NAG Library Routine Document

F07FHF (DPORFS)

+− 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

1 Purpose

F07FHF (DPORFS) returns error bounds for the solution of a real symmetric positive definite system of linear equations with multiple right-hand sides,

A X = B

. It improves the solution by iterative refinement, in order to reduce the backward error as much as possible.

2 Specification

SUBROUTINE F07FHF (

UPLO, N, NRHS, A, LDA, AF, LDAF, B, LDB, X, LDX, FERR, BERR, WORK, IWORK, INFO)

INTEGER	N, NRHS, LDA, LDAF, 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)	UPLO

The routine may be called by its LAPACK name dporfs.

3 Description

F07FHF (DPORFS) returns the backward errors and estimated bounds on the forward errors for the solution of a real symmetric positive definite system of linear equations with multiple right-hand sides

A 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 F07FHF (DPORFS) 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: UPLO – CHARACTER(1)Input

On entry: specifies whether the upper or lower triangular part of

A

is stored and how

A

is to be factorized.

$UPLO ='U'$: The upper triangular part of $A$ is stored and $A$ is factorized as $U^{T} U$ , where $U$ is upper triangular.
$UPLO ='L'$: The lower triangular part of $A$ is stored and $A$ is factorized as $L L^{T}$ , where $L$ is lower triangular.

Constraint:

UPLO ='U'

'L'

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 symmetric positive definite matrix

A

as supplied to F07FDF (DPOTRF).

5: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F07FHF (DPORFS) 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 Cholesky factor of

A

, as returned by F07FDF (DPOTRF).

7: LDAF – INTEGERInput

On entry: the first dimension of the array AF as declared in the (sub)program from which F07FHF (DPORFS) is called.

Constraint:

LDAF \geq \max (1, N)

8: 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

9: LDB – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F07FHF (DPORFS) is called.

Constraint:

LDB \geq \max (1, N)

10: 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 F07FEF (DPOTRS).

On exit: the improved solution matrix

X

11: LDX – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which F07FHF (DPORFS) is called.

Constraint:

LDX \geq \max (1, N)

12: 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

13: 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

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

15: IWORK(N) – INTEGER arrayWorkspace

16: 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

; 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 F07FVF (ZPORFS).

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} 4.16 & - 3.12 & 0.56 & - 0.10 \\ - 3.12 & 5.03 & - 0.83 & 1.18 \\ 0.56 & - 0.83 & 0.76 & 0.34 \\ - 0.10 & 1.18 & 0.34 & 1.18 \end{matrix}) and B = (\begin{matrix} 8.70 & 8.30 \\ - 13.35 & 2.13 \\ 1.89 & 1.61 \\ - 4.14 & 5.00 \end{matrix}) .

Here

A

is symmetric positive definite and must first be factorized by F07FDF (DPOTRF).

NAG Library Routine DocumentF07FHF (DPORFS)

+− 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

F07FHF (DPORFS)