NAG Library Routine Document

F11MFF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

F11MFF solves a real sparse system of linear equations with multiple right-hand sides given an

L U

factorization of the sparse matrix computed by F11MEF.

2 Specification

SUBROUTINE F11MFF (

TRANS, N, IPRM, IL, LVAL, IU, UVAL, NRHS, B, LDB, IFAIL)

INTEGER	N, IPRM(7N), IL(), IU(*), NRHS, LDB, IFAIL
REAL (KIND=nag_wp)	LVAL(), UVAL(), B(LDB,*)
CHARACTER(1)	TRANS

3 Description

F11MFF solves a real system of linear equations with multiple right-hand sides

A X = B

A^{T} X = B

, according to the value of the parameter TRANS, where the matrix factorization

P_{r} A P_{c} = L U

corresponds to an

L U

decomposition of a sparse matrix stored in compressed column (Harwell–Boeing) format, as computed by F11MEF.

In the above decomposition

L

is a lower triangular sparse matrix with unit diagonal elements and

U

is an upper triangular sparse matrix;

P_{r}

and

P_{c}

are permutation matrices.

4 References

None.

5 Parameters

1: $TRANS$ – CHARACTER(1)Input

On entry: specifies whether

A X = B

A^{T} X = B

is solved.

$TRANS ='N'$: $A X = B$ is solved.
$TRANS ='T'$: $A^{T} X = B$ is solved.

Constraint:

TRANS ='N'

'T'

2: $N$ – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 0

3: $IPRM (7 \times N)$ – INTEGER arrayInput

On entry: the column permutation which defines

P_{c}

, the row permutation which defines

P_{r}

, plus associated data structures as computed by F11MEF.

4: $IL (*)$ – INTEGER arrayInput

Note: the dimension of the array IL must be at least as large as the dimension of the array of the same name in F11MEF.

On entry: records the sparsity pattern of matrix

L

as computed by F11MEF.

5: $LVAL (*)$ – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array LVAL must be at least as large as the dimension of the array of the same name in F11MEF.

On entry: records the nonzero values of matrix

L

and some nonzero values of matrix

U

as computed by F11MEF.

6: $IU (*)$ – INTEGER arrayInput

Note: the dimension of the array IU must be at least as large as the dimension of the array of the same name in F11MEF.

On entry: records the sparsity pattern of matrix

U

as computed by F11MEF.

7: $UVAL (*)$ – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array UVAL must be at least as large as the dimension of the array of the same name in F11MEF.

On entry: records some nonzero values of matrix

U

as computed by F11MEF.

8: $NRHS$ – INTEGERInput

On entry:

nrhs

, the number of right-hand sides in

B

Constraint:

NRHS \geq 0

9: $B (LDB *)$ – 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

NRHS

right-hand side matrix

B

On exit: the

N

NRHS

solution matrix

X

10: $LDB$ – INTEGERInput

On entry: the first dimension of the array B as declared in the (sub)program from which F11MFF is called.

Constraint:

LDB \geq \max (1, N)

11: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$TRANS \neq'N'$ or $'T'$ ,
or	$N < 0$ ,
or	$NRHS < 0$ ,
or	$LDB < \max (1, N)$ .

$IFAIL = 2$: Ill-defined row permutation in array $IPRM$ . Internal checks have revealed that the $IPRM$ array is corrupted.

$IFAIL = 3$: Ill-defined column permutations in array $IPRM$ . Internal checks have revealed that the $IPRM$ array is corrupted.

$IFAIL = 301$: Unable to allocate required internal workspace.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

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 |L| |U|,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision, when partial pivoting is used.

\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 F11MHF, and an estimate for

κ_{\infty} (A)

can be obtained by calling F11MGF.

8 Parallelism and Performance

F11MFF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

F11MFF 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

F11MFF may be followed by a call to F11MHF to refine the solution and return an error estimate.

10 Example

This example solves the system of equations

A X = B

, where

A = (\begin{matrix} 2.00 & 1.00 & 0 & 0 & 0 \\ 0 & 0 & 1.00 & - 1.00 & 0 \\ 4.00 & 0 & 1.00 & 0 & 1.00 \\ 0 & 0 & 0 & 1.00 & 2.00 \\ 0 & - 2.00 & 0 & 0 & 3.00 \end{matrix}) and B = (\begin{matrix} 1.56 & 3.12 \\ - 0.25 & - 0.50 \\ 3.60 & 7.20 \\ 1.33 & 2.66 \\ 0.52 & 1.04 \end{matrix}) .

Here

A

is nonsymmetric and must first be factorized by F11MEF.

NAG Library Routine DocumentF11MFF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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

NAG Library Routine Document

F11MFF