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

A = L D L^{T},

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,

\hat{X}

, to the required equations. Letting

\hat{x}

denote a column of

\hat{X}

, nag_lapack_dptrfs (f07jh) computes a component-wise backward error,

β

, the smallest relative perturbation in each element of

A

and

b

such that

\hat{x}

is the exact solution of a perturbed system

(A + E) \hat{x} = b + f, with |e_{i j}| \leq β |a_{i j}|, and |f_{j}| \leq β |b_{j}| .

The function also estimates a bound for the component-wise forward error in the computed solution defined by

\max |x_{i} - \hat{x_{i}}| / \max |\hat{x_{i}}|

, 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 = U^{T} D U,

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

Must contain the $n$ diagonal elements of the matrix 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 matrix $A$ .
3: $df (:)$ – double array: The dimension of the array df 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 $A$ .
4: $ef (:)$ – double array: The dimension of the array ef must be at least $\max (1, n)$

Must contain the $(n - 1)$ subdiagonal elements of the unit bidiagonal matrix $L$ from the $L D L^{T}$ factorization of $A$ .
5: $b (ldb :)$ – 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 $B$ .
6: $x (ldx :)$ – double array: The first dimension of the array x must be at least $\max (1, n)$ .
The second dimension of the array x must be at least $\max (1, nrhs_p)$ .

The $n$ by $r$ initial solution matrix $X$ .

Optional Input Parameters

1: $n$ – 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: $n \geq 0$ .
2: $nrhs_p$ – 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: $nrhs_p \geq 0$ .

Output Parameters

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

The $n$ by $r$ refined solution matrix $X$ .
2: $ferr (nrhs_p)$ – double array: Estimate of the forward error bound for each computed solution vector, such that ${‖{\hat{x}}_{j} - x_{j}‖}_{\infty} / {‖{\hat{x}}_{j}‖}_{\infty} \leq ferr (j)$ , where ${\hat{x}}_{j}$ 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: $berr (nrhs_p)$ – double array: Estimate of the component-wise relative backward error of each computed solution vector ${\hat{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\hat{x}}_{j}$ an exact solution).
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.

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‖}_{\infty} = O (ε) {‖A‖}_{\infty}

and

ε

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

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

where

κ (A) = {‖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

Further Comments

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

A X = B

is proportional to

n r

. 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

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 & 10.0 \\ 9.0 & 4.0 \\ 2.0 & 9.0 \\ 14.0 & 65.0 \\ 7.0 & 23.0 \end{matrix}) .

Estimates for the backward errors and forward errors are also output.

Open in the MATLAB editor: f07jh_example

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

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

Chapter Contents

Chapter Introduction

NAG Toolbox