?trexc

Developer Reference for Intel® oneAPI Math Kernel Library for C

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ID 766684

Date 3/31/2023

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?trexc

Reorders the Schur factorization of a general matrix.

Syntax

lapack_int LAPACKE_strexc( int matrix_layout, char compq, lapack_int n, float* t, lapack_int ldt, float* q, lapack_int ldq, lapack_int* ifst, lapack_int* ilst );

lapack_int LAPACKE_dtrexc( int matrix_layout, char compq, lapack_int n, double* t, lapack_int ldt, double* q, lapack_int ldq, lapack_int* ifst, lapack_int* ilst );

lapack_int LAPACKE_ctrexc( int matrix_layout, char compq, lapack_int n, lapack_complex_float* t, lapack_int ldt, lapack_complex_float* q, lapack_int ldq, lapack_int ifst, lapack_int ilst );

lapack_int LAPACKE_ztrexc( int matrix_layout, char compq, lapack_int n, lapack_complex_double* t, lapack_int ldt, lapack_complex_double* q, lapack_int ldq, lapack_int ifst, lapack_int ilst );

Include Files

mkl.h

Description

The routine reorders the Schur factorization of a general matrix A = Q*T*Q^H, so that the diagonal element or block of T with row index ifst is moved to row ilst.

The reordered Schur form S is computed by an unitary (or, for real flavors, orthogonal) similarity transformation: S = Z^H*T*Z. Optionally the updated matrix P of Schur vectors is computed as P = Q*Z, giving A = P*S*P^H.

Input Parameters

matrix_layout

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

compq

Must be 'V' or 'N'.

If compq = 'V', then the Schur vectors (Q) are updated.

If compq = 'N', then no Schur vectors are updated.

n

The order of the matrix T (n≥ 0).

t, q

Arrays:

t (size max(1, ldt*n)) contains the n-by-n matrix T.

q (size max(1, ldq*n))

If compq = 'V', then q must contain Q (Schur vectors).

If compq = 'N', then q is not referenced.

ldt

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

ldq

The leading dimension of q;

If compq = 'N', then ldq≥ 1.

If compq = 'V', then ldq≥ max(1, n).

ifst, ilst

1 ≤ifst≤n; 1 ≤ilst≤n.

Must specify the reordering of the diagonal elements (or blocks, which is possible for real flavors) of the matrix T. The element (or block) with row index ifst is moved to row ilst by a sequence of exchanges between adjacent elements (or blocks).

Output Parameters

t

Overwritten by the updated matrix S.

q

If compq = 'V', q contains the updated matrix of Schur vectors.

ifst, ilst

Overwritten for real flavors only.

If ifst pointed to the second row of a 2 by 2 block on entry, it is changed to point to the first row; ilst always points to the first row of the block in its final position (which may differ from its input value by ±1).

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.

Application Notes

The computed matrix S is exactly similar to a matrix T+E, where ||E||₂ = O(ε)*||T||₂, and ε is the machine precision.

Note that if a 2 by 2 diagonal block is involved in the re-ordering, its off-diagonal elements are in general changed; the diagonal elements and the eigenvalues of the block are unchanged unless the block is sufficiently ill-conditioned, in which case they may be noticeably altered. It is possible for a 2 by 2 block to break into two 1 by 1 blocks, that is, for a pair of complex eigenvalues to become purely real.

The approximate number of floating-point operations is

for real flavors:

6n(ifst-ilst) if compq = 'N';

12n(ifst-ilst) if compq = 'V';

for complex flavors:

20n(ifst-ilst) if compq = 'N';

40n(ifst-ilst) if compq = 'V'.

Parent topic: Nonsymmetric Eigenvalue Problems: LAPACK Computational Routines

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Developer Reference for Intel® oneAPI Math Kernel Library for C

?trexc