NAG Library Function Document

nag_zpbcon (f07huc)

void	nag_zpbcon (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer kd, const Complex ab[], Integer pdab, double anorm, double rcond, NagError fail)

3 Description

nag_zpbcon (f07huc) 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 function actually returns an estimate of the reciprocal of

κ_{1} (A)

The function should be preceded by a call to nag_zhb_norm (f16uec) to compute

{‖A‖}_{1}

and a call to nag_zpbtrf (f07hrc) to compute the Cholesky factorization of

A

. The function 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: order – Nag_OrderTypeInput

On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by

order = Nag_RowMajor

. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: uplo – Nag_UploTypeInput

On entry: specifies how

A

has been factorized.

$uplo = Nag_Upper$: $A = U^{H} U$ , where $U$ is upper triangular.
$uplo = Nag_Lower$: $A = L L^{H}$ , where $L$ is lower triangular.

Constraint:

uplo = Nag_Upper

Nag_Lower

3: n – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

4: kd – IntegerInput

On entry:

k_{d}

, the number of superdiagonals or subdiagonals of the matrix

A

Constraint:

kd \geq 0

5: ab[ $\dim$ ] – const ComplexInput

Note: the dimension, dim, of the array ab must be at least

\max (1, pdab \times n)

On entry: the Cholesky factor of

A

, as returned by nag_zpbtrf (f07hrc).

6: pdab – IntegerInput

On entry: the stride separating row or column elements (depending on the value of order) of the matrix in the array ab.

Constraint:

pdab \geq kd + 1

7: anorm – doubleInput

On entry: the

1

-norm of the original matrix

A

, which may be computed by calling nag_zhb_norm (f16uec) with its argument

norm = Nag_OneNorm

. anorm must be computed either before calling nag_zpbtrf (f07hrc) or else from a copy of the original matrix

A

Constraint:

anorm \geq 0.0

8: rcond – double *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.

9: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $kd = ⟨value⟩$ .
Constraint: $kd \geq 0$ .
On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 0$ .
On entry, $pdab = ⟨value⟩$ .
Constraint: $pdab > 0$ .
NE_INT_2: On entry, $pdab = ⟨value⟩$ and $kd = ⟨value⟩$ .
Constraint: $pdab \geq kd + 1$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_REAL: On entry, $anorm = ⟨value⟩$ .
Constraint: $anorm \geq 0.0$ .

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

nag_zpbcon (f07huc) is not threaded by NAG in any implementation.

nag_zpbcon (f07huc) 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 Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

A call to nag_zpbcon (f07huc) 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 nag_zpbtrs (f07hsc) with one right-hand side, because extra care is taken to avoid overflow when

A

is approximately singular.

The real analogue of this function is nag_dpbcon (f07hgc).

10 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 nag_zpbtrf (f07hrc). The true condition number in the

1

-norm is

153.45

NAG Library Function Documentnag_zpbcon (f07huc)

+− 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

NAG Library Function Document

nag_zpbcon (f07huc)