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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_dptrfs (f07jh)

## Purpose

nag_lapack_dptrfs (f07jh) computes error bounds and refines 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, using the modified Cholesky factorization returned by nag_lapack_dpttrf (f07jd) and an initial solution returned by nag_lapack_dpttrs (f07je). Iterative refinement is used to reduce the backward error as much as possible.

## Syntax

[x, ferr, berr, info] = f07jh(d, e, df, ef, b, x, 'n', n, 'nrhs_p', nrhs_p)
[x, ferr, berr, info] = nag_lapack_dptrfs(d, e, df, ef, b, x, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_dptrfs (f07jh) should normally be preceded by calls to nag_lapack_dpttrf (f07jd) and nag_lapack_dpttrs (f07je). nag_lapack_dpttrf (f07jd) computes a modified Cholesky factorization of the matrix $A$ as
 $A=LDLT ,$
where $L$ is a unit lower bidiagonal matrix and $D$ is a diagonal matrix, with positive diagonal elements. nag_lapack_dpttrs (f07je) then utilizes the factorization to compute a solution, $\stackrel{^}{X}$, to the required equations. Letting $\stackrel{^}{x}$ denote a column of $\stackrel{^}{X}$, nag_lapack_dptrfs (f07jh) computes a component-wise backward error, $\beta$, the smallest relative perturbation in each element of $A$ and $b$ such that $\stackrel{^}{x}$ is the exact solution of a perturbed system
 $A+E x^ = b + f , with eij ≤ β aij , and fj ≤ β bj .$
The function also estimates a bound for the component-wise forward error in the computed solution defined by $\mathrm{max}\left|{x}_{i}-\stackrel{^}{{x}_{i}}\right|/\mathrm{max}\left|\stackrel{^}{{x}_{i}}\right|$, where $x$ is the corresponding column of the exact solution, $X$.
Note that the modified Cholesky factorization of $A$ can also be expressed as
 $A=UTDU ,$
where $U$ is unit upper bidiagonal.

## 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:     $\mathrm{d}\left(:\right)$ – double array
The dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Must contain the $n$ diagonal elements of the matrix of $A$.
2:     $\mathrm{e}\left(:\right)$ – double array
The dimension of the array e must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$
Must contain the $\left(n-1\right)$ subdiagonal elements of the matrix $A$.
3:     $\mathrm{df}\left(:\right)$ – double array
The dimension of the array df must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Must contain the $n$ diagonal elements of the diagonal matrix $D$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$.
4:     $\mathrm{ef}\left(:\right)$ – double array
The dimension of the array ef must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
Must contain the $\left(n-1\right)$ subdiagonal elements of the unit bidiagonal matrix $L$ from the $LD{L}^{\mathrm{T}}$ factorization of $A$.
5:     $\mathrm{b}\left(\mathit{ldb},:\right)$ – 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 $r$ matrix of right-hand sides $B$.
6:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – double array
The first dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $n$ by $r$ initial solution matrix $X$.

### Optional Input Parameters

1:     $\mathrm{n}$int64int32nag_int scalar
Default: the first dimension of the arrays b, x and the dimension of the arrays d, df, ef.
$n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathrm{nrhs_p}$int64int32nag_int scalar
Default: the second dimension of the arrays b, x.
$r$, 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:     $\mathrm{x}\left(\mathit{ldx},:\right)$ – double array
The first dimension of the array x will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The second dimension of the array x will be $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs_p}}\right)$.
The $n$ by $r$ refined solution matrix $X$.
2:     $\mathrm{ferr}\left({\mathbf{nrhs_p}}\right)$ – double array
Estimate of the forward error bound for each computed solution vector, such that ${‖{\stackrel{^}{x}}_{j}-{x}_{j}‖}_{\infty }/{‖{\stackrel{^}{x}}_{j}‖}_{\infty }\le {\mathbf{ferr}}\left(j\right)$, where ${\stackrel{^}{x}}_{j}$ is the $j$th column of the computed solution returned in the array x and ${x}_{j}$ is the corresponding column of the exact solution $X$. The estimate is almost always a slight overestimate of the true error.
3:     $\mathrm{berr}\left({\mathbf{nrhs_p}}\right)$ – double array
Estimate of the component-wise relative backward error of each computed solution vector ${\stackrel{^}{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\stackrel{^}{x}}_{j}$ an exact solution).
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.

## Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b ,$
where
 $E∞=OεA∞$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^ - x ∞ x∞ ≤ κA E∞ A∞ ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{\infty }{‖A‖}_{\infty }$, 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.
Function nag_lapack_dptcon (f07jg) can be used to compute the condition number of $A$.

The total number of floating-point operations required to solve the equations $AX=B$ is proportional to $nr$. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
The complex analogue of this function is nag_lapack_zptrfs (f07jv).

## Example

This example solves the equations
 $AX=B ,$
where $A$ is the symmetric positive definite tridiagonal matrix
 $A = 4.0 -2.0 0 0 0 -2.0 10.0 -6.0 0 0 0 -6.0 29.0 15.0 0 0 0 15.0 25.0 8.0 0 0 0 8.0 5.0 and B = 6.0 10.0 9.0 4.0 2.0 9.0 14.0 65.0 7.0 23.0 .$
Estimates for the backward errors and forward errors are also output.
```function f07jh_example

fprintf('f07jh 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);

% Refine
ef = [ef 0];
[x, ferr, berr, info] = f07jh( ...
d, e, df, ef, b, x);

disp('Solution');
disp(x);
fprintf('Forward  error bounds = %10.1e  %10.1e\n',ferr);
fprintf('Backward error bounds = %10.1e  %10.1e\n',berr);

```
```f07jh example results

Solution
2.5000    2.0000
2.0000   -1.0000
1.0000   -3.0000
-1.0000    6.0000
3.0000   -5.0000

Forward  error bounds =    2.4e-14     4.7e-14
Backward error bounds =    0.0e+00     7.4e-17
```