void	nag_superlu_condition_number_lu (Nag_NormType norm, Integer n, const Integer il[], const double lval[], const Integer iu[], const double uval[], double anorm, double rcond, NagError fail)

3 Description

nag_superlu_condition_number_lu (f11mgc) estimates the condition number of a real sparse 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 function actually returns an estimate of the reciprocal of the condition number.

The function should be preceded by a call to nag_superlu_matrix_norm (f11mlc) to compute

{‖A‖}_{1}

{‖A‖}_{\infty}

, and a call to nag_superlu_lu_factorize (f11mec) to compute the

L U

factorization of

A

. The function then estimates

{‖A^{- 1}‖}_{1}

{‖A^{- 1}‖}_{\infty}

and computes the reciprocal of the condition number.

4 References

None.

5 Arguments

1: $norm$ – Nag_NormTypeInput

On entry: indicates whether

κ_{1} (A)

κ_{\infty} (A)

is to be estimated.

$norm = Nag_RealOneNorm$: $κ_{1} (A)$ is estimated.
$norm = Nag_RealInfNorm$: $κ_{\infty} (A)$ is estimated.

Constraint:

norm = Nag_RealOneNorm

Nag_RealInfNorm

2: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $il [\dim]$ – const IntegerInput

Note: the dimension, dim, of the array il must be at least as large as the dimension of the array of the same name in nag_superlu_lu_factorize (f11mec).

On entry: records the sparsity pattern of matrix

L

as computed by nag_superlu_lu_factorize (f11mec).

4: $lval [\dim]$ – const doubleInput

Note: the dimension, dim, of the array lval must be at least as large as the dimension of the array of the same name in nag_superlu_lu_factorize (f11mec).

On entry: records the nonzero values of matrix

L

and some nonzero values of matrix

U

as computed by nag_superlu_lu_factorize (f11mec).

5: $iu [\dim]$ – const IntegerInput

Note: the dimension, dim, of the array iu must be at least as large as the dimension of the array of the same name in nag_superlu_lu_factorize (f11mec).

On entry: records the sparsity pattern of matrix

U

as computed by nag_superlu_lu_factorize (f11mec).

6: $uval [\dim]$ – const doubleInput

Note: the dimension, dim, of the array uval must be at least as large as the dimension of the array of the same name in nag_superlu_lu_factorize (f11mec).

On entry: records some nonzero values of matrix

U

as computed by nag_superlu_lu_factorize (f11mec).

7: $anorm$ – doubleInput

On entry: if

norm = Nag_RealOneNorm

, the

1

-norm of the matrix

A

norm = Nag_RealInfNorm

, the

\infty

-norm of the matrix

A

anorm may be computed by calling nag_superlu_matrix_norm (f11mlc) with the same value for the argument norm.

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 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 0$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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_superlu_condition_number_lu (f11mgc) 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

A call to nag_superlu_condition_number_lu (f11mgc) involves solving a number of systems of linear equations of the form

A x = b

A^{T} x = b

10 Example

This example estimates the condition number in the

1

-norm of the matrix

A

, where

A = (\begin{matrix} 2.00 & 1.00 & 0 & 0 & 0 \\ 0 & 0 & 1.00 & - 1.00 & 0 \\ 4.00 & 0 & 1.00 & 0 & 1.00 \\ 0 & 0 & 0 & 1.00 & 2.00 \\ 0 & - 2.00 & 0 & 0 & 3.00 \end{matrix}) .

Here

A

is nonsymmetric and must first be factorized by nag_superlu_lu_factorize (f11mec). The true condition number in the

1

-norm is

20.25

NAG Library Function Documentnag_superlu_condition_number_lu (f11mgc)

▸▿ 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_superlu_condition_number_lu (f11mgc)