NAG Library Routine Document

Integer, Intent (In)	::	n, kd, ldab
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)	::	uplo

C Header Interface

#include <nagmk26.h>

void	f07hgf_ (const char uplo, const Integer n, const Integer kd, const double ab[], const Integer ldab, const double anorm, double rcond, double work[], Integer iwork[], Integer *info, const Charlen length_uplo)

The routine may be called by its LAPACK name dpbcon.

3

Description

f07hgf (dpbcon) estimates the condition number (in the

1

-norm) of a real symmetric positive definite band 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 f06ref to compute

{‖A‖}_{1}

and a call to f07hdf (dpbtrf) 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: $kd$ – IntegerInput

On entry:

k_{d}

, the number of superdiagonals or subdiagonals of the matrix

A

Constraint:

kd \geq 0

4: $ab (ldab *)$ – Real (Kind=nag_wp) arrayInput

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

\max (1, n)

On entry: the Cholesky factor of

A

, as returned by f07hdf (dpbtrf).

5: $ldab$ – IntegerInput

On entry: the first dimension of the array ab as declared in the (sub)program from which f07hgf (dpbcon) is called.

Constraint:

ldab \geq kd + 1

6: $anorm$ – Real (Kind=nag_wp)Input

On entry: the

1

-norm of the original matrix

A

, which may be computed by calling f06ref with its argument

norm ='1'

. anorm must be computed either before calling f07hdf (dpbtrf) or else from a copy of the original matrix

A

Constraint:

anorm \geq 0.0

7: $rcond$ – 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.

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

9: $iwork (n)$ – Integer arrayWorkspace

10: $info$ – 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

f07hgf (dpbcon) 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 f07hgf (dpbcon) 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

4 n k

floating-point operations (assuming

n ≫ k

) but takes considerably longer than a call to f07hef (dpbtrs) 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 f07huf (zpbcon).

10

Example

This example estimates the condition number in the

1

-norm (or

\infty

-norm) of the matrix

A

, where

A = (\begin{matrix} 5.49 & 2.68 & 0.00 & 0.00 \\ 2.68 & 5.63 & - 2.39 & 0.00 \\ 0.00 & - 2.39 & 2.60 & - 2.22 \\ 0.00 & 0.00 & - 2.22 & 5.17 \end{matrix}) .

Here

A

is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by f07hdf (dpbtrf). The true condition number in the

1

-norm is

74.15

NAG Library Routine Document

f07hgf (dpbcon)

▸▿ 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