NAG Library Routine Document

F07FVF (ZPORFS)

+− 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

F07FVF (ZPORFS) returns error bounds for the solution of a complex Hermitian 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 F07FVF (

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

INTEGER	N, NRHS, LDA, LDAF, LDB, LDX, INFO
REAL (KIND=nag_wp)	FERR(NRHS), BERR(NRHS), RWORK(N)
COMPLEX (KIND=nag_wp)	A(LDA,), AF(LDAF,), B(LDB,), X(LDX,), WORK(2*N)
CHARACTER(1)	UPLO

The routine may be called by its LAPACK name zporfs.

3 Description

F07FVF (ZPORFS) returns the backward errors and estimated bounds on the forward errors for the solution of a complex Hermitian 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 F07FVF (ZPORFS) 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 $U^{H} U$ , where $U$ is upper triangular.
$UPLO ='L'$: The lower triangular part of $A$ is stored and $A$ is factorized as $L L^{H}$ , 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, $*$ ) – COMPLEX (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 Hermitian positive definite matrix

A

as supplied to F07FRF (ZPOTRF).

5: LDA – INTEGERInput

On entry: the first dimension of the array A as declared in the (sub)program from which F07FVF (ZPORFS) is called.

Constraint:

LDA \geq \max (1, N)

6: AF(LDAF, $*$ ) – COMPLEX (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 F07FRF (ZPOTRF).

7: LDAF – INTEGERInput

On entry: the first dimension of the array AF as declared in the (sub)program from which F07FVF (ZPORFS) is called.

Constraint:

LDAF \geq \max (1, N)

8: B(LDB, $*$ ) – COMPLEX (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 F07FVF (ZPORFS) is called.

Constraint:

LDB \geq \max (1, N)

10: X(LDX, $*$ ) – COMPLEX (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 F07FSF (ZPOTRS).

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 F07FVF (ZPORFS) 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( $2 \times N$ ) – COMPLEX (KIND=nag_wp) arrayWorkspace

15: RWORK(N) – REAL (KIND=nag_wp) 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

16 n^{2}

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

24 n^{2}

real 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

5

and never more than

11

. Each solution involves approximately

8 n^{2}

real operations.

The real analogue of this routine is F07FHF (DPORFS).

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{array}{r} 3.23 + 0.00 i & 1.51 - 1.92 i & 1.90 + 0.84 i & 0.42 + 2.50 i \\ 1.51 + 1.92 i & 3.58 + 0.00 i & - 0.23 + 1.11 i & - 1.18 + 1.37 i \\ 1.90 - 0.84 i & - 0.23 - 1.11 i & 4.09 + 0.00 i & 2.33 - 0.14 i \\ 0.42 - 2.50 i & - 1.18 - 1.37 i & 2.33 + 0.14 i & 4.29 + 0.00 i \end{array})

and

B = (\begin{array}{r} 3.93 - 6.14 i & 1.48 + 6.58 i \\ 6.17 + 9.42 i & 4.65 - 4.75 i \\ - 7.17 - 21.83 i & - 4.91 + 2.29 i \\ 1.99 - 14.38 i & 7.64 - 10.79 i \end{array}) .

Here

A

is Hermitian positive definite and must first be factorized by F07FRF (ZPOTRF).

NAG Library Routine DocumentF07FVF (ZPORFS)

+− 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

F07FVF (ZPORFS)