nag_lapack_dtbrfs (f07vh) returns the backward errors and estimated bounds on the forward errors for the solution of a real triangular band system of linear equations with multiple right-hand sides

A X = B

A^{T} 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_dtbrfs (f07vh) 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

A

is upper or lower triangular.

$uplo ='U'$: $A$ is upper triangular.
$uplo ='L'$: $A$ is lower triangular.

Constraint:

uplo ='U'

'L'

2: $trans$ – string (length ≥ 1)

Indicates the form of the equations.

$trans ='N'$: The equations are of the form $A X = B$ .
$trans ='T'$ or $'C'$: The equations are of the form $A^{T} X = B$ .

Constraint:

trans ='N'

'T'

'C'

3: $diag$ – string (length ≥ 1)

Indicates whether

A

is a nonunit or unit triangular matrix.

$diag ='N'$: $A$ is a nonunit triangular matrix.
$diag ='U'$: $A$ is a unit triangular matrix; the diagonal elements are not referenced and are assumed to be $1$ .

Constraint:

diag ='N'

'U'

4: $kd$ – int64int32nag_int scalar

k_{d}

, the number of superdiagonals of the matrix

A

uplo ='U'

, or the number of subdiagonals if

uplo ='L'

Constraint:

kd \geq 0

5: $ab (ldab :)$ – 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

triangular band matrix

A

The matrix is stored in rows

1

k_{d} + 1

, more precisely,

if $uplo ='U'$ , the elements of the upper triangle of $A$ within the band must be stored with element $A_{i j}$ in $ab (k_{d} + 1 + i - j, j) for \max (1, j - k_{d}) \leq i \leq j$ ;
if $uplo ='L'$ , the elements of the lower triangle of $A$ within the band must be stored with element $A_{i j}$ in $ab (1 + i - j, j) for j \leq i \leq \min (n, j + k_{d}) .$

diag ='U'

, the diagonal elements of

A

are assumed to be

1

, and are not referenced.

6: $b (ldb :)$ – 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

7: $x (ldx :)$ – 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_dtbtrs (f07ve).

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: $ferr (nrhs_p)$ – double array: $ferr (j)$ contains an estimated error bound for the $j$ th solution vector, that is, the $j$ th column of $X$ , for $j = 1, 2, \dots, r$ .
2: $berr (nrhs_p)$ – 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 $j = 1, 2, \dots, r$ .
3: $info$ – 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

A call to nag_lapack_dtbrfs (f07vh), for each right-hand side, involves solving a number of systems of linear equations of the form

A x = b

A^{T} x = b

; the number is usually

4

5

and never more than

11

. Each solution involves approximately

2 n k

floating-point operations (assuming

n ≫ k

The complex analogue of this function is nag_lapack_ztbrfs (f07vv).

Example

This example solves the system of equations

A X = B

and to compute forward and backward error bounds, where

A = (\begin{matrix} - 4.16 & 0.00 & 0.00 & 0.00 \\ - 2.25 & 4.78 & 0.00 & 0.00 \\ 0.00 & 5.86 & 6.32 & 0.00 \\ 0.00 & 0.00 & - 4.82 & 0.16 \end{matrix}) and B = (\begin{matrix} - 16.64 & - 4.16 \\ - 13.78 & - 16.59 \\ 13.10 & - 4.94 \\ - 14.14 & - 9.96 \end{matrix}) .

Open in the MATLAB editor: f07vh_example

function f07vh_example


fprintf('f07vh example results\n\n');

% Solve AX=B and compute error bounds, where A is lower triangular banded
% and stored in triangular/symmetric banded format 
kd = int64(1);
ab = [-4.16, 4.78,  6.32, 0.16;
      -2.25, 5.86, -4.82, 0.00];
b = [-16.64,  -4.16;
     -13.78, -16.59;
      13.10,  -4.94;
     -14.14,  -9.96];

% Solve
uplo  = 'L';
trans = 'N';
diag  = 'N';
[x, info] = f07ve( ...
                   uplo, trans, diag, kd, ab, b);

% Compute error bounds
[ferr, berr, info] = f07vh( ...
                            uplo, trans, diag, kd, ab, b, x);

% Display solution
[ifail] = x04ca( ...
                 'Gen', diag, x, 'Solution(s)');

fprintf('\nBackward errors (machine-dependent)\n   ')
fprintf('%11.1e', berr);
fprintf('\nEstimated forward error bounds (machine-dependent)\n   ')
fprintf('%11.1e', ferr);
fprintf('\n');

f07vh example results

 Solution(s)
             1          2
 1      4.0000     1.0000
 2     -1.0000    -3.0000
 3      3.0000     2.0000
 4      2.0000    -2.0000

Backward errors (machine-dependent)
       4.7e-17    2.5e-17
Estimated forward error bounds (machine-dependent)
       5.4e-14    5.8e-14

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

Chapter Contents

Chapter Introduction

NAG Toolbox