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: $uplo$ – string (length ≥ 1)

Specifies whether

A

is upper or lower triangular.

$uplo ='U'$: $A$ is upper triangular.
$uplo ='L'$: $A$ is lower triangular.

Constraint:

uplo ='U'

'L'

3: $diag$ – string (length ≥ 1)

Indicates whether

A

is a nonunit or unit triangular matrix.

$diag ='N'$: $A$ is a nonunit triangular matrix.
$diag ='U'$: $A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$ .

Constraint:

diag ='N'

'U'

4: $n$ – int64int32nag_int scalar

n

, the order of the matrix

A

Constraint:

n \geq 0

5: $ap (:)$ – double array

The dimension of the array ap must be at least

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

The

n

n

triangular matrix

A

, packed by columns.

More precisely,

if $uplo ='U'$ , the upper triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + j (j - 1) / 2)$ for $i \leq j$ ;
if $uplo ='L'$ , the lower triangle of $A$ must be stored with element $A_{i j}$ in $ap (i + (2 n - j) (j - 1) / 2)$ for $i \geq j$ .

diag ='U'

, the diagonal elements of

A

are assumed to be

1

, and are not referenced; the same storage scheme is used whether

diag ='N'

or ‘U’.

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_dtpcon (f07ug) 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

n^{2}

floating-point operations but takes considerably longer than a call to nag_lapack_dtptrs (f07ue) 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_ztpcon (f07uu).

Example

This example estimates the condition number in the

1

-norm of the matrix

A

, where

A = (\begin{matrix} 4.30 & 0.00 & 0.00 & 0.00 \\ - 3.96 & - 4.87 & 0.00 & 0.00 \\ 0.40 & 0.31 & - 8.02 & 0.00 \\ - 0.27 & 0.07 & - 5.95 & 0.12 \end{matrix}),

using packed storage. The true condition number in the

1

-norm is

116.41

Open in the MATLAB editor: f07ug_example

function f07ug_example


fprintf('f07ug example results\n\n');

% Estimate condition number of A, where A is Lower triangular and packed
n = int64(4);
ap = [ 4.30; -3.96;  0.40; -0.27;
             -4.87;  0.31;  0.07;
                    -8.02; -5.95;
                            0.12];

% Estimate condition number
norm_p = '1';
uplo = 'L';
diag = 'N';
[rcond, info] = f07ug( ...
                       norm_p, uplo, diag, n, ap);

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

f07ug example results

Estimate of condition number =  1.16e+02

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

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