NAG Library Routine Document

F11XNF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

+− 8 Further Comments

8.1 Timing

8.2 Use of CHECK

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

F11XNF computes a matrix-vector or conjugate transposed matrix-vector product involving a complex sparse non-Hermitian matrix stored in coordinate storage format.

2 Specification

SUBROUTINE F11XNF (

TRANS, N, NNZ, A, IROW, ICOL, CHECK, X, Y, IFAIL)

INTEGER	N, NNZ, IROW(NNZ), ICOL(NNZ), IFAIL
COMPLEX (KIND=nag_wp)	A(NNZ), X(N), Y(N)
CHARACTER(1)	TRANS, CHECK

3 Description

F11XNF computes either the matrix-vector product

y = A x

, or the conjugate transposed matrix-vector product

y = A^{H} x

, according to the value of the argument TRANS, where

A

is a complex

n

n

sparse non-Hermitian matrix, of arbitrary sparsity pattern. The matrix

A

is stored in coordinate storage (CS) format (see Section 2.1.1 in the F11 Chapter Introduction). The array A stores all the nonzero elements of

A

, while arrays IROW and ICOL store the corresponding row and column indices respectively.

It is envisaged that a common use of F11XNF will be to compute the matrix-vector product required in the application of F11BSF to sparse complex linear systems. This is illustrated in Section 9 in F11DRF.

4 References

None.

5 Parameters

1: TRANS – CHARACTER(1)Input

On entry: specifies whether or not the matrix

A

is conjugate transposed.

$TRANS ='N'$: $y = A x$ is computed.
$TRANS ='T'$: $y = A^{H} x$ is computed.

Constraint:

TRANS ='N'

'T'

2: N – INTEGERInput

On entry:

n

, the order of the matrix

A

Constraint:

N \geq 1

3: NNZ – INTEGERInput

On entry: the number of nonzero elements in the matrix

A

Constraint:

1 \leq NNZ \leq N^{2}

4: A(NNZ) – COMPLEX (KIND=nag_wp) arrayInput

On entry: the nonzero elements in the matrix

A

, ordered by increasing row index, and by increasing column index within each row. Multiple entries for the same row and column indices are not permitted. The routine F11ZNF may be used to order the elements in this way.

5: IROW(NNZ) – INTEGER arrayInput

6: ICOL(NNZ) – INTEGER arrayInput

On entry: the row and column indices of the nonzero elements supplied in array A.

Constraints:

$1 \leq IROW (i) \leq N$ and $1 \leq ICOL (i) \leq N$ , for $i = 1, 2, \dots, NNZ$ ;
$IROW (i - 1) < IROW (i)$ or $IROW (i - 1) = IROW (i)$ and $ICOL (i - 1) < ICOL (i)$ , for $i = 2, 3, \dots, NNZ$ .

7: CHECK – CHARACTER(1)Input

On entry: specifies whether or not the CS representation of the matrix

A

, values of N, NNZ, IROW and ICOL should be checked.

$CHECK ='C'$: Checks are carried on the values of N, NNZ, IROW and ICOL.
$CHECK ='N'$: None of these checks are carried out.

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	$CHECK \neq'C'$ or $'N'$ .

$IFAIL = 2$

On entry,	$N < 1$ ,
or	$NNZ < 1$ ,
or	$NNZ > N^{2}$ .

$IFAIL = 3$

On entry, the arrays IROW and ICOL fail to satisfy the following constraints:

$1 \leq IROW (i) \leq N$ and $1 \leq ICOL (i) \leq N$ , for $i = 1, 2, \dots, NNZ$ ;
$IROW (i - 1) < IROW (i)$ , or $IROW (i - 1) = IROW (i)$ and $ICOL (i - 1) < ICOL (i)$ , for $i = 2, 3, \dots, NNZ$ .

Therefore a nonzero element has been supplied which does not lie within the matrix

A

, is out of order, or has duplicate row and column indices. Call F11ZNF to reorder and sum or remove duplicates.

7 Accuracy

The computed vector

y

satisfies the error bound:

${‖y - A x‖}_{\infty} \leq c (n) ε {‖A‖}_{\infty} {‖x‖}_{\infty}$ , if $TRANS ='N'$ , or
${‖y - A^{H} x‖}_{\infty} \leq c (n) ε {‖A^{H}‖}_{\infty} {‖x‖}_{\infty}$ , if $TRANS ='T'$ ,

where

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

8 Further Comments

8.1 Timing

The time taken for a call to F11XNF is proportional to NNZ.

8.2 Use of CHECK

It is expected that a common use of F11XNF will be to compute the matrix-vector product required in the application of F11BSF to sparse complex linear systems. In this situation F11XNF is likely to be called many times with the same matrix

A

. In the interests of both reliability and efficiency you are recommended to set

CHECK ='C'

for the first of such calls, and to set

CHECK ='N'

for all subsequent calls.

9 Example

This example reads in a complex sparse matrix

A

and a vector

x

. It then calls F11XNF to compute the matrix-vector product

y = A x

and the conjugate transposed matrix-vector product

y = A^{H} x

NAG Library Routine DocumentF11XNF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

8.1 Timing

8.2 Use of CHECK

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

F11XNF