?ggev3

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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ID 766686

Date 3/31/2023

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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 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

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', ldvl≥n.

ldvr

INTEGER. The leading dimension of the matrix VR.

ldvr≥ 1, and if jobvr = 'V', ldvr≥n.

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 `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 =vl(:,`j`), the `j`-th column of vl. If the `j`-th and (`j`+1)-st eigenvalues form a complex conjugate pair, then `u`_j = vl(:,`j`)+ivl(:,`j`+1) and `u`_{j + 1} = vl(:,`j`)-ivl(:,`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 `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	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 `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 =vr(:,`j`), the `j`-th column of vr. If the `j`-th and (`j` + 1)-st eigenvalues form a complex conjugate pair, then `v`_j = vr(:,`j`) + ivr(:,`j` + 1) and `v`_{j + 1} = vr(:,`j`)-ivr(:,`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 `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'.
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.

Parent topic: Generalized Nonsymmetric Eigenvalue Problems: LAPACK Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

?ggev3