NAG Library Routine Document

F07BGF (DGBCON)

+− 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

F07BGF (DGBCON) estimates the condition number of a real band matrix

A

, where

A

has been factorized by F07BDF (DGBTRF).

2 Specification

SUBROUTINE F07BGF (

NORM, N, KL, KU, AB, LDAB, IPIV, ANORM, RCOND, WORK, IWORK, INFO)

INTEGER	N, KL, KU, LDAB, IPIV(*), IWORK(N), INFO
REAL (KIND=nag_wp)	AB(LDAB,), ANORM, RCOND, WORK(3N)
CHARACTER(1)	NORM

The routine may be called by its LAPACK name dgbcon.

3 Description

F07BGF (DGBCON) estimates the condition number of a real band matrix

A

, in either the

1

-norm or the

\infty

-norm:

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

Note that

κ_{\infty} (A) = κ_{1} (A^{T})

Because the condition number is infinite if

A

is singular, the routine actually returns an estimate of the reciprocal of the condition number.

The routine should be preceded by a call to F06RBF to compute

{‖A‖}_{1}

{‖A‖}_{\infty}

, and a call to F07BDF (DGBTRF) to compute the

L U

factorization of

A

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

{‖A^{- 1}‖}_{1}

{‖A^{- 1}‖}_{\infty}

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: NORM – CHARACTER(1)Input

On entry: indicates whether

κ_{1} (A)

κ_{\infty} (A)

is estimated.

$NORM ='1'$ or $'O'$: $κ_{1} (A)$ is estimated.
$NORM ='I'$: $κ_{\infty} (A)$ is estimated.

Constraint:

NORM ='1'

'O'

'I'

2: N – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

3: KL – INTEGERInput

On entry:

k_{l}

, the number of subdiagonals within the band of the matrix

A

Constraint:

KL \geq 0

4: KU – INTEGERInput

On entry:

k_{u}

, the number of superdiagonals within the band of the matrix

A

Constraint:

KU \geq 0

5: AB(LDAB, $*$ ) – REAL (KIND=nag_wp) arrayInput

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

\max (1, N)

On entry: the

L U

factorization of

A

, as returned by F07BDF (DGBTRF).

6: LDAB – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F07BGF (DGBCON) is called.

Constraint:

LDAB \geq 2 \times KL + KU + 1

7: IPIV( $*$ ) – INTEGER arrayInput

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

\max (1, N)

On entry: the pivot indices, as returned by F07BDF (DGBTRF).

8: ANORM – REAL (KIND=nag_wp)Input

On entry: if

NORM ='1'

'O'

, the

1

-norm of the original matrix

A

NORM ='I'

, the

\infty

-norm of the original matrix

A

ANORM may be computed by calling F06RBF with the same value for the parameter NORM.

ANORM must be computed either before calling F07BDF (DGBTRF) or else from a copy of the original matrix

A

(see Section 9).

Constraint:

ANORM \geq 0.0

9: 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.

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

11: IWORK(N) – INTEGER arrayWorkspace

12: 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 F07BGF (DGBCON) involves solving a number of systems of linear equations of the form

A x = b

A^{T} x = b

; the number is usually

4

5

and never more than

11

. Each solution involves approximately

2 n (2 k_{l} + k_{u})

floating point operations (assuming

n ≫ k_{l}

and

n ≫ k_{u}

) but takes considerably longer than a call to F07BEF (DGBTRS) 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 F07BUF (ZGBCON).

9 Example

This example estimates the condition number in the

1

-norm of the matrix

A

, where

A = (\begin{matrix} - 0.23 & 2.54 & - 3.66 & 0.00 \\ - 6.98 & 2.46 & - 2.73 & - 2.13 \\ 0.00 & 2.56 & 2.46 & 4.07 \\ 0.00 & 0.00 & - 4.78 & - 3.82 \end{matrix}) .

Here

A

is nonsymmetric and is treated as a band matrix, which must first be factorized by F07BDF (DGBTRF). The true condition number in the

1

-norm is

56.40

NAG Library Routine DocumentF07BGF (DGBCON)

+− 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

F07BGF (DGBCON)