$diag ='N'$: $A$ 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

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 array b.
$r$ , the number of right-hand sides.

Constraint: $nrhs_p \geq 0$ .

Output Parameters

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

The $n$ by $r$ solution matrix $X$ .
2: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

$info < 0$: If $info = - i$ , argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.

W $info > 0$: Element $_$ of the diagonal is exactly zero. $A$ is singular and the solution has not been computed.

Accuracy

The solutions of triangular systems of equations are usually computed to high accuracy. See Higham (1989).

For each right-hand side vector

b

, the computed solution

x

is the exact solution of a perturbed system of equations

(A + E) x = b

, where

|E| \leq c (k) ε |A|,

c (k)

is a modest linear function of

k

, and

ε

is the machine precision.

\hat{x}

is the true solution, then the computed solution

x

satisfies a forward error bound of the form

\frac{{‖x - \hat{x}‖}_{\infty}}{{‖x‖}_{\infty}} \leq c (k) cond (A, x) ε,   provided c (k) cond (A, x) ε < 1,

where

cond (A, x) = {‖|A^{- 1}| |A| |x|‖}_{\infty} / {‖x‖}_{\infty}

Note that

cond (A, x) \leq cond (A) = {‖|A^{- 1}| |A|‖}_{\infty} \leq κ_{\infty} (A)

;

cond (A, x)

can be much smaller than

cond (A)

and it is also possible for

cond (A^{T})

to be much larger (or smaller) than

cond (A)

Forward and backward error bounds can be computed by calling nag_lapack_dtbrfs (f07vh), and an estimate for

κ_{\infty} (A)

can be obtained by calling nag_lapack_dtbcon (f07vg) with

norm_p ='I'

Further Comments

The total number of floating-point operations is approximately

2 n k r

k ≪ n

The complex analogue of this function is nag_lapack_ztbtrs (f07vs).

Example

This example solves the system of equations

A X = B

, 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}) .

Here

A

is treated as a lower triangular band matrix with one subdiagonal.

Open in the MATLAB editor: f07ve_example

function f07ve_example


fprintf('f07ve example results\n\n');

% Solve AX=B, 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);

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

f07ve 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

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

Chapter Contents

Chapter Introduction

NAG Toolbox