, lapack_complex_float * vr, lapack_int ldvr);

lapack_int LAPACKE_zggev3 (int matrix_layout, char jobvl, char jobvr, lapack_int n, lapack_complex_double * a, lapack_int lda, lapack_complex_double * b, lapack_int ldb, lapack_complex_double * alpha, lapack_complex_double * beta, lapack_complex_double * vl, lapack_int ldvl, lapack_complex_double * vr, lapack_int ldvr);

Include Files

mkl.h

Description

For a pair of n-by-n real or complex nonsymmetric matrices (A, B), ?ggev3 computes the generalized eigenvalues, and optionally, the left and right generalized eigenvectors.

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.

For real flavors:

The right eigenvector v_j corresponding to the eigenvalue λ_j of (A, B) satisfies

A * v_j = λ_j * B * v_j.

The left eigenvector u_j corresponding to the eigenvalue λ_j of (A, B) satisfies

u_j^H * A = λ_j * u_j^H * B

where u_j^H is the conjugate-transpose of u_j.

For complex flavors:

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 eigenvalues λ_j of (A, B) satisfies

u_j^H * A = λ_j * u_j^H * B

where u_j^H is the conjugate-transpose of u_j.

Input Parameters

matrix_layout

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

jobvl

= 'N': do not compute the left generalized eigenvectors;

= 'V': compute the left generalized eigenvectors.

jobvr

= 'N': do not compute the right generalized eigenvectors;

= 'V': compute the right generalized eigenvectors.

n

The order of the matrices A, B, VL, and VR.

n≥ 0.

a

Array, size (lda*n).

On entry, the matrix A in the pair (A, B).

lda

The leading dimension of a.

lda≥ max(1,n).

b

Array, size (ldb*n).

On entry, the matrix B in the pair (A, B).

ldb

The leading dimension of b.

ldb≥ max(1,n).

ldvl

The leading dimension of the matrix VL.

ldvl≥ 1, and if jobvl = 'V', ldvl≥n.

ldvr

The leading dimension of the matrix VR.

ldvr≥ 1, and if jobvr = 'V', ldvr≥n.

Output Parameters

a	On exit, a is overwritten.
b	On exit, b is overwritten.
alphar	Array, size (n).
alphai	Array, size (n).
alpha	Array, size (n).
beta	Array, size (n). For real flavors: On exit, (alphar[`j`] + alphai[`j`]*i)/beta[`j`], `j`=0,...,n - 1, are the generalized eigenvalues. If alphai`[j - 1]` 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]` negative. Note: the quotients alphar`[j - 1]`/beta`[j - 1]` and alphai`[j - 1]`/beta`[j - 1]` can easily over- or underflow, and beta(`j`) might even be zero. Thus, you should avoid computing the ratio alpha/beta by simply dividing alpha by beta. However, alphar and alphai are always less than and usually comparable with norm(`A`) in magnitude, and beta is always less than and usually comparable with norm(`B`). For complex flavors: On exit, alpha[`j`]/beta[`j`], `j`=0,...,n - 1, are the generalized eigenvalues. Note: the quotients alpha`[j - 1]`/beta`[j - 1]` may easily over- or underflow, and beta(`j`) can even be zero. Thus, you should avoid computing the ratio alpha/beta by simply dividing alpha by beta. However, alpha is always less than and usually comparable with norm(`A`) in magnitude, and betais always less than and usually comparable with norm(`B`).
vl	Array, size (ldvl*n). For real flavors: If jobvl = 'V', the left eigenvectors `u`_j are stored one after another in the columns of vl, in the same order as their eigenvalues. If the `j`-th eigenvalue is real, then `u`_j = the `j`-th column of vl. If the `j`-th and (`j`+1)-st eigenvalues form a complex conjugate pair, then the real part of `u`_j = the `j`-th column of vl and the imaginary part of `v`_j = the (`j` + 1)-st column of vl. Each eigenvector is scaled so the largest component has abs(real part)+abs(imag. part)=1. Not referenced if jobvl = 'N'. For complex flavors: If jobvl = 'V', the left generalized eigenvectors `u`_j are stored one after another in the columns of vl, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has abs(real part) + abs(imag. part) = 1. Not referenced if jobvl = 'N'.
vr	Array, size (ldvr*n). For real flavors: If jobvr = 'V', the right eigenvectors `v`_j are stored one after another in the columns of vr, in the same order as their eigenvalues. If the `j`-th eigenvalue is real, then `v`_j = the `j`-th column of vr. If the `j`-th and (`j` + 1)-st eigenvalues form a complex conjugate pair, then the real part of `v`_j = the `j`-th column of vr and the imaginary part of `v`_j = the (`j` + 1)-st column of vr. Each eigenvector is scaled so the largest component has abs(real part)+abs(imag. part)=1. Not referenced if jobvr = 'N'. For complex flavors: If jobvr = 'V', the right generalized eigenvectors `v`_j are stored one after another in the columns of vr, in the same order as their eigenvalues. Each eigenvector is scaled so the largest component has abs(real part) + abs(imag. part) = 1. Not referenced if jobvr = 'N'.

Return Values

This function returns a value info.

= 0: successful exit

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

=1,...,n:

for real flavors:

The QZ iteration failed. No eigenvectors have been calculated, but alphar[j], alphar[j] and beta[j] should be correct for j=info,...,n - 1.

for complex flavors:

The QZ iteration failed. No eigenvectors have been calculated, but alpha[j] and beta[j] should be correct for j=info,...,n - 1.

> n:

=n + 1: other than QZ iteration failed in ?hgeqz,

=n + 2: error return from ?tgevc.

Parent topic: Generalized Nonsymmetric Eigenvalue Problems: LAPACK Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?ggev3