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

Parameters

Compulsory Input Parameters

1: $uplo$ – string (length ≥ 1)

Specifies how

A

has been factorized.

$uplo ='U'$: $A = P U D U^{T} P^{T}$ , where $U$ is upper triangular.
$uplo ='L'$: $A = P L D L^{T} P^{T}$ , where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

2: $ap (:)$ – double array

The dimension of the array ap must be at least

\max (1, n \times (n + 1) / 2)

The factorization of

A

stored in packed form, as returned by nag_lapack_dsptrf (f07pd).

3: $ipiv (:)$ – int64int32nag_int array

The dimension of the array ipiv must be at least

\max (1, n)

Details of the interchanges and the block structure of

D

, as returned by nag_lapack_dsptrf (f07pd).

4: $anorm$ – double scalar

The

1

-norm of the original matrix

A

. anorm must be computed either before calling nag_lapack_dsptrf (f07pd) or else from a copy of the original matrix

A

Constraint:

anorm \geq 0.0

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the dimension of the array ipiv.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $rcond$ – double scalar: 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.
2: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

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.

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.

Further Comments

A call to nag_lapack_dspcon (f07pg) 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

2 n^{2}

floating-point operations but takes considerably longer than a call to nag_lapack_dsptrs (f07pe) with one right-hand side, because extra care is taken to avoid overflow when

A

is approximately singular.

The complex analogues of this function are nag_lapack_zhpcon (f07pu) for Hermitian matrices and nag_lapack_zspcon (f07qu) for symmetric matrices.

Example

This example estimates the condition number in the

1

-norm (or

\infty

-norm) of the matrix

A

, where

A = (\begin{matrix} 2.07 & 3.87 & 4.20 & - 1.15 \\ 3.87 & - 0.21 & 1.87 & 0.63 \\ 4.20 & 1.87 & 1.15 & 2.06 \\ - 1.15 & 0.63 & 2.06 & - 1.81 \end{matrix}) .

Here

A

is symmetric indefinite, stored in packed form, and must first be factorized by nag_lapack_dsptrf (f07pd). The true condition number in the

1

-norm is

75.68

Open in the MATLAB editor: f07pg_example

function f07pg_example


fprintf('f07pg example results\n\n');

% Indefinite matrix A (lower triangular part stored in packed format)
uplo = 'L';
n = int64(4);
ap = [2.07;   3.87;   4.20;   -1.15;
             -0.21;   1.87;    0.63;
                      1.15;    2.06;
                              -1.81];

% Factorize
[apf, ipiv, info] = f07pd( ...
                           uplo, n, ap);

anorm = 11.29;
[rcond, info] = f07pg( ...
                       uplo, apf, ipiv, anorm);

fprintf('Estimate of condition number = %9.2e\n', 1/rcond);

f07pg example results

Estimate of condition number =  7.57e+01

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