NAG Library Routine Document

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

C Header Interface

#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.

3

Description

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 = Z T Z^{T}

(with orthogonal

Z

), from which

T

may have been derived.

f08qlf (dtrsna) computes the reciprocal of the condition number of an eigenvalue

λ_{i}

s_{i} = \frac{|v^{H} u|}{{‖u‖}_{E} {‖v‖}_{E}},

where

u

and

v

are the right and left eigenvectors of

T

, respectively, corresponding to

λ_{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

λ_{i}

is then given by

\frac{ε ‖T‖}{s_{i}},

where

ε

is the machine precision.

To estimate the reciprocal of the condition number of the right eigenvector corresponding to

λ_{i}

, the routine first calls f08qff (dtrexc) to reorder the eigenvalues so that

λ_{i}

is in the leading position:

T = Q (\begin{matrix} λ_{i} & c^{T} \\ 0 & T_{22} \end{matrix}) Q^{T} .

The reciprocal condition number of the eigenvector is then estimated as

{sep}_{i}

, the smallest singular value of the matrix

(T_{22} - λ_{i} I)

. 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

λ_{i}

is then given by

\frac{ε ‖T‖}{{sep}_{i}} .

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'

'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'

'S'

3: $select (*)$ – Logical arrayInput

Note: the dimension of the array select must be at least

\max (1, n)

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 real eigenvalue

λ_{j}

select (j)

must be set .TRUE.. To select condition numbers corresponding to a complex conjugate pair of eigenvalues

λ_{j}

and

λ_{j + 1}

select (j)

and/or

select (j + 1)

must be set to .TRUE..

howmny ='A'

, select is not referenced.

4: $n$ – IntegerInput

On entry:

n

, the order of the matrix

T

Constraint:

n \geq 0

5: $t (ldt *)$ – Real (Kind=nag_wp) arrayInput

Note: the second dimension of the array t must be at least

\max (1, n)

On entry: the

n

n

upper quasi-triangular matrix

T

in canonical Schur form, as returned by f08pef (dhseqr).

6: $ldt$ – IntegerInput

On entry: the first dimension of the array t as declared in the (sub)program from which f08qlf (dtrsna) is called.

Constraint:

ldt \geq \max (1, n)

7: $vl (ldvl *)$ – Real (Kind=nag_wp) arrayInput

Note: the second dimension of the array vl must be at least

\max (1, mm)

job ='E'

'B'

On entry: if

job ='E'

'B'

, vl must contain the left eigenvectors of

T

(or of any matrix

Q T Q^{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).

job ='V'

, vl is not referenced.

8: $ldvl$ – IntegerInput

On entry: the first dimension of the array vl as declared in the (sub)program from which f08qlf (dtrsna) is called.

Constraints:

if $job ='E'$ or $'B'$ , $ldvl \geq \max (1, n)$ ;
if $job ='V'$ , $ldvl \geq 1$ .

9: $vr (ldvr *)$ – Real (Kind=nag_wp) arrayInput

Note: the second dimension of the array vr must be at least

\max (1, mm)

job ='E'

'B'

On entry: if

job ='E'

'B'

, vr must contain the right eigenvectors of

T

(or of any matrix

Q T Q^{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).

job ='V'

, vr is not referenced.

10: $ldvr$ – IntegerInput

On entry: the first dimension of the array vr as declared in the (sub)program from which f08qlf (dtrsna) is called.

Constraints:

if $job ='E'$ or $'B'$ , $ldvr \geq \max (1, n)$ ;
if $job ='V'$ , $ldvr \geq 1$ .

11: $s (*)$ – Real (Kind=nag_wp) arrayOutput

Note: the dimension of the array s must be at least

\max (1, mm)

job ='E'

'B'

On exit: the reciprocal condition numbers of the selected eigenvalues if

job ='E'

'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 a complex conjugate pair of eigenvalues, two consecutive elements of s are set to the same value.

job ='V'

, s is not referenced.

12: $sep (*)$ – Real (Kind=nag_wp) arrayOutput

Note: the dimension of the array sep must be at least

\max (1, mm)

job ='V'

'B'

, and at least

1

otherwise.

On exit: the estimated reciprocal condition numbers of the selected right eigenvectors if

job ='V'

'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

sep (j)

sep (j)

is set to zero; this can only occur when the true value would be very small anyway.

job ='E'

, sep is not referenced.

13: $mm$ – 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

howmny ='A'

; if

howmny ='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 \leq m \leq n

Constraint:

mm \geq m

14: $m$ – IntegerOutput

On exit:

m

, 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

15: $work (ldwork *)$ – Real (Kind=nag_wp) arrayWorkspace

Note: the second dimension of the array work must be at least

\max (1, n + 6)

job ='V'

'B'

and at least

1

job ='E'

job ='E'

, work is not referenced.

16: $ldwork$ – IntegerInput

On entry: the first dimension of the array work as declared in the (sub)program from which f08qlf (dtrsna) is called.

Constraints:

if $job ='V'$ or $'B'$ , $ldwork \geq \max (1, n)$ ;
if $job ='E'$ , $ldwork \geq 1$ .

17: $iwork (*)$ – Integer arrayWorkspace

Note: the dimension of the array iwork must be at least

\max (1, 2 \times (n - 1))

18: $info$ – IntegerOutput

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

{sep}_{i}

may over estimate the true value, but seldom by a factor of more than

3

8

Parallelism 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.

9

Further Comments

For a description of canonical Schur form, see the document for f08pef (dhseqr).

The complex analogue of this routine is f08qyf (ztrsna).

10

Example

This example computes approximate error estimates for all the eigenvalues and right eigenvectors of the matrix

T

, where

T = (\begin{array}{r} 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 \end{array}) .

NAG Library Routine Document

f08qlf (dtrsna)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results