Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug

Parameters

Compulsory Input Parameters

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

Must contain the $(n - 1)$ subdiagonal elements of the matrix $A$ .
2: $d (:)$ – double array: The dimension of the array d must be at least $\max (1, n)$

Must contain the $n$ diagonal elements of the matrix $A$ .
3: $du (:)$ – double array: The dimension of the array du must be at least $\max (1, n - 1)$

Must contain the $(n - 1)$ superdiagonal 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: $dl (:)$ – double array: The dimension of the array dl will be $\max (1, n - 1)$

Stores the $(n - 1)$ multipliers that define the matrix $L$ of the $L U$ factorization of $A$ .
2: $d (:)$ – double array: The dimension of the array d will be $\max (1, n)$

Stores the $n$ diagonal elements of the upper triangular matrix $U$ from the $L U$ factorization of $A$ .
3: $du (:)$ – double array: The dimension of the array du will be $\max (1, n - 1)$

Stores the $(n - 1)$ elements of the first superdiagonal of $U$ .
4: $du2 (n - 2)$ – double array: Contains the $(n - 2)$ elements of the second superdiagonal of $U$ .
5: $ipiv (n)$ – int64int32nag_int array: Contains the $n$ pivot indices that define the permutation matrix $P$ . At the $i$ th step, row $i$ of the matrix was interchanged with row $ipiv (i)$ . $ipiv (i)$ will always be either $i$ or $(i + 1)$ , $ipiv (i) = i$ indicating that a row interchange was not performed.
6: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

W $info > 0$: Element $_$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

Accuracy

The computed factorization satisfies an equation of the form

A + E = P L U,

where

{‖E‖}_{\infty} = O (ε) {‖A‖}_{\infty}

and

ε

is the machine precision.

Following the use of this function, nag_lapack_dgttrs (f07ce) can be used to solve systems of equations

A X = B

A^{T} X = B

, and nag_lapack_dgtcon (f07cg) 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 complex analogue of this function is nag_lapack_zgttrf (f07cr).

Example

This example factorizes the tridiagonal matrix

A

given by

A = (\begin{array}{r} 3.0 & 2.1 & 0 & 0 & 0 \\ 3.4 & 2.3 & - 1.0 & 0 & 0 \\ 0 & 3.6 & - 5.0 & 1.9 & 0 \\ 0 & 0 & 7.0 & - 0.9 & 8.0 \\ 0 & 0 & 0 & - 6.0 & 7.1 \end{array}) .

Open in the MATLAB editor: f07cd_example

function f07cd_example


fprintf('f07cd example results\n\n');

% Tridiagonal matrix A stored as diagonals:
du = [        2.1    -1.0      1.9     8.0];
d  = [3.0     2.3    -5.0     -0.9     7.1];
dl = [3.4     3.6     7.0     -6.0        ];
n  = numel(d);

% Factorize A.
[dlf, df, duf, du2f, ipiv, info] = ...
  f07cd(dl, d, du);

disp('Details of factorization');
fprintf('\n');
disp(' Second super-diagonal of U');
fprintf('%9.4f', du2f);
fprintf('\n\n');
disp(' First super-diagonal of U');
fprintf('%9.4f', duf);
fprintf('\n\n');
disp(' Main diagonal of U');
fprintf('%9.4f', df);
fprintf('\n\n');
disp(' Multipliers');
fprintf('%9.4f', dlf);
fprintf('\n\n');
disp(' Vector of interchanges');
fprintf('%9d', ipiv);
fprintf('\n');

f07cd example results

Details of factorization

 Second super-diagonal of U
  -1.0000   1.9000   8.0000

 First super-diagonal of U
   2.3000  -5.0000  -0.9000   7.1000

 Main diagonal of U
   3.4000   3.6000   7.0000  -6.0000  -1.0154

 Multipliers
   0.8824   0.0196   0.1401  -0.0148

 Vector of interchanges
        2        3        4        5        5

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

Chapter Contents

Chapter Introduction

NAG Toolbox