# NAG Library Routine Document

## 1Purpose

f08qlf (dtrsna) estimates condition numbers for specified eigenvalues and/or right eigenvectors of a real upper quasi-triangular matrix.

## 2Specification

Fortran Interface
 Subroutine f08qlf ( job, n, t, ldt, vl, ldvl, vr, ldvr, s, sep, mm, m, work, info)
 Integer, Intent (In) :: n, ldt, ldvl, ldvr, mm, ldwork Integer, Intent (Inout) :: iwork(*) Integer, Intent (Out) :: m, info Real (Kind=nag_wp), Intent (In) :: t(ldt,*), vl(ldvl,*), vr(ldvr,*) Real (Kind=nag_wp), Intent (Inout) :: s(*), sep(*), work(ldwork,*) Logical, Intent (In) :: select(*) Character (1), Intent (In) :: job, howmny
#include nagmk26.h
 void f08qlf_ (const char *job, const char *howmny, const logical sel[], const Integer *n, const double t[], const Integer *ldt, const double vl[], const Integer *ldvl, const double vr[], const Integer *ldvr, double s[], double sep[], const Integer *mm, Integer *m, double work[], const Integer *ldwork, Integer iwork[], Integer *info, const Charlen length_job, const Charlen length_howmny)
The routine may be called by its LAPACK name dtrsna.

