?trsna

Estimates condition numbers for specified eigenvalues and right eigenvectors of an upper (quasi-) triangular matrix.

Syntax

FORTRAN 77:

call strsna(job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, iwork, info)

call dtrsna(job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, iwork, info)

call ctrsna(job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info)

call ztrsna(job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info)

FORTRAN 95:

call trsna(t [, s] [,sep] [,vl] [,vr] [,select] [,m] [,info])

lapack_int LAPACKE_strsna( int matrix_order, char job, char howmny, const lapack_logical* select, lapack_int n, const float* t, lapack_int ldt, const float* vl, lapack_int ldvl, const float* vr, lapack_int ldvr, float* s, float* sep, lapack_int mm, lapack_int* m );

lapack_int LAPACKE_dtrsna( int matrix_order, char job, char howmny, const lapack_logical* select, lapack_int n, const double* t, lapack_int ldt, const double* vl, lapack_int ldvl, const double* vr, lapack_int ldvr, double* s, double* sep, lapack_int mm, lapack_int* m );

lapack_int LAPACKE_ctrsna( int matrix_order, char job, char howmny, const lapack_logical* select, lapack_int n, const lapack_complex_float* t, lapack_int ldt, const lapack_complex_float* vl, lapack_int ldvl, const lapack_complex_float* vr, lapack_int ldvr, float* s, float* sep, lapack_int mm, lapack_int* m );

lapack_int LAPACKE_ztrsna( int matrix_order, char job, char howmny, const lapack_logical* select, lapack_int n, const lapack_complex_double* t, lapack_int ldt, const lapack_complex_double* vl, lapack_int ldvl, const lapack_complex_double* vr, lapack_int ldvr, double* s, double* sep, lapack_int mm, lapack_int* m );

Include Files

Fortran: mkl_lapack.fi and mkl_lapack.h
Fortran 95: lapack.f90
C: mkl_lapacke.h

Description

The routine estimates condition numbers for specified eigenvalues and/or right eigenvectors of an upper triangular matrix T (or, for real flavors, upper quasi-triangular matrix T in canonical Schur form). These are the same as the condition numbers of the eigenvalues and right eigenvectors of an original matrix A = Z*T*Z^H (with unitary or, for real flavors, orthogonal Z), from which T may have been derived.

The routine computes the reciprocal of the condition number of an eigenvalue lambda(i) as s_i = |v^T*u|/(||u||_E||v||_E) for real flavors and s_i = |v^H*u|/(||u||_E||v||_E) for complex flavors,

where:

u and v are the right and left eigenvectors of T, respectively, corresponding to lambda(i).
v^T/v^H denote transpose/conjugate transpose of v, respectively.

This reciprocal condition number always lies between zero (ill-conditioned) and one (well-conditioned).

An approximate error estimate for a computed eigenvalue lambda(i)is then given by ε*||T||/s_i, where ε is the machine precision.

To estimate the reciprocal of the condition number of the right eigenvector corresponding to lambda(i), the routine first calls trexc to reorder the eigenvalues so that lambda(i) is in the leading position:

The reciprocal condition number of the eigenvector is then estimated as sep_i, the smallest singular value of the matrix T₂₂ - lambda(i)*I. This number ranges from zero (ill-conditioned) to very large (well-conditioned).

An approximate error estimate for a computed right eigenvector u corresponding to lambda(i) is then given by ε*||T||/sep_i.

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.

job

CHARACTER*1. Must be 'E' or 'V' or 'B'.

If job = 'E', then condition numbers for eigenvalues only are computed.

If job = 'V', then condition numbers for eigenvectors only are computed.

If job = 'B', then condition numbers for both eigenvalues and eigenvectors are computed.

howmny

CHARACTER*1. Must be 'A' or 'S'.

If howmny = 'A', then the condition numbers for all eigenpairs are computed.

If howmny = 'S', then condition numbers for selected eigenpairs (as specified by select) are computed.

select

LOGICAL.

Array, DIMENSION at least max (1, n) if howmny = 'S' and at least 1 otherwise.

Specifies the eigenpairs for which condition numbers are to be computed if howmny= 'S'.

For real flavors:

To select condition numbers for the eigenpair corresponding to the real eigenvalue lambda(j), select(j) must be set .TRUE.;

to select condition numbers for the eigenpair corresponding to a complex conjugate pair of eigenvalues lambda(j) and lambda(j+1), select(j) and/or select(j+1) must be set .TRUE.

