NAG FL Interface
f08qyf (ztrsna)

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1 Purpose

f08qyf estimates condition numbers for specified eigenvalues and/or right eigenvectors of a complex upper triangular matrix.

2 Specification

Fortran Interface
Subroutine f08qyf ( job, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, ldwork, rwork, info)
Integer, Intent (In) :: n, ldt, ldvl, ldvr, mm, ldwork
Integer, Intent (Out) :: m, info
Real (Kind=nag_wp), Intent (Inout) :: s(*), sep(*), rwork(*)
Complex (Kind=nag_wp), Intent (In) :: t(ldt,*), vl(ldvl,*), vr(ldvr,*)
Complex (Kind=nag_wp), Intent (Inout) :: work(ldwork,*)
Logical, Intent (In) :: select(*)
Character (1), Intent (In) :: job, howmny
C Header Interface
#include <nag.h>
void  f08qyf_ (const char *job, const char *howmny, const logical sel[], const Integer *n, const Complex t[], const Integer *ldt, const Complex vl[], const Integer *ldvl, const Complex vr[], const Integer *ldvr, double s[], double sep[], const Integer *mm, Integer *m, Complex work[], const Integer *ldwork, double rwork[], Integer *info, const Charlen length_job, const Charlen length_howmny)
The routine may be called by the names f08qyf, nagf_lapackeig_ztrsna or its LAPACK name ztrsna.

3 Description

f08qyf estimates condition numbers for specified eigenvalues and/or right eigenvectors of a complex upper triangular matrix T. These are the same as the condition numbers of the eigenvalues and right eigenvectors of an original matrix A=ZTZH (with unitary Z), from which T may have been derived.
f08qyf computes the reciprocal of the condition number of an eigenvalue λi as
si = |vHu| uEvE ,  
where u and v are the right and left eigenvectors of T, respectively, corresponding to λi. This reciprocal condition number always lies between zero (i.e., ill-conditioned) and one (i.e., well-conditioned).
An approximate error estimate for a computed eigenvalue λi is then given by
εT si ,  
where ε is the machine precision.
To estimate the reciprocal of the condition number of the right eigenvector corresponding to λi, the routine first calls f08qtf to reorder the eigenvalues so that λi is in the leading position:
T =Q ( λi cH 0 T22 ) QH.  
The reciprocal condition number of the eigenvector is then estimated as sepi, the smallest singular value of the matrix (T22-λiI). This number ranges from zero (i.e., ill-conditioned) to very large (i.e., well-conditioned).
An approximate error estimate for a computed right eigenvector u corresponding to λi is then given by
εT sepi .  

