?tgsna

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?tgsna

Estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a pair of matrices in generalized real Schur canonical form.

Syntax

lapack_int LAPACKE_stgsna( int matrix_layout, char job, char howmny, const lapack_logical* select, lapack_int n, const float* a, lapack_int lda, const float* b, lapack_int ldb, const float* vl, lapack_int ldvl, const float* vr, lapack_int ldvr, float* s, float* dif, lapack_int mm, lapack_int* m );

lapack_int LAPACKE_dtgsna( int matrix_layout, char job, char howmny, const lapack_logical* select, lapack_int n, const double* a, lapack_int lda, const double* b, lapack_int ldb, const double* vl, lapack_int ldvl, const double* vr, lapack_int ldvr, double* s, double* dif, lapack_int mm, lapack_int* m );

lapack_int LAPACKE_ctgsna( int matrix_layout, char job, char howmny, const lapack_logical* select, lapack_int n, const lapack_complex_float* a, lapack_int lda, const lapack_complex_float* b, lapack_int ldb, const lapack_complex_float* vl, lapack_int ldvl, const lapack_complex_float* vr, lapack_int ldvr, float* s, float* dif, lapack_int mm, lapack_int* m );

lapack_int LAPACKE_ztgsna( int matrix_layout, char job, char howmny, const lapack_logical* select, lapack_int n, const lapack_complex_double* a, lapack_int lda, const lapack_complex_double* b, lapack_int ldb, const lapack_complex_double* vl, lapack_int ldvl, const lapack_complex_double* vr, lapack_int ldvr, double* s, double* dif, lapack_int mm, lapack_int* m );

Include Files

mkl.h

Description

The real flavors stgsna/dtgsna of this routine estimate reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (A, B) in generalized real Schur canonical form (or of any matrix pair (Q*A*Z^T, Q*B*Z^T) with orthogonal matrices Q and Z.

(A, B) must be in generalized real Schur form (as returned by gges/gges), that is, A is block upper triangular with 1-by-1 and 2-by-2 diagonal blocks. B is upper triangular.

The complex flavors ctgsna/ztgsna estimate reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (A, B). (A, B) must be in generalized Schur canonical form, that is, A and B are both upper triangular.

Input Parameters

matrix_layout

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

job

Specifies whether condition numbers are required for eigenvalues or eigenvectors. Must be 'E' or 'V' or 'B'.

If job = 'E', for eigenvalues only (compute s ).

If job = 'V', for eigenvectors only (compute dif ).

If job = 'B', for both eigenvalues and eigenvectors (compute both s and dif).

howmny

Must be 'A' or 'S'.

If howmny = 'A', compute condition numbers for all eigenpairs.

If howmny = 'S', compute condition numbers for selected eigenpairs specified by the logical array select.

select

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

If howmny = 'S', select specifies the eigenpairs for which condition numbers are required.

If howmny = 'A', select is not referenced.

For real flavors:

To select condition numbers for the eigenpair corresponding to a real eigenvalue ω_j, select[j - 1] must be set to 1; to select condition numbers corresponding to a complex conjugate pair of eigenvalues ω_j and ω_{j + 1}, either select[j - 1] or select[j] must be set to 1.

For complex flavors:

To select condition numbers for the corresponding j-th eigenvalue and/or eigenvector, select[j - 1] must be set to 1.

n

The order of the square matrix pair (A, B)

(n≥ 0).

a, b, vl, vr

Arrays:

a (size max(1, lda*n)) contains the upper quasi-triangular (for real flavors) or upper triangular (for complex flavors) matrix A in the pair (A, B).

b (size max(1, ldb*n)) contains the upper triangular matrix B in the pair (A, B).

If job = 'E' or 'B', vl(size max(1, ldvl*m) for column major layout and max(1, ldvl*n) for row major layout) must contain left eigenvectors of (A, B), corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vl, as returned by ?tgevc.

If job = 'V', vl is not referenced.

If job = 'E' or 'B', vr(size max(1, ldvr*m) for column major layout and max(1, ldvr*n) for row major layout) must contain right eigenvectors of (A, B), corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vr, as returned by ?tgevc.

If job = 'V', vr is not referenced.

lda

The leading dimension of a; at least max(1, n).

ldb

The leading dimension of b; at least max(1, n).

ldvl

The leading dimension of vl; ldvl≥ 1.

If job = 'E' or 'B', then ldvl≥ max(1, n) for column major layout and ldvl≥ max(1, m) for row major layout .

ldvr

The leading dimension of vr; ldvr≥ 1.

If job = 'E' or 'B', then ldvr≥ max(1, n) for column major layout and ldvr≥ max(1, m) for row major layout.

mm

The number of elements in the arrays s and dif (mm≥m).

Output Parameters

s

Array, size mm.

If job = 'E' or 'B', contains the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array.

If job = 'V', s is not referenced.

For real flavors:

For a complex conjugate pair of eigenvalues two consecutive elements of s are set to the same value. Thus, s[j - 1], dif[j - 1], 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).

dif

Array, size mm.

If job = 'V' or 'B', contains the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array.

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

If job = 'E', dif is not referenced.

For real flavors:

For a complex eigenvector, two consecutive elements of dif are set to the same value.

For complex flavors:

For each eigenvalue/vector specified by select, dif stores a Frobenius norm-based estimate of Difl.

m

The number of elements in the arrays s and dif used to store the specified condition numbers; for each selected eigenvalue one element is used.

If howmny = 'A', m is set to n.

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.

Parent topic: Generalized Nonsymmetric Eigenvalue Problems: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

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