?ggevx

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 12/16/2022

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

Syntax

lapack_int LAPACKE_sggevx( int matrix_layout, char balanc, char jobvl, char jobvr, char sense, 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* ilo, lapack_int* ihi, float* lscale, float* rscale, float* abnrm, float* bbnrm, float* rconde, float* rcondv );

lapack_int LAPACKE_dggevx( int matrix_layout, char balanc, char jobvl, char jobvr, char sense, 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* ilo, lapack_int* ihi, double* lscale, double* rscale, double* abnrm, double* bbnrm, double* rconde, double* rcondv );

lapack_int LAPACKE_cggevx( int matrix_layout, char balanc, char jobvl, char jobvr, char sense, 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* ilo, lapack_int* ihi, float* lscale, float* rscale, float* abnrm, float* bbnrm, float* rconde, float* rcondv );

lapack_int LAPACKE_zggevx( int matrix_layout, char balanc, char jobvl, char jobvr, char sense, 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, lapack_int* ilo, lapack_int* ihi, double* lscale, double* rscale, double* abnrm, double* bbnrm, double* rconde, double* rcondv );

Include Files

mkl.h

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

matrix_layout

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

balanc

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

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

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

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

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

a, b

Arrays:

a (size at least max(1, lda*n)) is an array containing the n-by-n matrix A (first of the pair of matrices).

b (size at least max(1, ldb*n)) is an array containing the n-by-n matrix B (second of the pair of matrices).

lda

The leading dimension of the array a.

Must be at least max(1, n).

ldb

The leading dimension of the array b.

Must be at least max(1, n).

ldvl, ldvr

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).

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

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

See beta.

alpha

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

beta

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

For real flavors:

On exit, (alphar[j] + alphai[j]*i)/beta[j], j=0,..., n - 1, 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=0,..., n - 1, will be the generalized eigenvalues.

See also Application Notes below.

vl, vr

Arrays:

vl (size at least max(1, ldvl*n)).

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 will be scaled so the largest component have abs(Re) + abs(Im) = 1.

If jobvl = 'N', vl is not referenced.

For real flavors:

If the j-th eigenvalue is real, then k-th component of j-th left eigenvector u_j is stored in vl[(k - 1) + (j - 1)*ldvl] for column major layout and in vl[(k - 1)*ldvl + (j - 1)] for row major layout..

If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then for i = sqrt(-1), the k-th components of the j-th left eigenvector u_j can be computed as vl[(k - 1) + (j - 1)*ldvl] + i*vl[(k - 1) + j*ldvl] for column major layout and vl[(k - 1)*ldvl + (j - 1)] + i*vl[(k - 1)*ldvl + j] for row major layout. Similarly, the k-th components of the left eigenvector j+1 u_j+1 can be computed as vl[(k - 1) + (j - 1)*ldvl] - i*vl[(k - 1) + j*ldvl] for column major layout and vl[(k - 1)*ldvl + (j - 1)] - i*vl[(k - 1)*ldvl + j] for row major layout..

For complex flavors:

The k-th component of the j-th left eigenvector u_j is stored in vl[(k - 1) + (j - 1)*ldvl] for column major layout and in vl[(k - 1)*ldvl + (j - 1)] for row major layout.

vr (size at least max(1, ldvr*n)).

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 will be scaled so the largest component have abs(Re) + abs(Im) = 1.

If jobvr = 'N', vr is not referenced.

For real flavors:

If the j-th eigenvalue is real, then the k-th component of the j-th right eigenvector v_j is stored in vr[(k - 1) + (j - 1)*ldvr] for column major layout and in vr[(k - 1)*ldvr + (j - 1)] for row major layout..

If the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then The k-th components of the j-th right eigenvector v_j can be computed as vr[(k - 1) + (j - 1)*ldvr] + i*vr[(k - 1) + j*ldvr] for column major layout and vr[(k - 1)*ldvr + (j - 1)] + i*vr[(k - 1)*ldvr + j] for row major layout. Respectively, the k-th components of right eigenvector j+1 v_{j + 1} can be computed as vr[(k - 1) + (j - 1)*ldvr] - i*vr[(k - 1) + j*ldvr] for column major layout and vr[(k - 1)*ldvr + (j - 1)] - i*vr[(k - 1)*ldvr + j] for row major layout..

For complex flavors:

The k-th component of the j-th right eigenvector v_j is stored in vr[(k - 1) + (j - 1)*ldvr] for column major layout and in vr[(k - 1)*ldvr + (j - 1)] for row major layout.

ilo, ihi

ilo and ihi are integer values such that on exit A_{i j} = 0 and B_{i j} = 0 if i > j and j = 1,..., ilo-1 or i = ihi+1,..., n.

If balanc = 'N' or 'S', ilo = 1 and ihi = n.

lscale, rscale

Arrays, size at least max(1, n) each.

lscale contains details of the permutations and scaling factors applied to the left side of A and B.

If PL(j) is the index of the row interchanged with row j, and DL(j) is the scaling factor applied to row j, then

lscale[j - 1] = PL(j), for j = 1,..., ilo-1

= DL(j), for j = ilo,...,ihi

= PL(j) for j = ihi+1,..., n.

The order in which the interchanges are made is n to ihi+1, then 1 to ilo-1.

rscale contains details of the permutations and scaling factors applied to the right side of A and B.

If PR(j) is the index of the column interchanged with column j, and DR(j) is the scaling factor applied to column j, then

rscale[j - 1] = PR(j), for j = 1,..., ilo-1

= DR(j), for j = ilo,...,ihi

= PR(j) for j = ihi+1,..., n.

The order in which the interchanges are made is n to ihi+1, then 1 to ilo-1.

abnrm, bbnrm

The one-norms of the balanced matrices A and B, respectively.

rconde, rcondv

Arrays, size at least max(1, n) each.

If sense = 'E', or 'B', rconde contains the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of rconde are set to the same value. Thus rconde[j], rcondv[j], and the j-th columns of vl and vr all correspond to the same eigenpair (but not in general the j-th eigenpair, unless all eigenpairs are selected).

If sense = 'N', or 'V', rconde is not referenced.

If sense = 'V', or 'B', rcondv contains the estimated reciprocal condition numbers of the eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of rcondv are set to the same value.

If the eigenvalues cannot be reordered to compute , rcondv[j] is set to 0; this can only occur when the true value would be very small anyway.

If sense = 'N', or 'E', rcondv is not referenced.

Return Values

This function returns a value info.

If info=0, the execution is successful.

If info = -i, the i-th parameter had an illegal value.

If info = i, and

i≤n: the QZ iteration failed. No eigenvectors have been calculated, but alphar[j], alphai[j] (for real flavors), or alpha[j] (for complex flavors), and beta[j], j=info,..., n - 1 should be correct.

i > n: errors that usually indicate LAPACK problems:

i = n+1: other than QZ iteration failed in hgeqz;

i = n+2: error return from tgevc.

Application Notes

The quotients alphar[j]/beta[j] and alphai[j]/beta[j] may easily over- or underflow, and beta[j] may even be zero. Thus, you should avoid simply computing the ratio. However, alphar and alphai (for real flavors) or alpha (for complex flavors) will be always less than and usually comparable with norm(A) in magnitude, and beta always less than and usually comparable with norm(B).

Parent topic: Generalized Nonsymmetric Eigenvalue Problems: LAPACK Driver Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?ggevx