$diag ='N'$: $A$ 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: $a (lda :)$ – complex array

The first dimension of the array a must be at least

\max (1, n)

The second dimension of the array a must be at least

\max (1, n)

The

n

n

triangular matrix

A

If $uplo ='U'$ , $a$ is upper triangular and the elements of the array below the diagonal are not referenced.
If $uplo ='L'$ , $a$ is lower triangular and the elements of the array above the diagonal are not referenced.
If $diag ='U'$ , the diagonal elements of $a$ are assumed to be $1$ , and are not referenced.

5: $b (ldb :)$ – complex 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 first dimension of the arrays a, b and the second dimension of the array a.
$n$ , 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: $b (ldb :)$ – complex 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 (n) ε |A|,

c (n)

is a modest linear function of

n

, 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 (n) cond (A, x) ε,   provided c (n) 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^{H})

, which is the same as

cond (A^{T})

, to be much larger (or smaller) than

cond (A)

Forward and backward error bounds can be computed by calling nag_lapack_ztrrfs (f07tv), and an estimate for

κ_{\infty} (A)

can be obtained by calling nag_lapack_ztrcon (f07tu) with

norm_p ='I'

Further Comments

The total number of real floating-point operations is approximately

4 n^{2} r

The real analogue of this function is nag_lapack_dtrtrs (f07te).

Example

This example solves the system of equations

A X = B

, where

A = (\begin{array}{r} 4.78 + 4.56 i & 0.00 + 0.00 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ 2.00 - 0.30 i & - 4.11 + 1.25 i & 0.00 + 0.00 i & 0.00 + 0.00 i \\ 2.89 - 1.34 i & 2.36 - 4.25 i & 4.15 + 0.80 i & 0.00 + 0.00 i \\ - 1.89 + 1.15 i & 0.04 - 3.69 i & - 0.02 + 0.46 i & 0.33 - 0.26 i \end{array})

and

B = (\begin{array}{r} - 14.78 - 32.36 i & - 18.02 + 28.46 i \\ 2.98 - 2.14 i & 14.22 + 15.42 i \\ - 20.96 + 17.06 i & 5.62 + 35.89 i \\ 9.54 + 9.91 i & - 16.46 - 1.73 i \end{array}) .

Open in the MATLAB editor: f07ts_example

function f07ts_example


fprintf('f07ts example results\n\n');

% Solve AX=B where A is Lower triangular
a = [  4.78 +  4.56i,   0    +  0i,     0    + 0i,    0    + 0i;
       2.00 -  0.30i,  -4.11 +  1.25i,  0    + 0i,    0    + 0i;
       2.89 -  1.34i,   2.36 -  4.25i,  4.15 + 0.8i,  0    + 0i;
      -1.89 +  1.15i,   0.04 -  3.69i, -0.02 + 0.46i, 0.33 - 0.26i];
b = [-14.78 - 32.36i, -18.02 + 28.46i;
       2.98 -  2.14i,  14.22 + 15.42i;
     -20.96 + 17.06i,   5.62 + 35.89i;
       9.54 +  9.91i, -16.46 -  1.73i];

% Solve
uplo  = 'L';
trans = 'N';
diag  = 'N';
[x, info] = f07ts( ...
                   uplo, trans, diag, a, b);

% Display solution
disp('Solution(s)');
disp(x);

f07ts example results

Solution(s)
  -5.0000 - 2.0000i   1.0000 + 5.0000i
  -3.0000 - 1.0000i  -2.0000 - 2.0000i
   2.0000 + 1.0000i   3.0000 + 4.0000i
   4.0000 + 3.0000i   4.0000 - 3.0000i

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

Chapter Contents

Chapter Introduction

NAG Toolbox