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.

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

Constraint:

uplo ='U'

'L'

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 Cholesky factor of

A

, as returned by nag_lapack_zpotrf (f07fr).

3: $anorm$ – double scalar

The

1

-norm of the original matrix

A

. anorm must be computed either before calling nag_lapack_zpotrf (f07fr) 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 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_zpocon (f07fu) 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

8 n^{2}

real floating-point operations but takes considerably longer than a call to nag_lapack_zpotrs (f07fs) 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_dpocon (f07fg).

Example

This example estimates the condition number in the

1

-norm (or

\infty

-norm) of the matrix

A

, where

A = (\begin{array}{r} 3.23 + 0.00 i & 1.51 - 1.92 i & 1.90 + 0.84 i & 0.42 + 2.50 i \\ 1.51 + 1.92 i & 3.58 + 0.00 i & - 0.23 + 1.11 i & - 1.18 + 1.37 i \\ 1.90 - 0.84 i & - 0.23 - 1.11 i & 4.09 + 0.00 i & 2.33 - 0.14 i \\ 0.42 - 2.50 i & - 1.18 - 1.37 i & 2.33 + 0.14 i & 4.29 + 0.00 i \end{array}) .

Here

A

is Hermitian positive definite and must first be factorized by nag_lapack_zpotrf (f07fr). The true condition number in the

1

-norm is

201.92

Open in the MATLAB editor: f07fu_example

function f07fu_example


fprintf('f07fu example results\n\n');

% Hermitian matrix A
uplo = 'L';
a = [ 3.23,          1.51 - 1.92i,  1.90 + 0.84i,  0.42 + 2.50i;
      1.51 + 1.92i,  3.58 + 0i,    -0.23 + 1.11i, -1.18 + 1.37i;
      1.90 - 0.84i, -0.23 - 1.11i,  4.09 + 0i,     2.33 - 0.14i;
      0.42 - 2.50i, -1.18 - 1.37i,  2.33 + 0.14i,  4.29 + 0i];
anorm = norm(a, 1);

% Factorize A
[L, info] = f07fr( ...
                   uplo, a);

% Condition number estimator
[rcond, info] = f07fu( ...
                       uplo, L, anorm);

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

f07fu example results

Estimate of condition number =  1.51e+02

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