void	f11jpc (Integer n, const Complex a[], Integer la, const Integer irow[], const Integer icol[], const Integer ipiv[], const Integer istr[], Nag_SparseSym_CheckData check, const Complex y[], Complex x[], NagError *fail)

The function may be called by the names: f11jpc, nag_sparse_complex_herm_precon_ilu_solve or nag_sparse_herm_precon_ichol_solve.

3 Description

f11jpc solves a system of linear equations

M x = y

involving the preconditioning matrix

M = P L D L^{H} P^{T}

, corresponding to an incomplete Cholesky decomposition of a complex sparse Hermitian matrix stored in symmetric coordinate storage (SCS) format (see Section 2.1.2 in the F11 Chapter Introduction), as generated by f11jnc.

In the above decomposition

L

is a complex lower triangular sparse matrix with unit diagonal,

D

is a real diagonal matrix and

P

is a permutation matrix.

L

and

D

are supplied to f11jpc through the matrix

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

which is a lower triangular

n

n

complex sparse matrix, stored in SCS format, as returned by f11jnc. The permutation matrix

P

is returned from f11jnc via the array ipiv.

f11jpc may also be used in combination with f11jnc to solve a sparse complex Hermitian positive definite system of linear equations directly (see f11jnc). This is illustrated in Section 10.

4 References

None.

5 Arguments

1: $n$ – 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 f11jnc.

Constraint:

n \geq 1

2: $a [la]$ – const Complex Input

On entry: the values returned in the array a by a previous call to f11jnc.

3: $la$ – Integer Input

On entry: the dimension of the arrays a, irow and icol. This must be the same value supplied in the preceding call to f11jnc.

4: $irow [la]$ – const Integer Input

5: $icol [la]$ – const Integer Input

6: $ipiv [n]$ – const Integer Input

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

On entry: the values returned in arrays irow, icol, ipiv and istr by a previous call to f11jnc.

8: $check$ – Nag_SparseSym_CheckData Input

On entry: specifies whether or not the input data should be checked.

$check = Nag_SparseSym_Check$: Checks are carried out on the values of n, irow, icol, ipiv and istr.
$check = Nag_SparseSym_NoCheck$: None of these checks are carried out.

Constraint:

check = Nag_SparseSym_Check

Nag_SparseSym_NoCheck

9: $y [n]$ – const Complex Input

On entry: the right-hand side vector

y

10: $x [n]$ – Complex Output

On exit: the solution vector

x

11: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

Check that a, irow, icol, ipiv and istr have not been corrupted between calls to f11jnc and f11jpc.
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_ROWCOL_PIVOT: On entry, $i = 〈value〉$ , $ipiv [i - 1] = 〈value〉$ , $n = 〈value〉$ .
Constraint: $ipiv [i - 1] \geq 1$ and $ipiv [i - 1] \leq n$ .

On entry, $ipiv [i - 1]$ is a repeated value: $i = 〈value〉$ .
NE_INVALID_SCS: On entry, $I = 〈value〉$ , $icol [I - 1] = 〈value〉$ and $irow [I - 1] = 〈value〉$ .
Constraint: $icol [I - 1] \geq 1$ and $icol [I - 1] \leq irow [I - 1]$ .

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

On entry, $istr [i - 1]$ is inconsistent with irow: $i = 〈value〉$ .
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

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| |L^{H}| P^{T},

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

8 Parallelism and Performance

f11jpc is not threaded in any implementation.

9 Further Comments

9.1 Timing

The time taken for a call to f11jpc is proportional to the value of nnzc returned from f11jnc.

10 Example

This example reads in a complex sparse Hermitian positive definite matrix

A

and a vector

y

. It then calls f11jnc, with

lfill = - 1

and

dtol = 0.0

, to compute the complete Cholesky decomposition of

A

A = P L D L^{H} P^{T} .

Finally it calls f11jpc to solve the system

P L D L^{H} P^{T} x = y .

Interfaces: FL CL AD

NAG CL Interface Introduction

F11 (Sparse) Chapter Contents

F11 (Sparse) Chapter Introduction

f11jp: FL CL AD

NAG CL Interfacef11jpc (complex_​herm_​precon_​ilu_​solve)

▸▿ 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

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
f11jpc (complex_herm_precon_ilu_solve)