## 3Description

f08qlf (dtrsna) estimates condition numbers for specified eigenvalues and/or right eigenvectors of a real 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 $A=ZT{Z}^{\mathrm{T}}$ (with orthogonal $Z$), from which $T$ may have been derived.
f08qlf (dtrsna) computes the reciprocal of the condition number of an eigenvalue ${\lambda }_{i}$ as
 $si = vHu uEvE ,$
where $u$ and $v$ are the right and left eigenvectors of $T$, respectively, corresponding to ${\lambda }_{i}$. 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 ${\lambda }_{i}$ is then given by
 $εT si ,$
where $\epsilon$ is the machine precision.
To estimate the reciprocal of the condition number of the right eigenvector corresponding to ${\lambda }_{i}$, the routine first calls f08qff (dtrexc) to reorder the eigenvalues so that ${\lambda }_{i}$ is in the leading position:
 $T =Q λi cT 0 T22 QT.$
The reciprocal condition number of the eigenvector is then estimated as ${\mathit{sep}}_{i}$, the smallest singular value of the matrix $\left({T}_{22}-{\lambda }_{i}I\right)$. 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 ${\lambda }_{i}$ is then given by
 $εT sepi .$
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1:     $\mathbf{job}$ – Character(1)Input
On entry: indicates whether condition numbers are required for eigenvalues and/or eigenvectors.
${\mathbf{job}}=\text{'E'}$
Condition numbers for eigenvalues only are computed.
${\mathbf{job}}=\text{'V'}$
Condition numbers for eigenvectors only are computed.
${\mathbf{job}}=\text{'B'}$
Condition numbers for both eigenvalues and eigenvectors are computed.
Constraint: ${\mathbf{job}}=\text{'E'}$, $\text{'V'}$ or $\text{'B'}$.
2:     $\mathbf{howmny}$ – Character(1)Input
On entry: indicates how many condition numbers are to be computed.
${\mathbf{howmny}}=\text{'A'}$
Condition numbers for all eigenpairs are computed.
${\mathbf{howmny}}=\text{'S'}$
Condition numbers for selected eigenpairs (as specified by select) are computed.
Constraint: ${\mathbf{howmny}}=\text{'A'}$ or $\text{'S'}$.
3:     $\mathbf{select}\left(*\right)$ – Logical arrayInput
Note: the dimension of the array select must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{howmny}}=\text{'S'}$, and at least $1$ otherwise.
On entry: specifies the eigenpairs for which condition numbers are to be computed if ${\mathbf{howmny}}=\text{'S'}$. To select condition numbers for the eigenpair corresponding to the real eigenvalue ${\lambda }_{j}$, ${\mathbf{select}}\left(j\right)$ must be set .TRUE.. To select condition numbers corresponding to a complex conjugate pair of eigenvalues ${\lambda }_{j}$ and ${\lambda }_{j+1}$, ${\mathbf{select}}\left(j\right)$ and/or ${\mathbf{select}}\left(j+1\right)$ must be set to .TRUE..
If ${\mathbf{howmny}}=\text{'A'}$, select is not referenced.
4:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     $\mathbf{t}\left({\mathbf{ldt}},*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array t must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ by $n$ upper quasi-triangular matrix $T$ in canonical Schur form, as returned by f08pef (dhseqr).
6:     $\mathbf{ldt}$ – IntegerInput
On entry: the first dimension of the array t as declared in the (sub)program from which f08qlf (dtrsna) is called.
Constraint: ${\mathbf{ldt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7:     $\mathbf{vl}\left({\mathbf{ldvl}},*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array vl must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{mm}}\right)$ if ${\mathbf{job}}=\text{'E'}$ or $\text{'B'}$ and at least $1$ if ${\mathbf{job}}=\text{'V'}$.
On entry: if ${\mathbf{job}}=\text{'E'}$ or $\text{'B'}$, vl must contain the left eigenvectors of $T$ (or of any matrix $QT{Q}^{\mathrm{T}}$ with $Q$ orthogonal) corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vl, as returned by f08pkf (dhsein) or f08qkf (dtrevc).
If ${\mathbf{job}}=\text{'V'}$, vl is not referenced.
8:     $\mathbf{ldvl}$ – IntegerInput
On entry: the first dimension of the array vl as declared in the (sub)program from which f08qlf (dtrsna) is called.
Constraints:
• if ${\mathbf{job}}=\text{'E'}$ or $\text{'B'}$, ${\mathbf{ldvl}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{job}}=\text{'V'}$, ${\mathbf{ldvl}}\ge 1$.
9:     $\mathbf{vr}\left({\mathbf{ldvr}},*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array vr must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{mm}}\right)$ if ${\mathbf{job}}=\text{'E'}$ or $\text{'B'}$ and at least $1$ if ${\mathbf{job}}=\text{'V'}$.
On entry: if ${\mathbf{job}}=\text{'E'}$ or $\text{'B'}$, vr must contain the right eigenvectors of $T$ (or of any matrix $QT{Q}^{\mathrm{T}}$ with $Q$ orthogonal) corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vr, as returned by f08pkf (dhsein) or f08qkf (dtrevc).
If ${\mathbf{job}}=\text{'V'}$, vr is not referenced.
10:   $\mathbf{ldvr}$ – IntegerInput
On entry: the first dimension of the array vr as declared in the (sub)program from which f08qlf (dtrsna) is called.
Constraints:
• if ${\mathbf{job}}=\text{'E'}$ or $\text{'B'}$, ${\mathbf{ldvr}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{job}}=\text{'V'}$, ${\mathbf{ldvr}}\ge 1$.
11:   $\mathbf{s}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array s must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{mm}}\right)$ if ${\mathbf{job}}=\text{'E'}$ or $\text{'B'}$, and at least $1$ otherwise.
On exit: the reciprocal condition numbers of the selected eigenvalues if ${\mathbf{job}}=\text{'E'}$ or $\text{'B'}$, stored in consecutive elements of the array. Thus ${\mathbf{s}}\left(j\right)$, ${\mathbf{sep}}\left(j\right)$ 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 a complex conjugate pair of eigenvalues, two consecutive elements of s are set to the same value.
If ${\mathbf{job}}=\text{'V'}$, s is not referenced.
12:   $\mathbf{sep}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the array sep must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{mm}}\right)$ if ${\mathbf{job}}=\text{'V'}$ or $\text{'B'}$, and at least $1$ otherwise.
On exit: the estimated reciprocal condition numbers of the selected right eigenvectors if ${\mathbf{job}}=\text{'V'}$ or $\text{'B'}$, stored in consecutive elements of the array. For a complex eigenvector, two consecutive elements of sep are set to the same value. If the eigenvalues cannot be reordered to compute ${\mathbf{sep}}\left(j\right)$, ${\mathbf{sep}}\left(j\right)$ is set to zero; this can only occur when the true value would be very small anyway.
If ${\mathbf{job}}=\text{'E'}$, sep is not referenced.
13:   $\mathbf{mm}$ – IntegerInput
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 ${\mathbf{howmny}}=\text{'A'}$; if ${\mathbf{howmny}}=\text{'S'}$, $m$ is obtained by counting $1$ for each selected real eigenvalue, and $2$ for each selected complex conjugate pair of eigenvalues (see select), in which case $0\le m\le n$.
Constraint: ${\mathbf{mm}}\ge {\mathbf{m}}$.
14:   $\mathbf{m}$ – IntegerOutput
On exit: $m$, the number of elements of s and/or sep actually used to store the estimated condition numbers. If ${\mathbf{howmny}}=\text{'A'}$, m is set to $n$.
15:   $\mathbf{work}\left({\mathbf{ldwork}},*\right)$ – Real (Kind=nag_wp) arrayWorkspace
Note: the second dimension of the array work must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}+6\right)$ if ${\mathbf{job}}=\text{'V'}$ or $\text{'B'}$ and at least $1$ if ${\mathbf{job}}=\text{'E'}$.
If ${\mathbf{job}}=\text{'E'}$, work is not referenced.
16:   $\mathbf{ldwork}$ – IntegerInput
On entry: the first dimension of the array work as declared in the (sub)program from which f08qlf (dtrsna) is called.
Constraints:
• if ${\mathbf{job}}=\text{'V'}$ or $\text{'B'}$, ${\mathbf{ldwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{job}}=\text{'E'}$, ${\mathbf{ldwork}}\ge 1$.
17:   $\mathbf{iwork}\left(*\right)$ – Integer arrayWorkspace
Note: the dimension of the array iwork must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×\left({\mathbf{n}}-1\right)\right)$.
18:   $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

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

## 7Accuracy

The computed values ${\mathit{sep}}_{i}$ may over estimate the true value, but seldom by a factor of more than $3$.

## 8Parallelism and Performance

f08qlf (dtrsna) 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.

For a description of canonical Schur form, see the document for f08pef (dhseqr).
The complex analogue of this routine is f08qyf (ztrsna).

## 10Example

This example computes approximate error estimates for all the eigenvalues and right eigenvectors of the matrix $T$, where
 $T = 0.7995 -0.1144 0.0060 0.0336 0.0000 -0.0994 0.2478 0.3474 0.0000 -0.6483 -0.0994 0.2026 0.0000 0.0000 0.0000 -0.1007 .$

### 10.1Program Text

Program Text (f08qlfe.f90)

### 10.2Program Data

Program Data (f08qlfe.d)

### 10.3Program Results

Program Results (f08qlfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017