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: $kl$ – int64int32nag_int scalar

k_{l}

, the number of subdiagonals within the band of the matrix

A

Constraint:

kl \geq 0

3: $ku$ – int64int32nag_int scalar

k_{u}

, the number of superdiagonals within the band of the matrix

A

Constraint:

ku \geq 0

4: $ab (ldab :)$ – double array

The first dimension of the array ab must be at least

2 \times kl + ku + 1

The second dimension of the array ab must be at least

\max (1, n)

The

L U

factorization of

A

, as returned by nag_lapack_dgbtrf (f07bd).

5: $ipiv (:)$ – int64int32nag_int array

The dimension of the array ipiv must be at least

\max (1, n)

The pivot indices, as returned by nag_lapack_dgbtrf (f07bd).

6: $anorm$ – double scalar

norm_p ='1'

'O'

, the

1

-norm of the original matrix

A

norm_p ='I'

, the

\infty

-norm of the original matrix

A

anorm must be computed either before calling nag_lapack_dgbtrf (f07bd) or else from a copy of the original matrix

A

(see Example).

Constraint:

anorm \geq 0.0

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the second dimension of the array ab.
$n$ , 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_dgbcon (f07bg) 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

2 n (2 k_{l} + k_{u})

floating-point operations (assuming

n ≫ k_{l}

and

n ≫ k_{u}

) but takes considerably longer than a call to nag_lapack_dgbtrs (f07be) 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_zgbcon (f07bu).

Example

This example estimates the condition number in the

1

-norm of the matrix

A

, where

A = (\begin{matrix} - 0.23 & 2.54 & - 3.66 & 0.00 \\ - 6.98 & 2.46 & - 2.73 & - 2.13 \\ 0.00 & 2.56 & 2.46 & 4.07 \\ 0.00 & 0.00 & - 4.78 & - 3.82 \end{matrix}) .

Here

A

is nonsymmetric and is treated as a band matrix, which must first be factorized by nag_lapack_dgbtrf (f07bd). The true condition number in the

1

-norm is

56.40

Open in the MATLAB editor: f07bg_example

function f07bg_example


fprintf('f07bg example results\n\n');

m  = int64(4);
kl = int64(1);
ku = int64(2);
ab = [ 0,    0,    -3.66, -2.13;
       0,    2.54, -2.73,  4.07;
      -0.23, 2.46,  2.46, -3.82;
      -6.98, 2.56, -4.78,  0];

norm_p = 'O';
anorm = f16rb(norm_p, m, kl, ku, ab);

ab = [zeros(kl,m); ab];
  
% Factorize A
[abf, ipiv, info] = f07bd( ...
                           m, kl, ku, ab);

% Estimate condition number
[rcond, info] = f07bg( ...
                       norm_p, kl, ku, abf, ipiv, anorm);

if rcond > x02aj
  fprintf('Estimate of condition number = %10.2e\n', 1/rcond);
else
  fprintf('A is singular to working precision\n');
end

f07bg example results

Estimate of condition number =   5.64e+01

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

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