NAG Library Routine Document

F07GGF (DPPCON)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F07GGF (DPPCON) estimates the condition number of a real symmetric positive definite matrix

A

, where

A

has been factorized by F07GDF (DPPTRF), using packed storage.

2 Specification

SUBROUTINE F07GGF (

UPLO, N, AP, ANORM, RCOND, WORK, IWORK, INFO)

INTEGER	N, IWORK(N), INFO
REAL (KIND=nag_wp)	AP(), ANORM, RCOND, WORK(3N)
CHARACTER(1)	UPLO

The routine may be called by its LAPACK name dppcon.

3 Description

F07GGF (DPPCON) 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 F06RDF to compute

{‖A‖}_{1}

and a call to F07GDF (DPPTRF) 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 Parameters

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: AP( $*$ ) – REAL (KIND=nag_wp) arrayInput

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

\max (1, N \times (N + 1) / 2)

On entry: the Cholesky factor of

A

stored in packed form, as returned by F07GDF (DPPTRF).

4: ANORM – REAL (KIND=nag_wp)Input

On entry: the

1

-norm of the original matrix

A

, which may be computed by calling F06RDF with its parameter

NORM ='1'

. ANORM must be computed either before calling F07GDF (DPPTRF) or else from a copy of the original matrix

A

Constraint:

ANORM \geq 0.0

5: 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.

6: WORK( $3 \times N$ ) – REAL (KIND=nag_wp) arrayWorkspace

7: IWORK(N) – INTEGER arrayWorkspace

8: INFO – INTEGEROutput

On exit:

INFO = 0

unless the routine detects an error (see Section 6).

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

$INFO < 0$: If $INFO = - i$ , the $i$ th parameter 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 Further Comments

A call to F07GGF (DPPCON) 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 F07GEF (DPPTRS) 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 F07GUF (ZPPCON).

9 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, stored in packed form, and must first be factorized by F07GDF (DPPTRF). The true condition number in the

1

-norm is

97.32

NAG Library Routine DocumentF07GGF (DPPCON)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F07GGF (DPPCON)