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: $ab (ldab :)$ – complex 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_zgbtrf (f07br).

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_zgbtrf (f07br).

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_zgbtrf (f07br) 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_zgbcon (f07bu) 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

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

real floating-point operations (assuming

n ≫ k_{l}

and

n ≫ k_{u}

) but takes considerably longer than a call to nag_lapack_zgbtrs (f07bs) 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_dgbcon (f07bg).

Example

This example estimates the condition number in the

1

-norm of the matrix

A

, where

A = (\begin{array}{r} - 1.65 + 2.26 i & - 2.05 - 0.85 i & 0.97 - 2.84 i & 0.00 + 0.00 i \\ 0.00 + 6.30 i & - 1.48 - 1.75 i & - 3.99 + 4.01 i & 0.59 - 0.48 i \\ 0.00 + 0.00 i & - 0.77 + 2.83 i & - 1.06 + 1.94 i & 3.33 - 1.04 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & 4.48 - 1.09 i & - 0.46 - 1.72 i \end{array}) .

Open in the MATLAB editor: f07bu_example

function f07bu_example


fprintf('f07bu example results\n\n');

m = int64(4);
kl = int64(1);
ku = int64(2);
ab =  [0    + 0i,     0    + 0i,     0.97 - 2.84i,  0.59 - 0.48i;
       0    + 0i,    -2.05 - 0.85i, -3.99 + 4.01i,  3.33 - 1.04i;
      -1.65 + 2.26i, -1.48 - 1.75i, -1.06 + 1.94i, -0.46 - 1.72i;
       0    + 6.3i,  -0.77 + 2.83i,  4.48 - 1.09i,  0    + 0i];

norm_p = 'one';
anorm = f16ub( ...
               norm_p, m, kl, ku, ab);

% Fcatorize
abf = [ complex(zeros(kl,m)); ab];
[abf, ipiv, info] = f07br( ...
                           m, kl, ku, abf);

[rcond, info] = f07bu( ...
                       norm_p, kl, ku, abf, ipiv, anorm);

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

f07bu example results

Estimate of condition number =  1.04e+02

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

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