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 matrix $A$ .
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 matrix $A$ .

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

Stores 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 will be $\max (1, n - 1)$

Stores the $(n - 1)$ subdiagonal elements of the lower bidiagonal matrix $L$ . (e can also be regarded as containing the $(n - 1)$ superdiagonal elements of the upper bidiagonal matrix $U$ .)
3: $info$ – 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.

$info > 0 and info < n$: The leading minor of order $_$ is not positive definite, the factorization could not be completed.

$info > 0 and info = n$: The leading minor of order $n$ is not positive definite, the factorization was completed, but $d (n) \leq 0$ .

Accuracy

The computed factorization satisfies an equation of the form

A + E = L D L^{H},

where

{‖E‖}_{\infty} = O (ε) {‖A‖}_{\infty}

and

ε

is the machine precision.

Following the use of this function, nag_lapack_zpttrs (f07js) can be used to solve systems of equations

A X = B

, and nag_lapack_zptcon (f07ju) can be used to estimate the condition number of

A

Further Comments

The total number of floating-point operations required to factorize the matrix

A

is proportional to

n

The real analogue of this function is nag_lapack_dpttrf (f07jd).

Example

This example factorizes 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: f07jr_example

function f07jr_example


fprintf('f07jr 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);

disp('Details of factorization');
disp('The diagonal elements of D');
disp(df);
disp('Sub-diagonal elements of the Cholesky factor L');
disp(ef);

f07jr example results

Details of factorization
The diagonal elements of D
    16     9     1     4

Sub-diagonal elements of the Cholesky factor L
   1.0000 + 1.0000i   2.0000 - 1.0000i   1.0000 - 4.0000i

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

Chapter Contents

Chapter Introduction

NAG Toolbox