void	f11dbc (Nag_TransType trans, Integer n, const double 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 double y[], double x[], NagError *fail)

The function may be called by the names: f11dbc, nag_sparse_real_gen_precon_ilu_solve or nag_sparse_nsym_precon_ilu_solve.

3 Description

f11dbc solves a system of linear equations

\begin{matrix} M x = y, & or & M^{T} x = y, \end{matrix}

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 sparse matrix stored in coordinate storage (CS) format (see Section 2.1.1 in the F11 Chapter Introduction), as generated by f11dac.

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 f11dbc 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 f11dac. The permutation matrices

P

and

Q

are returned from f11dac via the arrays ipivp and ipivq.

It is envisaged that a common use of f11dbc will be to carry out the preconditioning step required in the application of f11bec to sparse linear systems. f11dbc is used for this purpose by the Black Box function f11dcc.

f11dbc may also be used in combination with f11dac to solve a sparse system of linear equations directly (see Section 9.5 in f11dac). This use of f11dbc is demonstrated in Section 10.

4 References

None.

5 Arguments

1: $trans$ – Nag_TransType Input

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: $n$ – Integer Input

On entry:

n

, the order of the matrix

M

. This must be the same value as was supplied in the preceding call to f11dac.

Constraint:

n \geq 1

3: $a [la]$ – const double Input

On entry: the values returned in the array a by a previous call to f11dac.

4: $la$ – Integer Input

On entry: the dimension of the arrays a, irow and icol. This must be the same value returned by the preceding call to f11dac.

5: $irow [la]$ – const Integer Input

6: $icol [la]$ – const Integer Input

7: $ipivp [n]$ – const Integer Input

8: $ipivq [n]$ – const Integer Input

9: $istr [n + 1]$ – const Integer Input

10: $idiag [n]$ – const Integer Input

On entry: the values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to f11dac.

11: $check$ – Nag_SparseNsym_CheckData Input

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

Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dbc and f11dac.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_CS: On entry, $i = 〈value〉$ , $icol [i - 1] = 〈value〉$ , and $n = 〈value〉$ .
Constraint: $icol [i - 1] \geq 1$ and $icol [i - 1] \leq n$ .

On entry, $i = 〈value〉$ , $irow [i - 1] = 〈value〉$ , $n = 〈value〉$ .
Constraint: $irow [i - 1] \geq 1$ and $irow [i - 1] \leq n$ .
NE_INVALID_CS_PRECOND: On entry, $idiag [i - 1]$ appears to be incorrect: $i = 〈value〉$ .

On entry, istr appears to be invalid.

On entry, $istr [i - 1]$ is inconsistent with irow: $i = 〈value〉$ .
NE_INVALID_ROWCOL_PIVOT: On entry, $i = 〈value〉$ , $ipivp [i - 1] = 〈value〉$ , $n = 〈value〉$ .
Constraint: $ipivp [i - 1] \geq 1$ and $ipivp [i - 1] \leq n$ .

On entry, $i = 〈value〉$ , $ipivq [i - 1] = 〈value〉$ , $n = 〈value〉$ .
Constraint: $ipivq [i - 1] \geq 1$ and $ipivq [i - 1] \leq n$ .

On entry, $ipivp [i - 1]$ is a repeated value: $i = 〈value〉$ .

On entry, $ipivq [i - 1]$ is a repeated value: $i = 〈value〉$ .
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_STRICTLY_INCREASING: On entry, $a [i - 1]$ is out of order: $i = 〈value〉$ .

On entry, the location ( $irow [i - 1], icol [i - 1]$ ) is a duplicate: $i = 〈value〉$ .

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

f11dbc is not threaded in any implementation.

9 Further Comments

9.1 Timing

The time taken for a call to f11dbc is proportional to the value of nnzc returned from f11dac.

9.2 Use of check

It is expected that a common use of f11dbc will be to carry out the preconditioning step required in the application of f11bec to sparse linear systems. In this situation f11dbc 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 for all subsequent calls set

check = Nag_SparseNsym_NoCheck

10 Example

This example reads in a sparse nonsymmetric matrix

A

and a vector

y

. It then calls f11dac, with

lfill = - 1

and

dtol = 0.0

, to compute the complete

L U

decomposition

A = P L D U Q .

Finally it calls f11dbc to solve the system

P L D U Q x = y .

f11db: FL CL CPP AD

NAG CL Interfacef11dbc (real_​gen_​precon_​ilu_​solve)

▸▿ Contents

1 Purpose

2 Specification