is a diagonal matrix, with positive diagonal elements. nag_lapack_dpttrs (f07je) then utilizes the factorization to solve the required equations. Note that the factorization may also be regarded as having the form

U^{T} D U

, where

U

is a unit upper bidiagonal matrix.

References

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: $d (:)$ – 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 $L D L^{T}$ factorization of $A$ .
2: $e (:)$ – double 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 $U^{T} D U$ factorization of $A$ .)
3: $b (ldb :)$ – double array: The first dimension of the array b must be at least $\max (1, n)$ .
The second dimension of the array b must be at least $\max (1, nrhs_p)$ .

The $n$ by $r$ matrix of right-hand sides $B$ .

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the array b and the dimension of the array d.
$n$ , the order of the matrix $A$ .

Constraint: $n \geq 0$ .
2: $nrhs_p$ – int64int32nag_int scalar: Default: the second dimension of the array b.
$r$ , the number of right-hand sides, i.e., the number of columns of the matrix $B$ .

Constraint: $nrhs_p \geq 0$ .

Output Parameters

1: $b (ldb :)$ – double array: The first dimension of the array b will be $\max (1, n)$ .
The second dimension of the array b will be $\max (1, nrhs_p)$ .

The $n$ by $r$ solution matrix $X$ .
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 solution for a single right-hand side,

\hat{x}

, satisfies an equation of the form

(A + E) \hat{x} = b,

where

{‖E‖}_{1} = O (ε) {‖A‖}_{1}

and

ε

is the machine precision. An approximate error bound for the computed solution is given by

\frac{{‖\hat{x} - x‖}_{1}}{{‖x‖}_{1}} \leq κ (A) \frac{{‖E‖}_{1}}{{‖A‖}_{1}},

where

κ (A) = {‖A^{- 1}‖}_{1} {‖A‖}_{1}

, the condition number of

A

with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.

Following the use of this function nag_lapack_dptcon (f07jg) can be used to estimate the condition number of

A

and nag_lapack_dptrfs (f07jh) can be used to obtain approximate error bounds.

Further Comments

The total number of floating-point operations required to solve the equations

A X = B

is proportional to

n r

The complex analogue of this function is nag_lapack_zpttrs (f07js).

Example

This example solves the equations

A X = B,

where

A

is the symmetric positive definite tridiagonal matrix

A = (\begin{array}{r} 4.0 & - 2.0 & 0 & 0 & 0 \\ - 2.0 & 10.0 & - 6.0 & 0 & 0 \\ 0 & - 6.0 & 29.0 & 15.0 & 0 \\ 0 & 0 & 15.0 & 25.0 & 8.0 \\ 0 & 0 & 0 & 8.0 & 5.0 \end{array}) and B = (\begin{array}{r} 6.0 & 10.0 \\ 9.0 & 4.0 \\ 2.0 & 9.0 \\ 14.0 & 65.0 \\ 7.0 & 23.0 \end{array}) .

Open in the MATLAB editor: f07je_example

function f07je_example


fprintf('f07je example results\n\n');

% Symmetric tridiagonal A stored as two diagonals
d = [ 4     10     29     25     5];
e = [-2     -6     15     8       ];

% RHS
b = [ 6, 10;
      9,  4;
      2,  9;
     14, 65;
      7, 23];

% Factorize
[df, ef, info] = f07jd( ...
                        d, e);

%Solve
[x, info] = f07je( ...
                   df, ef, b);

disp('Solution(s)');
disp(x);

f07je example results

Solution(s)
    2.5000    2.0000
    2.0000   -1.0000
    1.0000   -3.0000
   -1.0000    6.0000
    3.0000   -5.0000

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

Chapter Contents

Chapter Introduction

NAG Toolbox