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.

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

Constraint:

uplo ='U'

'L'

2: $n$ – int64int32nag_int scalar

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $ap (:)$ – double array

The dimension of the array ap must be at least

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

The Cholesky factor of

A

stored in packed form, as returned by nag_lapack_dpptrf (f07gd).

4: $anorm$ – double scalar

The

1

-norm of the original matrix

A

. anorm must be computed either before calling nag_lapack_dpptrf (f07gd) or else from a copy of the original matrix

A

Constraint:

anorm \geq 0.0

Optional Input Parameters

None.

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_dppcon (f07gg) 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_dpptrs (f07ge) 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_zppcon (f07gu).

Example

This example estimates the condition number in the

1

-norm (or

\infty

-norm) of the matrix

A

, where

A = (\begin{matrix} 4.16 & - 3.12 & 0.56 & - 0.10 \\ - 3.12 & 5.03 & - 0.83 & 1.18 \\ 0.56 & - 0.83 & 0.76 & 0.34 \\ - 0.10 & 1.18 & 0.34 & 1.18 \end{matrix}) .

Here

A

is symmetric positive definite, stored in packed form, and must first be factorized by nag_lapack_dpptrf (f07gd). The true condition number in the

1

-norm is

97.32

Open in the MATLAB editor: f07gg_example

function f07gg_example


fprintf('f07gg example results\n\n');

% Symmetric matrix A, lower triangular part packed in ap
uplo = 'L';
n = int64(4);
ap = [4.16 -3.12  0.56 -0.10 ...
            5.03 -0.83  1.18 ...
                  0.76  0.34 ...
                        1.18];

% Factorize
[L, info] = f07gd( ...
                   uplo, n, ap);

% 1-norm given by second row/column
anorm = 10.16;

% Condition number estimator
[rcond, info] = f07gg( ...
                       uplo, n, ap, anorm);

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

f07gg example results

Estimate of condition number =  2.65e+01

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

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