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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_zgetrs (f07as)

▸▿ Contents

1 Purpose

2 Syntax

3 Description

4 References

▸▿ 5 Parameters

5.1 Compulsory Input Parameters

5.2 Optional Input Parameters

5.3 Output Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

Purpose

nag_lapack_zgetrs (f07as) solves a complex system of linear equations with multiple right-hand sides,

A X = B, A^{T} X = B or A^{H} X = B,

where

A

has been factorized by nag_lapack_zgetrf (f07ar).

Syntax

[b, info] = f07as(trans, a, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

[b, info] = nag_lapack_zgetrs(trans, a, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

Description

nag_lapack_zgetrs (f07as) is used to solve a complex system of linear equations

A X = B

A^{T} X = B

A^{H} X = B

, the function must be preceded by a call to nag_lapack_zgetrf (f07ar) which computes the

L U

factorization of

A

A = P L U

. The solution is computed by forward and backward substitution.

trans ='N'

, the solution is computed by solving

P L Y = B

and then

U X = Y

trans ='T'

, the solution is computed by solving

U^{T} Y = B

and then

L^{T} P^{T} X = Y

trans ='C'

, the solution is computed by solving

U^{H} Y = B

and then

L^{H} P^{T} X = Y

References

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

Parameters

Compulsory Input Parameters

1: $trans$ – string (length ≥ 1)

Indicates the form of the equations.

$trans ='N'$: $A X = B$ is solved for $X$ .
$trans ='T'$: $A^{T} X = B$ is solved for $X$ .
$trans ='C'$: $A^{H} X = B$ is solved for $X$ .

Constraint:

trans ='N'

'T'

'C'

2: $a (lda :)$ – 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

L U

factorization of

A

, as returned by nag_lapack_zgetrf (f07ar).

3: $ipiv (:)$ – int64int32nag_int array

The dimension of the array ipiv must be at least

\max (1, n)

The pivot indices, as returned by nag_lapack_zgetrf (f07ar).

4: $b (ldb :)$ – 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 arrays a, ipiv.
$n$ , 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

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

Accuracy

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 P |L| |U|,

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

where

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

Note that

cond (A, x)

can be much smaller than

cond (A)

, and

cond (A^{H})

(which is the same as

cond (A^{T})

) can be much larger (or smaller) than

cond (A)

Forward and backward error bounds can be computed by calling nag_lapack_zgerfs (f07av), and an estimate for

κ_{\infty} (A)

can be obtained by calling nag_lapack_zgecon (f07au) with

norm_p ='I'

Further Comments

The total number of real floating-point operations is approximately

8 n^{2} r

This function may be followed by a call to nag_lapack_zgerfs (f07av) to refine the solution and return an error estimate.

The real analogue of this function is nag_lapack_dgetrs (f07ae).

Example

This example solves the system of equations

A X = B

, where

A = (\begin{array}{r} - 1.34 + 2.55 i & 0.28 + 3.17 i & - 6.39 - 2.20 i & 0.72 - 0.92 i \\ - 0.17 - 1.41 i & 3.31 - 0.15 i & - 0.15 + 1.34 i & 1.29 + 1.38 i \\ - 3.29 - 2.39 i & - 1.91 + 4.42 i & - 0.14 - 1.35 i & 1.72 + 1.35 i \\ 2.41 + 0.39 i & - 0.56 + 1.47 i & - 0.83 - 0.69 i & - 1.96 + 0.67 i \end{array})

and

B = (\begin{array}{r} 26.26 + 51.78 i & 31.32 - 6.70 i \\ 6.43 - 8.68 i & 15.86 - 1.42 i \\ - 5.75 + 25.31 i & - 2.15 + 30.19 i \\ 1.16 + 2.57 i & - 2.56 + 7.55 i \end{array}) .

Here

A

is nonsymmetric and must first be factorized by nag_lapack_zgetrf (f07ar).

Open in the MATLAB editor: f07as_example

function f07as_example


fprintf('f07as example results\n\n');

trans = 'N';
a = [-1.34 + 2.55i,  0.28 + 3.17i, -6.39 - 2.20i,  0.72 - 0.92i;
     -0.17 - 1.41i,  3.31 - 0.15i, -0.15 + 1.34i,  1.29 + 1.38i;
     -3.29 - 2.39i, -1.91 + 4.42i, -0.14 - 1.35i,  1.72 + 1.35i;
      2.41 + 0.39i, -0.56 + 1.47i, -0.83 - 0.69i, -1.96 + 0.67i];
b = [26.26 + 51.78i, 31.32 -  6.70i;
      6.43 -  8.68i, 15.86 -  1.42i;
     -5.75 + 25.31i, -2.15 + 30.19i;
      1.16 +  2.57i, -2.56 +  7.55i];

% Factorize a
[a, ipiv, info] = f07ar(a);

% Compute solution
[x, info] = f07as(trans, a, ipiv, b);

disp('Solution(s)');
fprintf('%11d        ', [1:size(b,2)]);
fprintf('\n');
disp(x);

f07as example results

Solution(s)
          1                  2        
   1.0000 + 1.0000i  -1.0000 - 2.0000i
   2.0000 - 3.0000i   5.0000 + 1.0000i
  -4.0000 - 5.0000i  -3.0000 + 4.0000i
   0.0000 + 6.0000i   2.0000 - 3.0000i

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

Chapter Contents

Chapter Introduction

NAG Toolbox