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: $a (lda :)$ – 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)

Details of the factorization of

A

, as returned by nag_lapack_dsytrf (f07md).

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_dsytrf (f07md).

4: $anorm$ – double scalar

The

1

-norm of the original matrix

A

. anorm must be computed either before calling nag_lapack_dsytrf (f07md) 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 arrays a, 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_dsycon (f07mg) 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_dsytrs (f07me) 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_zhecon (f07mu) for Hermitian matrices and nag_lapack_zsycon (f07nu) 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 and must first be factorized by nag_lapack_dsytrf (f07md). The true condition number in the

1

-norm is

75.68

Open in the MATLAB editor: f07mg_example

function f07mg_example


fprintf('f07mg example results\n\n');

% Symmetric indefinite matrix A, lower triangle stored.
uplo = 'L';
a    = [ 2.07   0      0      0;
         3.87  -0.21   0      0;
         4.20   1.87   1.15   0;
        -1.15   0.63   2.06  -1.81];

% Get 1-norm of A
afull = a  + a' - diag(diag(a));
anorm = norm(afull,1);

% Factorize A
[L, ipiv, info] = f07md( ...
                         uplo, a);

% Get estimate of repciprocal of condition number
[rcond, info] = f07mg( ...
                       uplo, L, ipiv, anorm);

fprintf('Estimate of condition number = %8.2e\n',1/rcond);

f07mg example results

Estimate of condition number = 7.57e+01

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

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