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: $a (lda :)$ – 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

n

n

triangular matrix

A

If $uplo ='U'$ , $a$ is upper triangular and the elements of the array below the diagonal are not referenced.
If $uplo ='L'$ , $a$ is lower triangular and the elements of the array above the diagonal are not referenced.
If $diag ='U'$ , the diagonal elements of $a$ are assumed to be $1$ , and are not referenced.

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_ztrcon (f07tu) 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

4 n^{2}

real floating-point operations but takes considerably longer than a call to nag_lapack_ztrtrs (f07ts) 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_dtrcon (f07tg).

Example

This example estimates the condition number in the

1

-norm of the matrix

A

, where

A = (\begin{array}{r} 4.78 + 4.56 i & 0.00 + 0.00 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ 2.00 - 0.30 i & - 4.11 + 1.25 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ 2.89 - 1.34 i & 2.36 - 4.25 i & 4.15 + 0.80 i & 0.00 + 0.00 i \\ - 1.89 + 1.15 i & 0.04 - 3.69 i & - 0.02 + 0.46 i & 0.33 - 0.26 i \end{array}) .

The true condition number in the

1

-norm is

70.27

Open in the MATLAB editor: f07tu_example

function f07tu_example


fprintf('f07tu example results\n\n');

% Estimate condition number of A, where A is Lower triangular
a = [  4.78 +  4.56i,   0    +  0i,     0    + 0i,    0    + 0i;
       2.00 -  0.30i,  -4.11 +  1.25i,  0    + 0i,    0    + 0i;
       2.89 -  1.34i,   2.36 -  4.25i,  4.15 + 0.8i,  0    + 0i;
      -1.89 +  1.15i,   0.04 -  3.69i, -0.02 + 0.46i, 0.33 - 0.26i];


% Get reciprocal condition number
norm_p = '1';
uplo = 'L';
diag = 'N';
[rcond, info] = f07tu( ...
                       norm_p, uplo, diag, a);

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

f07tu example results

Estimate of condition number =  3.74e+01

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

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