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.

$uplo ='U'$: $A = U^{H} U$ , where $U$ is upper triangular.
$uplo ='L'$: $A = L L^{H}$ , where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

2: $kd$ – int64int32nag_int scalar

k_{d}

, the number of superdiagonals or subdiagonals of the matrix

A

Constraint:

kd \geq 0

3: $ab (ldab :)$ – complex array

The first dimension of the array ab must be at least

kd + 1

The second dimension of the array ab must be at least

\max (1, n)

The Cholesky factor of

A

, as returned by nag_lapack_zpbtrf (f07hr).

4: $anorm$ – double scalar

The

1

-norm of the original matrix

A

. anorm must be computed either before calling nag_lapack_zpbtrf (f07hr) 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 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_zpbcon (f07hu) involves solving a number of systems of linear equations of the form

A x = b

; the number is usually

5

and never more than

11

. Each solution involves approximately

16 n k

real floating-point operations (assuming

n ≫ k

) but takes considerably longer than a call to nag_lapack_zpbtrs (f07hs) 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_dpbcon (f07hg).

Example

This example estimates the condition number in the

1

-norm (or

\infty

-norm) of the matrix

A

, where

A = (\begin{array}{r} 9.39 + 0.00 i & 1.08 - 1.73 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ 1.08 + 1.73 i & 1.69 + 0.00 i & - 0.04 + 0.29 i & 0.00 + 0.00 i \\ 0.00 + 0.00 i & - 0.04 - 0.29 i & 2.65 + 0.00 i & - 0.33 + 2.24 i \\ 0.00 + 0.00 i & 0.00 + 0.00 i & - 0.33 - 2.24 i & 2.17 + 0.00 i \end{array}) .

Here

A

is Hermitian positive definite, and is treated as a band matrix, which must first be factorized by nag_lapack_zpbtrf (f07hr). The true condition number in the

1

-norm is

153.45

Open in the MATLAB editor: f07hu_example

function f07hu_example


fprintf('f07hu example results\n\n');

uplo = 'L';
kd = int64(1);
n  = int64(4);
ab = [ 9.39 + 0i,     1.69 + 0i,      2.65 + 0i,      2.17 + 0i;
       1.08 - 1.73i, -0.04 + 0.29i,  -0.33 + 2.24i    0    + 0i];

% Factorize
[abf, info] = f07hr( ...
                     uplo, kd, ab);

% To calculate 1-norm here, need to add superdiagonal
abn = [0 + 0i,  ab(2,1:n-1);
       ab];
% 1-norm of A = 1-norm of abn
  anorm = norm(abn,1);

% Get reciprocal condition number
[rcond, info] = f07hu( ...
                       uplo, kd, abf, anorm);

fprintf('Estimate of Condition number = %7.2e\n',1/rcond);

f07hu example results

Estimate of Condition number = 1.32e+02

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

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