NAG Library Routine Document

F07HUF (ZPBCON)

+− 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

F07HUF (ZPBCON) estimates the condition number of a complex Hermitian positive definite band matrix

A

, where

A

has been factorized by F07HRF (ZPBTRF).

2 Specification

SUBROUTINE F07HUF (

UPLO, N, KD, AB, LDAB, ANORM, RCOND, WORK, RWORK, INFO)

INTEGER	N, KD, LDAB, INFO
REAL (KIND=nag_wp)	ANORM, RCOND, RWORK(N)
COMPLEX (KIND=nag_wp)	AB(LDAB,), WORK(2N)
CHARACTER(1)	UPLO

The routine may be called by its LAPACK name zpbcon.

3 Description

F07HUF (ZPBCON) estimates the condition number (in the

1

-norm) of a complex Hermitian positive definite band matrix

A

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

Since

A

is Hermitian,

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 F06UEF to compute

{‖A‖}_{1}

and a call to F07HRF (ZPBTRF) 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.

$UPLO ='U'$: $A = U^{H} U$ , where $U$ is upper triangular.
$UPLO ='L'$: $A = L L^{H}$ , 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, $*$ ) – COMPLEX (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 F07HRF (ZPBTRF).

5: LDAB – INTEGERInput

On entry: the first dimension of the array AB as declared in the (sub)program from which F07HUF (ZPBCON) 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 F06UEF with its parameter

NORM ='1'

. ANORM must be computed either before calling F07HRF (ZPBTRF) 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( $2 \times N$ ) – COMPLEX (KIND=nag_wp) arrayWorkspace

9: RWORK(N) – REAL (KIND=nag_wp) arrayWorkspace

10: 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 F07HUF (ZPBCON) involves solving a number of systems of linear equations of the form

A x = b

; the number is usually

5

and never more than

11

. Each solution involves approximately

16 n k

real floating point operations (assuming

n ≫ k

) but takes considerably longer than a call to F07HSF (ZPBTRS) with one right-hand side, because extra care is taken to avoid overflow when

A

is approximately singular.

The real analogue of this routine is F07HGF (DPBCON).

9 Example

This example estimates the condition number in the

1

-norm (or

\infty

-norm) of the matrix

A

, where

A = (\begin{array}{r} 9.39 + 0.00 i & 1.08 - 1.73 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ 1.08 + 1.73 i & 1.69 + 0.00 i & - 0.04 + 0.29 i & 0.00 + 0.00 i \\ 0.00 + 0.00 i & - 0.04 - 0.29 i & 2.65 + 0.00 i & - 0.33 + 2.24 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & - 0.33 - 2.24 i & 2.17 + 0.00 i \end{array}) .

Here

A

is Hermitian positive definite, and is treated as a band matrix, which must first be factorized by F07HRF (ZPBTRF). The true condition number in the

1

-norm is

153.45

NAG Library Routine DocumentF07HUF (ZPBCON)

+− 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

F07HUF (ZPBCON)