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

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

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

The $n$ diagonal elements of the tridiagonal matrix $A$ .
2: $e (:)$ – double array: The dimension of the array e must be at least $\max (1, n - 1)$

The $(n - 1)$ subdiagonal elements of the tridiagonal matrix $A$ .
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 $r$ right-hand side matrix $B$ .

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

The $n$ diagonal elements of the diagonal matrix $D$ from the factorization $A = L D L^{T}$ .
2: $e (:)$ – double array: The dimension of the array e will be $\max (1, n - 1)$

The $(n - 1)$ subdiagonal elements of the unit bidiagonal factor $L$ from the $L D L^{T}$ factorization of $A$ . (e can also be regarded as the superdiagonal of the unit bidiagonal factor $U$ from the $U^{T} D U$ factorization of $A$ .)
3: $b (ldb :)$ – double array: The first dimension of the array b will be $\max (1, n)$ .
The second dimension of the array b will be $\max (1, nrhs_p)$ .

If $info = 0$ , the $n$ by $r$ solution matrix $X$ .
4: $info$ – 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.

$info > 0$: The leading minor of order $_$ is not positive definite, and the solution has not been computed.

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.

nag_lapack_dptsvx (f07jb) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_real_posdef_tridiag_solve (f04bg) solves

A x = b

and returns a forward error bound and condition estimate. nag_linsys_real_posdef_tridiag_solve (f04bg) calls nag_lapack_dptsv (f07ja) to solve the equations.

Further Comments

The number of floating-point operations required for the factorization of

A

is proportional to

n

, and the number of floating-point operations required for the solution of the equations is proportional to

n r

, where

r

is the number of right-hand sides.

The complex analogue of this function is nag_lapack_zptsv (f07jn).

Example

This example solves the equations

A x = b,

where

A

is the symmetric positive definite tridiagonal matrix

A = (\begin{matrix} 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{matrix}) and b = (\begin{matrix} 6.0 \\ 9.0 \\ 2.0 \\ 14.0 \\ 7.0 \end{matrix}) .

Details of the

L D L^{T}

factorization of

A

are also output.

Open in the MATLAB editor: f07ja_example

function f07ja_example


fprintf('f07ja 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;
      9;
      2;
     14;
      7];

% Solve
[df, ef, x, info] = f07ja( ...
                           d, e, b);

disp('Solution');
disp(x');
disp('Diagonal elements of the diagonal matrix D');
disp(df);
disp('Sub-diagonal elements of the Cholesky factor L');
disp(ef);

f07ja example results

Solution
    2.5000    2.0000    1.0000   -1.0000    3.0000

Diagonal elements of the diagonal matrix D
     4     9    25    16     1

Sub-diagonal elements of the Cholesky factor L
   -0.5000   -0.6667    0.6000    0.5000

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

Chapter Contents

Chapter Introduction

NAG Toolbox