4 References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5 Arguments

1: job Character(1) Input
On entry: indicates whether condition numbers are required for eigenvalues and/or eigenvectors.
job='E'
Condition numbers for eigenvalues only are computed.
job='V'
Condition numbers for eigenvectors only are computed.
job='B'
Condition numbers for both eigenvalues and eigenvectors are computed.
Constraint: job='E', 'V' or 'B'.
2: howmny Character(1) Input
On entry: indicates how many condition numbers are to be computed.
howmny='A'
Condition numbers for all eigenpairs are computed.
howmny='S'
Condition numbers for selected eigenpairs (as specified by select) are computed.
Constraint: howmny='A' or 'S'.
3: select(*) Logical array Input
Note: the dimension of the array select must be at least max(1,n) if howmny='S', and at least 1 otherwise.
On entry: specifies the eigenpairs for which condition numbers are to be computed if howmny='S'. To select condition numbers for the eigenpair corresponding to the eigenvalue λj, select(j) must be set to .TRUE..
If howmny='A', select is not referenced.
4: n Integer Input
On entry: n, the order of the matrix T.
Constraint: n0.
5: t(ldt,*) Complex (Kind=nag_wp) array Input
Note: the second dimension of the array t must be at least max(1,n).
On entry: the n×n upper triangular matrix T, as returned by f08psf.
6: ldt Integer Input
On entry: the first dimension of the array t as declared in the (sub)program from which f08qyf is called.
Constraint: ldt max(1,n) .
7: vl(ldvl,*) Complex (Kind=nag_wp) array Input
Note: the second dimension of the array vl must be at least max(1,mm) if job='E' or 'B'.
On entry: if job='E' or 'B', vl must contain the left eigenvectors of T (or of any matrix QTQH with Q unitary) corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vl, as returned by f08pxf or f08qxf.
If job='V', vl is not referenced.
8: ldvl Integer Input
On entry: the first dimension of the array vl as declared in the (sub)program from which f08qyf is called.
Constraints:
  • if job='E' or 'B', ldvl max(1,n) ;
  • if job='V', ldvl1.
9: vr(ldvr,*) Complex (Kind=nag_wp) array Input
Note: the second dimension of the array vr must be at least max(1,mm) if job='E' or 'B'.
On entry: if job='E' or 'B', vr must contain the right eigenvectors of T (or of any matrix QTQH with Q unitary) corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vr, as returned by f08pxf or f08qxf.
If job='V', vr is not referenced.
10: ldvr Integer Input
On entry: the first dimension of the array vr as declared in the (sub)program from which f08qyf is called.
Constraints:
  • if job='E' or 'B', ldvr max(1,n) ;
  • if job='V', ldvr1.
11: s(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array s must be at least max(1,mm) if job='E' or 'B'.
On exit: 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 jth columns of vl and vr all correspond to the same eigenpair (but not in general the jth eigenpair unless all eigenpairs have been selected).
If job='V', s is not referenced.
12: sep(*) Real (Kind=nag_wp) array Output
Note: the dimension of the array sep must be at least max(1,mm) if job='V' or 'B', and at least 1 otherwise.
On exit: the estimated reciprocal condition numbers of the selected right eigenvectors if job='V' or 'B', stored in consecutive elements of the array.
If job='E', sep is not referenced.
13: mm Integer Input
On entry: the number of elements in the arrays s and sep, and the number of columns in the arrays vl and vr (if used). The precise number required, m, is n if howmny='A'; if howmny='S', m is the number of selected eigenpairs (see select), in which case 0mn.
Constraints:
  • if howmny='A', mmn;
  • otherwise mmm.
14: m Integer Output
On exit: m, the number of selected eigenpairs. If howmny='A', m is set to n.
15: work(ldwork,*) Complex (Kind=nag_wp) array Workspace
Note: the second dimension of the array work must be at least max(1,n+1) if job='V' or 'B' and at least 1 if job='E'.
If job='E', work is not referenced.
16: ldwork Integer Input
On entry: the first dimension of the array work as declared in the (sub)program from which f08qyf is called.
Constraints:
  • if job='V' or 'B', ldwork max(1,n) ;
  • if job='E', ldwork1.
17: rwork(*) Real (Kind=nag_wp) array Workspace
Note: the dimension of the array rwork must be at least max(1,n).
18: info Integer Output
On exit: info=0 unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

info<0
If info=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

7 Accuracy

The computed values sepi may over estimate the true value, but seldom by a factor of more than 3.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f08qyf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

The real analogue of this routine is f08qlf.

10 Example

This example computes approximate error estimates for all the eigenvalues and right eigenvectors of the matrix T, where
T = ( -6.0004-6.9999i 0.3637-0.3656i -0.1880+0.4787i 0.8785-0.2539i 0.0000+0.0000i -5.0000+2.0060i -0.0307-0.7217i -0.2290+0.1313i 0.0000+0.0000i 0.0000+0.0000i 7.9982-0.9964i 0.9357+0.5359i 0.0000+0.0000i 0.0000+0.0000i 0.0000+0.0000i 3.0023-3.9998i ) .  

10.1 Program Text

Program Text (f08qyfe.f90)

10.2 Program Data

Program Data (f08qyfe.d)

10.3 Program Results

Program Results (f08qyfe.r)