nag_sparse_nherm_precon_ilu_solve (Nag_TransType trans, Integer n, const Complex a[], Integer la, const Integer irow[], const Integer icol[], const Integer ipivp[], const Integer ipivq[], const Integer istr[], const Integer idiag[], Nag_SparseNsym_CheckData check, const Complex y[], Complex x[], NagError *fail)

3

Description

nag_sparse_nherm_precon_ilu_solve (f11dpc) solves a system of complex linear equations

M x = y,   or M^{T} x = y,

according to the value of the argument trans, where the matrix

M = P L D U Q

corresponds to an incomplete

L U

decomposition of a complex sparse matrix stored in coordinate storage (CS) format (see Section 2.1.1 in the f11 Chapter Introduction), as generated by nag_sparse_nherm_fac (f11dnc).

In the above decomposition

L

is a lower triangular sparse matrix with unit diagonal elements,

D

is a diagonal matrix,

U

is an upper triangular sparse matrix with unit diagonal elements and,

P

and

Q

are permutation matrices.

L

D

and

U

are supplied to nag_sparse_nherm_precon_ilu_solve (f11dpc) through the matrix

C = L + D^{- 1} + U - 2 I

which is an n by n sparse matrix, stored in CS format, as returned by nag_sparse_nherm_fac (f11dnc). The permutation matrices

P

and

Q

are returned from nag_sparse_nherm_fac (f11dnc) via the arrays ipivp and ipivq.

It is envisaged that a common use of nag_sparse_nherm_precon_ilu_solve (f11dpc) will be to carry out the preconditioning step required in the application of nag_sparse_nherm_basic_solver (f11bsc) to sparse complex linear systems. nag_sparse_nherm_precon_ilu_solve (f11dpc) is used for this purpose by the Black Box function nag_sparse_nherm_fac_sol (f11dqc).

nag_sparse_nherm_precon_ilu_solve (f11dpc) may also be used in combination with nag_sparse_nherm_fac (f11dnc) to solve a sparse system of complex linear equations directly (see Section 9.5 in nag_sparse_nherm_fac (f11dnc)). This use of nag_sparse_nherm_precon_ilu_solve (f11dpc) is illustrated in Section 10.

4

References

None.

5

Arguments

1: $trans$ – Nag_TransTypeInput

On entry: specifies whether or not the matrix

M

is transposed.

$trans = Nag_NoTrans$: $M x = y$ is solved.
$trans = Nag_Trans$: $M^{T} x = y$ is solved.

Constraint:

trans = Nag_NoTrans

Nag_Trans

2: $n$ – IntegerInput

On entry:

n

, the order of the matrix

M

. This must be the same value as was supplied in the preceding call to nag_sparse_nherm_fac (f11dnc).

Constraint:

n \geq 1

3: $a [la]$ – const ComplexInput

On entry: the values returned in the array a by a previous call to nag_sparse_nherm_fac (f11dnc).

4: $la$ – IntegerInput

On entry: the dimension of the arrays a, irow and icol. This must be the same value supplied in the preceding call to nag_sparse_nherm_fac (f11dnc).

5: $irow [la]$ – const IntegerInput

6: $icol [la]$ – const IntegerInput

7: $ipivp [n]$ – const IntegerInput

8: $ipivq [n]$ – const IntegerInput

9: $istr [n + 1]$ – const IntegerInput

10: $idiag [n]$ – const IntegerInput

On entry: the values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to nag_sparse_nherm_fac (f11dnc).

11: $check$ – Nag_SparseNsym_CheckDataInput

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

M

should be checked.

$check = Nag_SparseNsym_Check$: Checks are carried on the values of n, irow, icol, ipivp, ipivq, istr and idiag.
$check = Nag_SparseNsym_NoCheck$: None of these checks are carried out.

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 1$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_INVALID_CS: On entry, $i = 〈value〉$ , $icol [i - 1] = 〈value〉$ , and $n = 〈value〉$ .
Constraint: $icol [i - 1] \geq 1$ and $icol [i - 1] \leq n$ .
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_precon_ilu_solve (f11dpc) and nag_sparse_nherm_fac (f11dnc).

