?trexc

Syntax

FORTRAN 77:

call strexc(compq, n, t, ldt, q, ldq, ifst, ilst, work, info)

call dtrexc(compq, n, t, ldt, q, ldq, ifst, ilst, work, info)

call ctrexc(compq, n, t, ldt, q, ldq, ifst, ilst, info)

call ztrexc(compq, n, t, ldt, q, ldq, ifst, ilst, info)

Fortran 95:

call trexc(t, ifst, ilst [,q] [,info])

C:

lapack_int LAPACKE_strexc( int matrix_order, 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_order, 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_order, 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_order, 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

The FORTRAN 77 interfaces are specified in the mkl_lapack.fi and mkl_lapack.h include files, the Fortran 95 interfaces are specified in the lapack.f90 include file, and the C interfaces are specified in the mkl_lapacke.h include file.

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

The data types are given for the Fortran interface. A <datatype> placeholder, if present, is used for the C interface data types in the C interface section above. See C Interface Conventions for the C interface principal conventions and type definitions.

compq

CHARACTER*1. 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

INTEGER. The order of the matrix T (n ≥ 0).

t, q

REAL for strexc

DOUBLE PRECISION for dtrexc

COMPLEX for ctrexc

DOUBLE COMPLEX for ztrexc.

Arrays:

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

The second dimension of t must be at least max(1, n).

q(ldq,*)

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

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

The second dimension of q must be at least max(1, n)

if compq = 'V' and at least 1 if compq = 'N'.

ldt

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

ldq

INTEGER. The leading dimension of q;

If compq = 'N', then ldq≥ 1.

If compq = 'V', then ldq≥ max(1, n).

ifst, ilst

INTEGER. 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).

work

REAL for strexc

DOUBLE PRECISION for dtrexc.

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

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).

info

INTEGER.

If info = 0, the execution is successful.

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

Fortran 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 Fortran 95 Interface Conventions.

Specific details for the routine trexc interface are the following:

t

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

q

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

compq

Restored based on the presence of the argument q as follows:

compq = 'V', if q is present,

compq = 'N', if q is omitted.

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 values of eigenvalues however are never changed by the re-ordering.

The approximate number of floating-point operations is