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Chapter Contents

Chapter Introduction

NAG Toolbox

NAG Toolbox: nag_lapack_dgetrs (f07ae)

▸▿ 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_dgetrs (f07ae) solves a real system of linear equations with multiple right-hand sides,

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

where

A

has been factorized by nag_lapack_dgetrf (f07ad).

Syntax

[b, info] = f07ae(trans, a, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

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

Description

nag_lapack_dgetrs (f07ae) is used to solve a real system of linear equations

A X = B

A^{T} X = B

, the function must be preceded by a call to nag_lapack_dgetrf (f07ad) 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'

'C'

, the solution is computed by solving

U^{T} Y = B

and then

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

Constraint:

trans ='N'

'T'

'C'

2: $a (lda :)$ – double 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_dgetrf (f07ad).

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_dgetrf (f07ad).

4: $b (ldb :)$ – 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 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: $b (ldb :)$ – 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

$info < 0$: 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^{T})

can be much larger (or smaller) than

cond (A)

Forward and backward error bounds can be computed by calling nag_lapack_dgerfs (f07ah), and an estimate for

κ_{\infty} (A)

can be obtained by calling nag_lapack_dgecon (f07ag) with

norm_p ='I'

Further Comments

The total number of floating-point operations is approximately

2 n^{2} r

This function may be followed by a call to nag_lapack_dgerfs (f07ah) to refine the solution and return an error estimate.

The complex analogue of this function is nag_lapack_zgetrs (f07as).

Example

This example solves the system of equations

A X = B

, where

A = (\begin{matrix} 1.80 & 2.88 & 2.05 & - 0.89 \\ 5.25 & - 2.95 & - 0.95 & - 3.80 \\ 1.58 & - 2.69 & - 2.90 & - 1.04 \\ - 1.11 & - 0.66 & - 0.59 & 0.80 \end{matrix}) and B = (\begin{matrix} 9.52 & 18.47 \\ 24.35 & 2.25 \\ 0.77 & - 13.28 \\ - 6.22 & - 6.21 \end{matrix}) .

Here

A

is nonsymmetric and must first be factorized by nag_lapack_dgetrf (f07ad).

Open in the MATLAB editor: f07ae_example

function f07ae_example


fprintf('f07ae example results\n\n');

trans = 'N';
a = [1.8, 2.88, 2.05, -0.89;
     5.25, -2.95, -0.95, -3.8;
     1.58, -2.69, -2.9, -1.04;
     -1.11, -0.66, -0.59, 0.8];
b = [9.52, 18.47;
     24.35, 2.25;
     0.77, -13.28;
     -6.22, -6.21];

% Factorize A
[LU, ipiv, info] = f07ad(a);

% Compute Solution
[x, info] = f07ae(trans, LU, ipiv, b);

[ifail] = x04ca( ...
                 'General', ' ', x, 'Solution(s)');

f07ae example results

 Solution(s)
             1          2
 1      1.0000     3.0000
 2     -1.0000     2.0000
 3      3.0000     4.0000
 4     -5.0000     1.0000

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Chapter Contents

Chapter Introduction

NAG Toolbox