Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

ID 766686
Date 6/24/2024
Public

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?ggev3

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

Syntax

call sggev3 (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info )

call dggev3 (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info )

call cggev3 (jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info )

call zggev3 (jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info )

Include Files

  • mkl.fi

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 vj corresponding to the eigenvalue λj of (A, B) satisfies

A * vj = λj * B * vj.

The left eigenvector uj corresponding to the eigenvalue λj of (A, B) satisfies

ujH * A = λj * ujH * B

where ujH is the conjugate-transpose of uj.

For complex flavors:

The right generalized eigenvector vj corresponding to the generalized eigenvalue λj of (A, B) satisfies

A * vj = λj * B * vj.

The left generalized eigenvector uj corresponding to the generalized eigenvalues λj of (A, B) satisfies

ujH * A = λj * ujH * B

where ujH is the conjugate-transpose of uj.

Input Parameters

jobvl

CHARACTER*1. = 'N': do not compute the left generalized eigenvectors;

= 'V': compute the left generalized eigenvectors.

jobvr

CHARACTER*1. = 'N': do not compute the right generalized eigenvectors;

= 'V': compute the right generalized eigenvectors.

n

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

n 0.

a

REAL for sggev3

DOUBLE PRECISION for dggev3

COMPLEX for cggev3

DOUBLE COMPLEX for zggev3

Array, size (lda, n).

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

lda

INTEGER. The leading dimension of a.

lda max(1,n).

b

REAL for sggev3

DOUBLE PRECISION for dggev3

COMPLEX for cggev3

DOUBLE COMPLEX for zggev3

Array, size (ldb, n).

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

ldb

INTEGER. The leading dimension of b.

ldb max(1,n).

ldvl

INTEGER. The leading dimension of the matrix VL.

ldvl 1, and if jobvl = 'V', ldvln.

ldvr

INTEGER. The leading dimension of the matrix VR.

ldvr 1, and if jobvr = 'V', ldvrn.

work

REAL for sggev3

DOUBLE PRECISION for dggev3

COMPLEX for cggev3

DOUBLE COMPLEX for zggev3

Array, size (MAX(1,lwork))

On exit, if info = 0, work(1) returns the optimal lwork.

lwork

INTEGER. The dimension of the array work.

If lwork = -1, then a workspace query is assumed; the routine only calculates the optimal (A, B) 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 cggev3

DOUBLE PRECISION for zggev3

Array, size (8*n).

Output Parameters

a

On exit, a is overwritten.

b

On exit, b is overwritten.

alphar

REAL for sggev3

DOUBLE PRECISION for dggev3

Array, size (n).

alphai

REAL for sggev3

DOUBLE PRECISION for dggev3

Array, size (n).

alpha

COMPLEX for cggev3

DOUBLE COMPLEX for zggev3

Array, size (n).

beta

REAL for sggev3

DOUBLE PRECISION for dggev3

COMPLEX for cggev3

DOUBLE COMPLEX for zggev3

Array, size (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.

Note: the quotients alphar(j)/beta(j) and alphai(j)/beta(j) 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=1,...,n, are the generalized eigenvalues.

Note: the quotients alpha(j)/beta(j) 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

REAL for sggev3

DOUBLE PRECISION for dggev3

COMPLEX for cggev3

DOUBLE COMPLEX for zggev3

Array, size (ldvl, n).

For real flavors:

If jobvl = 'V', the left eigenvectors uj 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 uj =vl(:,j), the j-th column of vl. If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then uj = vl(:,j)+i*vl(:,j+1) and uj + 1 = vl(:,j)-i*vl(:,j+1).

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 uj 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

REAL for sggev3

DOUBLE PRECISION for dggev3

COMPLEX for cggev3

DOUBLE COMPLEX for zggev3

Array, size (ldvr, n).

For real flavors:

If jobvr = 'V', the right eigenvectors vj 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 vj =vr(:,j), the j-th column of vr. If the j-th and (j + 1)-st eigenvalues form a complex conjugate pair, then vj = vr(:,j) + i*vr(:,j + 1) and vj + 1 = vr(:,j)-i*vr(:,j+1).

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 vj 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'.

info

INTEGER. = 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 + 1,...,n.

  • for complex flavors:

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

> n:

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

  • =n + 2: error return from ?tgevc.