nag_lapack_dpbrfs (f07hh) returns error bounds for the solution of a real symmetric positive definite band system of linear equations with multiple right-hand sides,

A X = B

. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

Syntax

[x, ferr, berr, info] = f07hh(uplo, kd, ab, afb, b, x, 'n', n, 'nrhs_p', nrhs_p)

[x, ferr, berr, info] = nag_lapack_dpbrfs(uplo, kd, ab, afb, b, x, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_dpbrfs (f07hh) returns the backward errors and estimated bounds on the forward errors for the solution of a real symmetric positive definite band system of linear equations with multiple right-hand sides

A X = B

. The function handles each right-hand side vector (stored as a column of the matrix

B

) independently, so we describe the function of nag_lapack_dpbrfs (f07hh) in terms of a single right-hand side

b

and solution

x

Given a computed solution

x

, the function computes the component-wise backward error

β

. This is the size of the smallest relative perturbation in each element of

A

and

b

such that

x

is the exact solution of a perturbed system

\begin{matrix} (A + δ A) x = b + δ b \\ |δ a_{i j}| \leq β |a_{i j}| and |δ b_{i}| \leq β |b_{i}| . \end{matrix}

Then the function estimates a bound for the component-wise forward error in the computed solution, defined by:

\max_{i} |x_{i} - {\hat{x}}_{i}| / \max_{i} |x_{i}|

where

\hat{x}

is the true solution.

For details of the method, see the F07 Chapter Introduction.

References

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

Parameters

Compulsory Input Parameters

1: $uplo$ – string (length ≥ 1)

Specifies whether the upper or lower triangular part of

A

is stored and how

A

is to be factorized.

$uplo ='U'$: The upper triangular part of $A$ is stored and $A$ is factorized as $U^{T} U$ , where $U$ is upper triangular.
$uplo ='L'$: The lower triangular part of $A$ is stored and $A$ is factorized as $L L^{T}$ , where $L$ is lower triangular.

Constraint:

uplo ='U'

'L'

2: $kd$ – int64int32nag_int scalar

k_{d}

, the number of superdiagonals or subdiagonals of the matrix

A

Constraint:

kd \geq 0

3: $ab (ldab :)$ – double array

The first dimension of the array ab must be at least

kd + 1

The second dimension of the array ab must be at least

\max (1, n)

The

n

n

original symmetric positive definite band matrix

A

as supplied to nag_lapack_dpbtrf (f07hd).

4: $afb (ldafb :)$ – double array

The first dimension of the array afb must be at least

kd + 1

The second dimension of the array afb must be at least

\max (1, n)

The Cholesky factor of

A

, as returned by nag_lapack_dpbtrf (f07hd).

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

r

right-hand side matrix

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

r

solution matrix

X

, as returned by nag_lapack_dpbtrs (f07he).

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the second dimension of the array ab.
$n$ , 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.

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 improved solution matrix $X$ .
2: $ferr (nrhs_p)$ – double array: $ferr (j)$ contains an estimated error bound for the $j$ th solution vector, that is, the $j$ th column of $X$ , for $j = 1, 2, \dots, r$ .
3: $berr (nrhs_p)$ – double array: $berr (j)$ contains the component-wise backward error bound $β$ for the $j$ th solution vector, that is, the $j$ th column of $X$ , for $j = 1, 2, \dots, r$ .
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 bounds returned in ferr are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.

Further Comments

For each right-hand side, computation of the backward error involves a minimum of

8 n k

floating-point operations. Each step of iterative refinement involves an additional

12 n k

operations. This assumes

n ≫ k

. At most five steps of iterative refinement are performed, but usually only one or two steps are required.

Estimating the forward error involves solving a number of systems of linear equations of the form

A x = b

; the number is usually

4

5

and never more than

11

. Each solution involves approximately

4 n k

operations.

The complex analogue of this function is nag_lapack_zpbrfs (f07hv).

Example

This example solves the system of equations

A X = B

using iterative refinement and to compute the forward and backward error bounds, where

A = (\begin{matrix} 5.49 & 2.68 & 0.00 & 0.00 \\ 2.68 & 5.63 & - 2.39 & 0.00 \\ 0.00 & - 2.39 & 2.60 & - 2.22 \\ 0.00 & 0.00 & - 2.22 & 5.17 \end{matrix}) and B = (\begin{matrix} 22.09 & 5.10 \\ 9.31 & 30.81 \\ - 5.24 & - 25.82 \\ 11.83 & 22.90 \end{matrix}) .

Here

A

is symmetric and positive definite, and is treated as a band matrix, which must first be factorized by nag_lapack_dpbtrf (f07hd).

Open in the MATLAB editor: f07hh_example

function f07hh_example


fprintf('f07hh example results\n\n');

% Symmetric banded matrix A in ab.
uplo = 'L';
kd = int64(1);
ab = [5.49,  5.63,  2.6,  5.17;
      2.68, -2.39, -2.22, 0   ];

% Factorize A
[abf, info] = f07hd( ...
                     uplo, kd, ab);

% RHS
b = [22.09,   5.10;
      9.31,  30.81;
     -5.24, -25.82;
     11.83,  22.90];
% Solve Ax = B
[x, info] = f07he( ...
                   uplo, kd, abf, b);

% Iterative refinement
[x, ferr, berr, info] = f07hh(...
                              uplo, kd, ab, abf, 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);

f07hh example results

Solution
    5.0000   -2.0000
   -2.0000    6.0000
   -3.0000   -1.0000
    1.0000    4.0000

Forward  error bounds =    2.1e-14     2.8e-14
Backward error bounds =    8.6e-17     1.1e-16

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

Chapter Contents

Chapter Introduction

NAG Toolbox