NAG Library Routine Document

Integer, Intent (In)	::	n, kl, ku, ldab, ipiv(*)
Integer, Intent (Out)	::	iwork(n), info
Real (Kind=nag_wp), Intent (In)	::	ab(ldab,*), anorm
Real (Kind=nag_wp), Intent (Out)	::	rcond, work(3*n)
Character (1), Intent (In)	::	norm

C Header Interface

#include <nagmk26.h>

void	f07bgf_ (const char norm, const Integer n, const Integer kl, const Integer ku, const double ab[], const Integer ldab, const Integer ipiv[], const double anorm, double rcond, double work[], Integer iwork[], Integer info, const Charlen length_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

Arguments

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 argument norm.

anorm must be computed either before calling f07bdf (dgbtrf) or else from a copy of the original matrix

A

(see Section 10).

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

$info < 0$: 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

f07bgf (dgbcon) 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 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).

10

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 Document

f07bgf (dgbcon)

▸▿ 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