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: $norm_p$ – string (length ≥ 1)

Indicates whether

κ_{1} (A)

κ_{\infty} (A)

is estimated.

$norm_p ='1'$ or $'O'$: $κ_{1} (A)$ is estimated.
$norm_p ='I'$: $κ_{\infty} (A)$ is estimated.

Constraint:

norm_p ='1'

'O'

'I'

2: $a (lda :)$ – double array

The first dimension of the array a must be at least

\max (1, n)

The second dimension of the array a must be at least

\max (1, n)

The

L U

factorization of

A

, as returned by nag_lapack_dgetrf (f07ad).

3: $anorm$ – double scalar

norm_p ='1'

'O'

, the

1

-norm of the original matrix

A

norm_p ='I'

, the

\infty

-norm of the original matrix

A

anorm must be computed either before calling nag_lapack_dgetrf (f07ad) or else from a copy of the original matrix

A

(see Example).

Constraint:

anorm \geq 0.0

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array a and the second dimension of the array a.
$n$ , 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_dgecon (f07ag) involves solving a number of systems of linear equations of the form

A x = b

A^{T} 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_dgetrs (f07ae) with one right-hand side, because extra care is taken to avoid overflow when

A

is approximately singular.

The complex analogue of this function is nag_lapack_zgecon (f07au).

Example

This example estimates the condition number in the

1

-norm of the matrix

A

, where

A = (\begin{matrix} 1.80 & 2.88 & 2.05 & - 0.89 \\ 5.25 & - 2.95 & - 0.95 & - 3.80 \\ 1.58 & - 2.69 & - 2.90 & - 1.04 \\ - 1.11 & - 0.66 & - 0.59 & 0.80 \end{matrix}) .

Here

A

is nonsymmetric and must first be factorized by nag_lapack_dgetrf (f07ad). The true condition number in the

1

-norm is

152.16

Open in the MATLAB editor: f07ag_example

function f07ag_example


fprintf('f07ag example results\n\n');

a = [ 1.80,  2.88,  2.05, -0.89;
      5.25, -2.95, -0.95, -3.80;
      1.58, -2.69, -2.90, -1.04;
     -1.11, -0.66, -0.59,  0.80];

anorm = norm(a, 1);

% Factorize A
[a, ipiv, info] = f07ad(a);

% Estimate condition number
norm_p = '1';
[rcond, info] = f07ag(norm_p, a, anorm);

if rcond > x02aj
  fprintf('Estimate of condition number = %10.2e\n', 1/rcond);
else
  fprintf('A is singular to working precision\n');
end

f07ag example results

Estimate of condition number =   1.52e+02

PDF version (NAG web site, 64-bit version, 64-bit version)

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