On entry, $i = 〈value〉$ , $irow [i - 1] = 〈value〉$ , $n = 〈value〉$ .
Constraint: $irow [i - 1] \geq 1$ and $irow [i - 1] \leq n$ .
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_precon_ilu_solve (f11dpc) and nag_sparse_nherm_fac (f11dnc).
NE_INVALID_CS_PRECOND: On entry, $idiag [i - 1]$ appears to be incorrect: $i = 〈value〉$ .
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_precon_ilu_solve (f11dpc) and nag_sparse_nherm_fac (f11dnc).

On entry, istr appears to be invalid.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_precon_ilu_solve (f11dpc) and nag_sparse_nherm_fac (f11dnc).

On entry, $istr [i - 1]$ is inconsistent with irow: $i = 〈value〉$ .
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_precon_ilu_solve (f11dpc) and nag_sparse_nherm_fac (f11dnc).
NE_INVALID_ROWCOL_PIVOT: On entry, $i = 〈value〉$ , $ipivp [i - 1] = 〈value〉$ , $n = 〈value〉$ .
Constraint: $ipivp [i - 1] \geq 1$ and $ipivp [i - 1] \leq n$ .
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_precon_ilu_solve (f11dpc) and nag_sparse_nherm_fac (f11dnc).

On entry, $i = 〈value〉$ , $ipivq [i - 1] = 〈value〉$ , $n = 〈value〉$ .
Constraint: $ipivq [i - 1] \geq 1$ and $ipivq [i - 1] \leq n$ .
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_precon_ilu_solve (f11dpc) and nag_sparse_nherm_fac (f11dnc).

On entry, $ipivp [i - 1]$ is a repeated value: $i = 〈value〉$ .
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_precon_ilu_solve (f11dpc) and nag_sparse_nherm_fac (f11dnc).

On entry, $ipivq [i - 1]$ is a repeated value: $i = 〈value〉$ .
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_precon_ilu_solve (f11dpc) and nag_sparse_nherm_fac (f11dnc).
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_NOT_STRICTLY_INCREASING: On entry, $a [i - 1]$ is out of order: $i = 〈value〉$ .
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_precon_ilu_solve (f11dpc) and nag_sparse_nherm_fac (f11dnc).

On entry, the location ( $irow [i - 1], icol [i - 1]$ ) is a duplicate: $i = 〈value〉$ .
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nherm_precon_ilu_solve (f11dpc) and nag_sparse_nherm_fac (f11dnc).

7

Accuracy

trans = Nag_NoTrans

the computed solution

x

is the exact solution of a perturbed system of equations

(M + δ M) x = y

, where

|δ M| \leq c (n) ε P |L| |D| |U| Q,

c (n)

is a modest linear function of

n

, and

ε

is the machine precision. An equivalent result holds when

trans = Nag_Trans

8

Parallelism and Performance

nag_sparse_nherm_precon_ilu_solve (f11dpc) is not threaded in any implementation.

9

Further Comments

9.1

Timing

The time taken for a call to nag_sparse_nherm_precon_ilu_solve (f11dpc) is proportional to the value of nnzc returned from nag_sparse_nherm_fac (f11dnc).

9.2

Use of check

It is expected that a common use of nag_sparse_nherm_precon_ilu_solve (f11dpc) will be to carry out the preconditioning step required in the application of nag_sparse_nherm_basic_solver (f11bsc) to sparse complex linear systems. In this situation nag_sparse_nherm_precon_ilu_solve (f11dpc) is likely to be called many times with the same matrix

M

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

check = Nag_SparseNsym_Check

for the first of such calls, and to set

check = Nag_SparseNsym_NoCheck

for all subsequent calls.

10

Example

This example reads in a complex sparse non-Hermitian matrix

A

and a vector

y

. It then calls nag_sparse_nherm_fac (f11dnc), with

lfill = - 1

and

dtol = 0.0

, to compute the complete

L U

decomposition

A = P L D U Q .

Finally it calls nag_sparse_nherm_precon_ilu_solve (f11dpc) to solve the system

P L D U Q x = y .

NAG Library Function Document

nag_sparse_nherm_precon_ilu_solve (f11dpc)

▸▿ 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

9.1 Timing

9.2 Use of check

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification