NAG Library Routine Document

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

Fortran Interface

Subroutine f07fhf (

uplo, n, nrhs, a, lda, af, ldaf, b, ldb, x, ldx, ferr, berr, work, iwork, info)

Integer, Intent (In)	::	n, nrhs, lda, ldaf, ldb, ldx
Integer, Intent (Out)	::	iwork(n), info
Real (Kind=nag_wp), Intent (In)	::	a(lda,), af(ldaf,), b(ldb,*)
Real (Kind=nag_wp), Intent (Inout)	::	x(ldx,*)
Real (Kind=nag_wp), Intent (Out)	::	ferr(nrhs), berr(nrhs), work(3*n)
Character (1), Intent (In)	::	uplo

C Header Interface

#include <nagmk26.h>

void

f07fhf_ (const char *uplo, const Integer *n, const Integer *nrhs, const double a[], const Integer *lda, const double af[], const Integer *ldaf, const double b[], const Integer *ldb, double x[], const Integer *ldx, double ferr[], double berr[], double work[], Integer iwork[], Integer *info, const Charlen length_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

Arguments

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

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

7

Accuracy

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

Parallelism and Performance

f07fhf (dporfs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

f07fhf (dporfs) 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

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).

10

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 Document

f07fhf (dporfs)

▸▿ 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