?ggevx

Developer Reference for Intel® oneAPI Math Kernel Library for Fortran

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Date 6/24/2024

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?ggevx

Computes the generalized eigenvalues, and, optionally, the left and/or right generalized eigenvectors.

Syntax

call sggevx(balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, iwork, bwork, info)

call dggevx(balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, iwork, bwork, info)

call cggevx(balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, rwork, iwork, bwork, info)

call zggevx(balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, rwork, iwork, bwork, info)

call ggevx(a, b, alphar, alphai, beta [,vl] [,vr] [,balanc] [,ilo] [,ihi] [, lscale] [,rscale] [,abnrm] [,bbnrm] [,rconde] [,rcondv] [,info])

call ggevx(a, b, alpha, beta [, vl] [,vr] [,balanc] [,ilo] [,ihi] [,lscale] [, rscale] [,abnrm] [,bbnrm] [,rconde] [,rcondv] [,info])

Include Files

mkl.fi, lapack.f90

Description

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

Optionally also, it computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors (ilo, ihi, lscale, rscale, abnrm, and bbnrm), reciprocal condition numbers for the eigenvalues (rconde), and reciprocal condition numbers for the right eigenvectors (rcondv).

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).

Input Parameters

balanc

CHARACTER*1. Must be 'N', 'P', 'S', or 'B'. Specifies the balance option to be performed.

If balanc = 'N', do not diagonally scale or permute;

If balanc = 'P', permute only;

If balanc = 'S', scale only;

If balanc = 'B', both permute and scale.

Computed reciprocal condition numbers will be for the matrices after balancing and/or permuting. Permuting does not change condition numbers (in exact arithmetic), but balancing does.

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.

sense

CHARACTER*1. Must be 'N', 'E', 'V', or 'B'. Determines which reciprocal condition number are computed.

If sense = 'N', none are computed;

If sense = 'E', computed for eigenvalues only;

If sense = 'V', computed for eigenvectors only;

If sense = 'B', computed for eigenvalues and eigenvectors.

n

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

a, b, work

REAL for sggevx

DOUBLE PRECISION for dggevx

COMPLEX for cggevx

DOUBLE COMPLEX for zggevx.

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, 2*n);

For real flavors:

If balanc = 'S', or 'B', or jobvl = 'V', or jobvr = 'V', then lwork≥ max(1, 6*n);

if sense = 'E', or 'B', then lwork≥ max(1, 10*n);

if sense = 'V', or 'B', lwork≥ (2n²+ 8*n+16).

For complex flavors:

if sense = 'E', lwork≥ max(1, 4*n);

if sense = 'V', or 'B', lwork≥max(1, 2*n²+ 2*n).

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 cggevx

DOUBLE PRECISION for zggevx

Workspace array, size at least max(1, 6*n) if balanc = 'S', or 'B', and at least max(1, 2*n) otherwise.

This array is used in complex flavors only.

iwork

INTEGER.

Workspace array, size at least (n+6) for real flavors and at least (n+2) for complex flavors.

Not referenced if sense = 'E'.

bwork

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

Not referenced if sense = 'N'.

Output Parameters

a, b

On exit, these arrays have been overwritten.

If jobvl = 'V' or jobvr = 'V' or both, then a contains the first part of the real Schur form of the "balanced" versions of the input A and B, and b contains its second part.

alphar, alphai

REAL for sggevx;

DOUBLE PRECISION for dggevx.

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

See beta.

alpha

COMPLEX for cggevx;

DOUBLE COMPLEX for zggevx.

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

beta

REAL for sggevx

DOUBLE PRECISION for dggevx

COMPLEX for cggevx

DOUBLE COMPLEX for zggevx.

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

For real flavors:

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

LAPACK 95 Interface Notes

Routines in Fortran 95 interface have fewer arguments in the calling sequence than their FORTRAN 77 counterparts. For general conventions applied to skip redundant or restorable arguments, see LAPACK 95 Interface Conventions.

Specific details for the routine ggevx interface are the following:

a

Holds the matrix A of size (n, n).

b

Holds the matrix B of size (n, n).

alphar

Holds the vector of length n. Used in real flavors only.

alphai

Holds the vector of length n. Used in real flavors only.

alpha

Holds the vector of length n. Used in complex flavors only.

beta