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.

$uplo ='U'$: $A = U^{T} U$ , where $U$ is upper triangular.
$uplo ='L'$: $A = L L^{T}$ , 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: $ab (ldab :)$ – double 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_dpbtrf (f07hd).

4: $anorm$ – double scalar

The

1

-norm of the original matrix

A

. anorm must be computed either before calling nag_lapack_dpbtrf (f07hd) 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_dpbcon (f07hg) involves solving a number of systems of linear equations of the form

A x = b

; the number is usually

4

5

and never more than

11

. Each solution involves approximately

4 n k

floating-point operations (assuming

n ≫ k

) but takes considerably longer than a call to nag_lapack_dpbtrs (f07he) 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_zpbcon (f07hu).

Example

This example estimates the condition number in the

1

-norm (or

\infty

-norm) of the matrix

A

, where

A = (\begin{matrix} 5.49 & 2.68 & 0.00 & 0.00 \\ 2.68 & 5.63 & - 2.39 & 0.00 \\ 0.00 & - 2.39 & 2.60 & - 2.22 \\ 0.00 & 0.00 & - 2.22 & 5.17 \end{matrix}) .

Here

A

is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by nag_lapack_dpbtrf (f07hd). The true condition number in the

1

-norm is

74.15

Open in the MATLAB editor: f07hg_example

function f07hg_example


fprintf('f07hg example results\n\n');

% Symmetric banded matrix A in ab.
uplo = 'L';
kd  = int64(1);
ab  = [5.49,  5.63,  2.60, 5.17;
       2.68, -2.39, -2.22, 0.00];

% To calculate 1-norm here, need to add superdiagonal
abn = [0.00,  2.68, -2.39, -2.22;
       ab];
% 1-norm of A = 1-norm of abn
anorm = norm(abn,1);

% Factorize A
[abf, info] = f07hd( ...
                     uplo, kd, ab);

% Get reciprocal condition number
[rcond, info] = f07hg( ...
                       uplo, kd, abf, anorm);

fprintf('Condition number of A = %7.1f\n',1/rcond);

f07hg example results

Condition number of A =    74.2

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

Chapter Contents

Chapter Introduction

NAG Toolbox