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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dptsv (f07ja)

## Purpose

nag_lapack_dptsv (f07ja) computes the solution to a real system of linear equations
 $AX=B ,$
where $A$ is an $n$ by $n$ symmetric positive definite tridiagonal matrix, and $X$ and $B$ are $n$ by $r$ matrices.

## Syntax

[d, e, b, info] = f07ja(d, e, b, 'n', n, 'nrhs_p', nrhs_p)
[d, e, b, info] = nag_lapack_dptsv(d, e, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dptsv (f07ja) factors $A$ as $A=LD{L}^{\mathrm{T}}$. The factored form of $A$ is then used to solve the system of equations.

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

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{d}\left(:\right)$ – double array
The dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The $n$ diagonal elements of the tridiagonal matrix $A$.
2:     $\mathrm{e}\left(:\right)$ – double array
The dimension of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
The $\left(n-1\right)$ subdiagonal elements of the tridiagonal matrix $A$.
3:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array
The first dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $n$ by $r$ right-hand side matrix $B$.

### Optional Input Parameters

1:     $\mathrm{n}$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: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{nrhs_p}$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: ${\mathbf{nrhs_p}}\ge 0$.

### Output Parameters

1:     $\mathrm{d}\left(:\right)$ – double array
The dimension of the array d will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The $n$ diagonal elements of the diagonal matrix $D$ from the factorization $A=LD{L}^{\mathrm{T}}$.
2:     $\mathrm{e}\left(:\right)$ – double array
The dimension of the array e will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
The $\left(n-1\right)$ subdiagonal elements of the unit bidiagonal factor $L$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$. (e can also be regarded as the superdiagonal of the unit bidiagonal factor $U$ from the ${U}^{\mathrm{T}}DU$ factorization of $A$.)
3:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – double array
The first dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array b will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
If ${\mathbf{info}}={\mathbf{0}}$, the $n$ by $r$ solution matrix $X$.
4:     $\mathrm{info}$int64int32nag_int scalar
${\mathbf{info}}=0$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
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, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b ,$
where
 $E1 = Oε A1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, 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 $Ax=b$ 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.

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 $nr$, 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
 $Ax=b ,$
where $A$ is the symmetric positive definite tridiagonal matrix
 $A = 4.0 -2.0 0 0 0 -2.0 10.0 -6.0 0 0 0 -6.0 29.0 15.0 0 0 0 15.0 25.0 8.0 0 0 0 8.0 5.0 and b = 6.0 9.0 2.0 14.0 7.0 .$
Details of the $LD{L}^{\mathrm{T}}$ factorization of $A$ are also output.
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