NAG Library Routine Document

Integer, Intent (In)	::	n, lda
Integer, Intent (Out)	::	iwork(n), info
Real (Kind=nag_wp), Intent (In)	::	a(lda,*), anorm
Real (Kind=nag_wp), Intent (Out)	::	rcond, work(3*n)
Character (1), Intent (In)	::	uplo

C Header Interface

#include <nagmk26.h>

void	f07fgf_ (const char uplo, const Integer n, const double a[], const Integer lda, const double anorm, double rcond, double work[], Integer iwork[], Integer info, const Charlen length_uplo)

The routine may be called by its LAPACK name dpocon.

3

Description

f07fgf (dpocon) estimates the condition number (in the

1

-norm) of a real symmetric positive definite matrix

A

κ_{1} (A) = {‖A‖}_{1} {‖A^{- 1}‖}_{1} .

Since

A

is symmetric,

κ_{1} (A) = κ_{\infty} (A) = {‖A‖}_{\infty} {‖A^{- 1}‖}_{\infty}

Because

κ_{1} (A)

is infinite if

A

is singular, the routine actually returns an estimate of the reciprocal of

κ_{1} (A)

The routine should be preceded by a call to f06rcf to compute

{‖A‖}_{1}

and a call to f07fdf (dpotrf) to compute the Cholesky factorization of

A

. The routine then uses Higham's implementation of Hager's method (see Higham (1988)) to estimate

{‖A^{- 1}‖}_{1}

4

References

Higham N J (1988) FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396

5

Arguments

1: $uplo$ – Character(1)Input

On entry: specifies how

A

has been factorized.

$uplo ='U'$: $A = U^{T} U$ , where $U$ is upper triangular.
$uplo ='L'$: $A = L L^{T}$ , where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

2: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $a (lda *)$ – Real (Kind=nag_wp) arrayInput

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

\max (1, n)

On entry: the Cholesky factor of

A

, as returned by f07fdf (dpotrf).

4: $lda$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f07fgf (dpocon) is called.

Constraint:

lda \geq \max (1, n)

5: $anorm$ – Real (Kind=nag_wp)Input

On entry: the

1

-norm of the original matrix

A

, which may be computed by calling f06rcf with its argument

norm ='1'

. anorm must be computed either before calling f07fdf (dpotrf) or else from a copy of the original matrix

A

Constraint:

anorm \geq 0.0

6: $rcond$ – Real (Kind=nag_wp)Output

On exit: an estimate of the reciprocal of the condition number of

A

. rcond is set to zero if exact singularity is detected or the estimate underflows. If rcond is less than machine precision,

A

is singular to working precision.

7: $work (3 \times n)$ – Real (Kind=nag_wp) arrayWorkspace

8: $iwork (n)$ – Integer arrayWorkspace

9: $info$ – 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 estimate rcond is never less than the true value

ρ

, and in practice is nearly always less than

10 ρ

, although examples can be constructed where rcond is much larger.

8

Parallelism and Performance

f07fgf (dpocon) 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

A call to f07fgf (dpocon) involves solving a number of systems of linear equations of the form

A x = b

; the number is usually

4

5

and never more than

11

. Each solution involves approximately

2 n^{2}

floating-point operations but takes considerably longer than a call to f07fef (dpotrs) with one right-hand side, because extra care is taken to avoid overflow when

A

is approximately singular.

The complex analogue of this routine is f07fuf (zpocon).

10

Example

This example estimates the condition number in the

1

-norm (or

\infty

-norm) of the matrix

A

, where

A = (\begin{matrix} 4.16 & - 3.12 & 0.56 & - 0.10 \\ - 3.12 & 5.03 & - 0.83 & 1.18 \\ 0.56 & - 0.83 & 0.76 & 0.34 \\ - 0.10 & 1.18 & 0.34 & 1.18 \end{matrix}) .

Here

A

is symmetric positive definite and must first be factorized by f07fdf (dpotrf). The true condition number in the

1

-norm is

97.32

NAG Library Routine Document

f07fgf (dpocon)

▸▿ 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