For complex flavors

To select condition numbers for the eigenpair corresponding to the eigenvalue lambda(j), select(j) must be set .TRUE. select is not referenced if howmny = 'A'.

n

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

t, vl, vr, work

REAL for strsna

DOUBLE PRECISION for dtrsna

COMPLEX for ctrsna

DOUBLE COMPLEX for ztrsna.

Arrays:

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

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

vl(ldvl,*)

If job = 'E' or 'B', then vl must contain the left eigenvectors of T (or of any matrix Q*T*Q^H with Q unitary or orthogonal) corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vl, as returned by trevc or hsein.

The second dimension of vl must be at least max(1, mm) if job = 'E' or 'B' and at least 1 if job = 'V'.

The array vl is not referenced if job = 'V'.

vr(ldvr,*)

If job = 'E' or 'B', then vr must contain the right eigenvectors of T (or of any matrix Q*T*Q^H with Q unitary or orthogonal) corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vr, as returned by trevc or hsein.

The second dimension of vr must be at least max(1, mm) if job = 'E' or 'B' and at least 1 if job = 'V'.

The array vr is not referenced if job = 'V'.

work is a workspace array, its dimension (ldwork,n+6).

The array work is not referenced if job = 'E'.

ldt

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

ldvl

INTEGER. The leading dimension of vl.

If job = 'E' or 'B', ldvl ≥ max(1,n).

If job = 'V', ldvl ≥ 1.

ldvr

INTEGER. The leading dimension of vr.

If job = 'E' or 'B', ldvr ≥ max(1,n).

If job = 'R', ldvr ≥ 1.

mm

INTEGER. The number of elements in the arrays s and sep, and the number of columns in vl and vr (if used). Must be at least m (the precise number required).

If howmny = 'A', m = n;

if howmny = 'S', for real flavors m is obtained by counting 1 for each selected real eigenvalue and 2 for each selected complex conjugate pair of eigenvalues.

for complex flavors m is the number of selected eigenpairs (see select). Constraint:

0 ≤ m ≤ n.

ldwork

INTEGER. The leading dimension of work.

If job = 'V' or 'B', ldwork ≥ max(1,n).

If job = 'E', ldwork ≥ 1.

rwork

REAL for ctrsna, ztrsna.

Array, DIMENSION at least max (1, n). The array is not referenced if job = 'E'.

iwork

INTEGER for strsna, dtrsna.

Array, DIMENSION at least max (1, 2*(n - 1)). The array is not referenced if job = 'E'.

Output Parameters

s

REAL for single-precision flavors

DOUBLE PRECISION for double-precision flavors.

Array, DIMENSION at least max(1, mm) if job = 'E' or 'B' and at least 1 if job = 'V'.

Contains the reciprocal condition numbers of the selected eigenvalues if job = 'E' or 'B', stored in consecutive elements of the array. Thus s(j), sep(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 have been selected).

For real flavors: for a complex conjugate pair of eigenvalues, two consecutive elements of S are set to the same value. The array s is not referenced if job = 'V'.

sep

REAL for single-precision flavors

DOUBLE PRECISION for double-precision flavors.

Array, DIMENSION at least max(1, mm) if job = 'V' or 'B' and at least 1 if job = 'E'. Contains the estimated reciprocal condition numbers of the selected right eigenvectors if job = 'V' or 'B', stored in consecutive elements of the array.

For real flavors: for a complex eigenvector, two consecutive elements of sep are set to the same value; if the eigenvalues cannot be reordered to compute sep(j), then sep(j) is set to zero; this can only occur when the true value would be very small anyway. The array sep is not referenced if job = 'E'.

m

INTEGER.

For complex flavors: the number of selected eigenpairs.

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

For real flavors: the number of elements of s and/or sep actually used to store the estimated condition numbers.

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

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 trsna interface are the following:

t

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

s

Holds the vector of length (mm).

sep

Holds the vector of length (mm).

vl

Holds the matrix VL of size (n,mm).

vr

Holds the matrix VR of size (n,mm).

select

Holds the vector of length n.

job

Restored based on the presence of arguments s and sep as follows:

job = 'B', if both s and sep are present,

job = 'E', if s is present and sep omitted,

job = 'V', if s is omitted and sep present.

Note an error condition if both s and sep are omitted.

howmny

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

howmny = 'S', if select is present,

howmny = 'A', if select is omitted.

Note that the arguments s, vl, and vr must either be all present or all omitted.

Otherwise, an error condition is observed.

Application Notes

The computed values sep_i may overestimate the true value, but seldom by a factor of more than 3.