NAG Library Routine Document

Integer, Intent (In)	::	n, nrhs, lda, ipiv(*), ldb
Integer, Intent (Out)	::	info
Real (Kind=nag_wp), Intent (In)	::	a(lda,*)
Real (Kind=nag_wp), Intent (Inout)	::	b(ldb,*)
Character (1), Intent (In)	::	trans

C Header Interface

#include <nagmk26.h>

void	f07aef_ (const char trans, const Integer n, const Integer nrhs, const double a[], const Integer lda, const Integer ipiv[], double b[], const Integer ldb, Integer info, const Charlen length_trans)

The routine may be called by its LAPACK name dgetrs.

3

Description

f07aef (dgetrs) is used to solve a real system of linear equations

A X = B

A^{T} X = B

, the routine must be preceded by a call to f07adf (dgetrf) 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

4

References

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

5

Arguments

1: $trans$ – Character(1)Input

On entry: 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: $n$ – IntegerInput

On entry:

n

, the order of the matrix

A

Constraint:

n \geq 0

3: $nrhs$ – IntegerInput

On entry:

r

, the number of right-hand sides.

Constraint:

nrhs \geq 0

4: $a (lda *)$ – Real (Kind=nag_wp) arrayInput

Note: the second dimension of the array a must be at least

\max (1, n)

On entry: the

L U

factorization of

A

, as returned by f07adf (dgetrf).

5: $lda$ – IntegerInput

On entry: the first dimension of the array a as declared in the (sub)program from which f07aef (dgetrs) is called.

Constraint:

lda \geq \max (1, n)

6: $ipiv (*)$ – Integer arrayInput

Note: the dimension of the array ipiv must be at least

\max (1, n)

On entry: the pivot indices, as returned by f07adf (dgetrf).

7: $b (ldb *)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the second dimension of the array b must be at least

\max (1, nrhs)

On entry: the

n

r

right-hand side matrix

B

On exit: the

n

r

solution matrix

X

8: $ldb$ – IntegerInput

On entry: the first dimension of the array b as declared in the (sub)program from which f07aef (dgetrs) is called.

Constraint:

ldb \geq \max (1, n)

9: $info$ – IntegerOutput

On exit:

info = 0

unless the routine detects an error (see Section 6).

6

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.

7

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 f07ahf (dgerfs), and an estimate for

κ_{\infty} (A)

can be obtained by calling f07agf (dgecon) with

norm ='I'

8

Parallelism and Performance

f07aef (dgetrs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07aef (dgetrs) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

The total number of floating-point operations is approximately

2 n^{2} r

This routine may be followed by a call to f07ahf (dgerfs) to refine the solution and return an error estimate.

The complex analogue of this routine is f07asf (zgetrs).

10

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 f07adf (dgetrf).

NAG Library Routine Document

f07aef (dgetrs)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

6

Error Indicators and Warnings

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

10.2

Program Data

10.3

Program Results