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: $a (lda :)$ – complex 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_zgetrf (f07ar).

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_zgetrf (f07ar) 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_zgecon (f07au) involves solving a number of systems of linear equations of the form

A x = b

A^{H} x = b

; the number is usually

5

and never more than

11

. Each solution involves approximately

8 n^{2}

real floating-point operations but takes considerably longer than a call to nag_lapack_zgetrs (f07as) 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_lapack_dgecon (f07ag).

Example

This example estimates the condition number in the

1

-norm of the matrix

A

, where

A = (\begin{array}{r} - 1.34 + 2.55 i & 0.28 + 3.17 i & - 6.39 - 2.20 i & 0.72 - 0.92 i \\ - 0.17 - 1.41 i & 3.31 - 0.15 i & - 0.15 + 1.34 i & 1.29 + 1.38 i \\ - 3.29 - 2.39 i & - 1.91 + 4.42 i & - 0.14 - 1.35 i & 1.72 + 1.35 i \\ 2.41 + 0.39 i & - 0.56 + 1.47 i & - 0.83 - 0.69 i & - 1.96 + 0.67 i \end{array}) .

Here

A

is nonsymmetric and must first be factorized by nag_lapack_zgetrf (f07ar). The true condition number in the

1

-norm is

231.86

Open in the MATLAB editor: f07au_example

function f07au_example


fprintf('f07au example results\n\n');

a = [-1.34 + 2.55i,  0.28 + 3.17i, -6.39 - 2.20i,  0.72 - 0.92i;
     -0.17 - 1.41i,  3.31 - 0.15i, -0.15 + 1.34i,  1.29 + 1.38i;
     -3.29 - 2.39i, -1.91 + 4.42i, -0.14 - 1.35i,  1.72 + 1.35i;
      2.41 + 0.39i, -0.56 + 1.47i, -0.83 - 0.69i, -1.96 + 0.67i];

norm_p = '1';
anorm = norm(a, 1);

% Factorise a
[LU, ipiv, info] = f07ar(a);

% Estimate condition number
[rcond, info] = f07au( ...
                       norm_p, LU, anorm);

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

f07au example results


Estimate of condition number =   1.50e+02

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