1: $d (:)$ – double array: The dimension of the array d must be at least $\max (1, n)$

Must contain the $n$ diagonal elements of the diagonal matrix $D$ from the $L D L^{H}$ factorization of $A$ .
2: $e (:)$ – complex array: The dimension of the array e must be at least $\max (1, n - 1)$

Must contain the $(n - 1)$ subdiagonal elements of the unit lower bidiagonal matrix $L$ . (e can also be regarded as the superdiagonal of the unit upper bidiagonal matrix $U$ from the $U^{H} D U$ factorization of $A$ .)
3: $anorm$ – double scalar: The $1$ -norm of the original matrix $A$ . anorm must be computed either before calling nag_lapack_zpttrf (f07jr) 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 dimension of the array d.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $rcond$ – double scalar: The reciprocal condition number, $1 / κ_{1} (A) = 1 / ({‖A‖}_{1} {‖A^{- 1}‖}_{1})$ .
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 condition number will be the exact condition number for a closely neighbouring matrix.

Further Comments

The condition number estimation requires

O (n)

floating-point operations.

See Section 15.6 of Higham (2002) for further details on computing the condition number of tridiagonal matrices.

The real analogue of this function is nag_lapack_dptcon (f07jg).

Example

This example computes the condition number of the Hermitian positive definite tridiagonal matrix

A

given by

A = (\begin{array}{r} 16.0 & 16.0 - 16.0 i & 0 & 0 \\ 16.0 + 16.0 i & 41.0 & 18.0 + 9.0 i & 0 \\ 0 & 18.0 - 9.0 i & 46.0 & 1.0 + 4.0 i \\ 0 & 0 & 1.0 - 4.0 i & 21.0 \end{array}) .

Open in the MATLAB editor: f07ju_example

function f07ju_example


fprintf('f07ju example results\n\n');

% Hermitian tridiagonal A stored as two diagonals
d = [ 16            41          46            21];
e = [ 16 + 16i      18 - 9i      1 - 4i         ];

% Factorize
[df, ef, info] = f07jr( ...
                        d, e);

% Construct matrix an with same 1-norm
an = [0 e; d; e 0];
anorm = norm(an,1);

% Get reciprocal condition number
[rcond, info] = f07ju( ...
                       df, ef, anorm);

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

f07ju example results

Condition number of A = 9.21e+03

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

Chapter Contents

Chapter Introduction

NAG Toolbox