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: $dl (:)$ – complex 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: $d (:)$ – complex 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: $du (:)$ – complex 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: $dl (:)$ – complex 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: $d (:)$ – complex 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: $du (:)$ – complex 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: $du2 (n - 2)$ – complex 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_zgttrs (f07cs) can be used to solve systems of equations

A X = B

A^{T} X = B

A^{H} X = B

, and nag_lapack_zgtcon (f07cu) 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_dgttrf (f07cd).

Example

This example factorizes the tridiagonal matrix

A

given by

A = (\begin{array}{r} - 1.3 + 1.3 i & 2.0 - 1.0 i & 0 & 0 & 0 \\ 1.0 - 2.0 i & - 1.3 + 1.3 i & 2.0 + 1.0 i & 0 & 0 \\ 0 & 1.0 + 1.0 i & - 1.3 + 3.3 i & - 1.0 + 1.0 i & 0 \\ 0 & 0 & 2.0 - 3.0 i & - 0.3 + 4.3 i & 1.0 - 1.0 i \\ 0 & 0 & 0 & 1.0 + 1.0 i & - 3.3 + 1.3 i \end{array}) .

Open in the MATLAB editor: f07cr_example

function f07cr_example


fprintf('f07cr example results\n\n');

% Tridiagonal matrix stored by diagonals
du = [              2   - 1i     2   + 1i    -1   + 1i     1   - 1i  ];
d  = [-1.3 + 1.3i  -1.3 + 1.3i  -1.3 + 3.3i  -0.3 + 4.3i  -3.3 + 1.3i];
dl = [ 1   - 2i     1   + 1i     2   - 3i     1   + 1i               ];

% Factorize.
[dlf, df, duf, du2, ipiv, info] = ...
  f07cr(dl, d, du);

disp('Details of factorization');
fprintf('\n');
disp(' Second super-diagonal of U');
disp(du2');
disp(' First super-diagonal of U');
disp(duf);
disp(' Main diagonal of U');
disp(df(1:4));
disp(df(5:end));
disp(' Multipliers');
disp(dlf);
disp(' Vector of interchanges');
disp(double(ipiv)');

f07cr example results

Details of factorization

 Second super-diagonal of U
   2.0000 - 1.0000i  -1.0000 - 1.0000i   1.0000 + 1.0000i

 First super-diagonal of U
  -1.3000 + 1.3000i  -1.3000 + 3.3000i  -0.3000 + 4.3000i  -3.3000 + 1.3000i

 Main diagonal of U
   1.0000 - 2.0000i   1.0000 + 1.0000i   2.0000 - 3.0000i   1.0000 + 1.0000i

  -1.3399 + 0.2875i

 Multipliers
  -0.7800 - 0.2600i   0.1620 - 0.4860i  -0.0452 - 0.0010i  -0.3979 - 0.0562i

 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