NAG Library Routine Document

F07AHF (DGERFS)

INTEGER	N, NRHS, LDA, LDAF, IPIV(*), LDB, LDX, IWORK(N), INFO
REAL (KIND=nag_wp)	A(LDA,), AF(LDAF,), B(LDB,), X(LDX,), FERR(NRHS), BERR(NRHS), WORK(3*N)
CHARACTER(1)	TRANS

The routine may be called by its LAPACK name dgerfs.

3 Description

F07AHF (DGERFS) returns the backward errors and estimated bounds on the forward errors for the solution of a real system of linear equations with multiple right-hand sides

A X = B

A^{T} X = B

. The routine handles each right-hand side vector (stored as a column of the matrix

B

) independently, so we describe the function of F07AHF (DGERFS) in terms of a single right-hand side

b

and solution

x

Given a computed solution

x

, the routine computes the component-wise backward error

β

. This is the size of the smallest relative perturbation in each element of

A

and

b

such that

x

is the exact solution of a perturbed system

\begin{matrix} (A + δ A) x = b + δ b \\ |δ a_{i j}| \leq β |a_{i j}| and |δ b_{i}| \leq β |b_{i}| . \end{matrix}

Then the routine estimates a bound for the component-wise forward error in the computed solution, defined by:

\max_{i} |x_{i} - {\hat{x}}_{i}| / \max_{i} |x_{i}|

where

\hat{x}

is the true solution.

For details of the method, see the F07 Chapter Introduction.

4 References

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

5 Parameters

1: TRANS – CHARACTER(1)Input

On entry: indicates the form of the linear equations for which

X

is the computed solution.

$TRANS ='N'$: The linear equations are of the form $A X = B$ .
$TRANS ='T'$ or $'C'$: The linear equations are of the form $A^{T} X = B$ .

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

n

n

original matrix

A

as supplied to F07ADF (DGETRF).

5: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F07AHF (DGERFS) is called.

Constraint:

LDA \geq \max (1, N)

6: AF(LDAF, $*$ ) – REAL (KIND=nag_wp) arrayInput

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

\max (1, N)

On entry: the

L U

factorization of

A

, as returned by F07ADF (DGETRF).

7: LDAF – INTEGERInput

On entry: the first dimension of the array AF as declared in the (sub)program from which F07AHF (DGERFS) is called.

Constraint:

LDAF \geq \max (1, N)

8: 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).

9: B(LDB, $*$ ) – REAL (KIND=nag_wp) arrayInput

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

10: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07AHF (DGERFS) is called.

Constraint:

LDB \geq \max (1, N)

11: X(LDX, $*$ ) – REAL (KIND=nag_wp) arrayInput/Output

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

\max (1, NRHS)

On entry: the

n

r

solution matrix

X

, as returned by F07AEF (DGETRS).

On exit: the improved solution matrix

X

12: LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which F07AHF (DGERFS) is called.

Constraint:

LDX \geq \max (1, N)

13: FERR(NRHS) – REAL (KIND=nag_wp) arrayOutput

On exit:

FERR (j)

contains an estimated error bound for the

j

th solution vector, that is, the

j

th column of

X

, for

j = 1, 2, \dots, r

14: BERR(NRHS) – REAL (KIND=nag_wp) arrayOutput

On exit:

BERR (j)

contains the component-wise backward error bound

β

for the

j

th solution vector, that is, the

j

th column of

X

, for

j = 1, 2, \dots, r

15: WORK( $3 \times N$ ) – REAL (KIND=nag_wp) arrayWorkspace

16: IWORK(N) – INTEGER arrayWorkspace

17: INFO – INTEGEROutput

On exit:

INFO = 0

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

6 Error Indicators and Warnings

Errors or warnings detected by the routine:

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

7 Accuracy

The bounds returned in FERR are not rigorous, because they are estimated, not computed exactly; but in practice they almost always overestimate the actual error.

8 Further Comments

For each right-hand side, computation of the backward error involves a minimum of

4 n^{2}

floating point operations. Each step of iterative refinement involves an additional

6 n^{2}

operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required.

Estimating the forward error involves solving a number of systems of linear equations of the form

A x = b

A^{T} x = b

; the number is usually

4

5

and never more than

11

. Each solution involves approximately

2 n^{2}

operations.

The complex analogue of this routine is F07AVF (ZGERFS).

9 Example

This example solves the system of equations

A X = B

using iterative refinement and to compute the forward and backward error bounds, 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 DocumentF07AHF (DGERFS)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F07AHF (DGERFS)