void	f11jbc (Integer n, const double a[], Integer la, const Integer irow[], const Integer icol[], const Integer ipiv[], const Integer istr[], Nag_SparseSym_CheckData check, const double y[], double x[], NagError *fail)

The function may be called by the names: f11jbc, nag_sparse_real_symm_precon_ichol_solve or nag_sparse_sym_precon_ichol_solve.

3 Description

f11jbc solves a system of linear equations

M x = y

involving the preconditioning matrix

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

, corresponding to an incomplete Cholesky decomposition of a sparse symmetric matrix stored in symmetric coordinate storage (SCS) format (see Section 2.1.2 in the F11 Chapter Introduction), as generated by f11jac.

In the above decomposition

L

is a lower triangular sparse matrix with unit diagonal,

D

is a diagonal matrix and

P

is a permutation matrix.

L

and

D

are supplied to f11jbc through the matrix

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

which is a lower triangular n by n sparse matrix, stored in SCS format, as returned by f11jac. The permutation matrix

P

is returned from f11jac via the array ipiv.

It is envisaged that a common use of f11jbc will be to carry out the preconditioning step required in the application of f11gec to sparse symmetric linear systems. f11jbc is used for this purpose by the Black Box function f11jcc.

f11jbc may also be used in combination with f11jac to solve a sparse symmetric positive definite system of linear equations directly (see Section 9.4 in f11jac). This use of f11jbc is demonstrated 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 f11jac.

Constraint:

n \geq 1

2: $a [la]$ – const double Input

On entry: the values returned in the array a by a previous call to f11jac.

3: $la$ – 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 f11jac.

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 f11jac.

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.
$check = Nag_SparseSym_NoCheck$: No checks are carried out.

6 Error Indicators and Warnings

Check that a, irow, icol, ipiv and istr have not been corrupted between calls to f11jac and f11jbc.
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^{T}| P^{T},

c (n)

is a modest linear function of

n

, and

ε

is the machine precision.

8 Parallelism and Performance

f11jbc is not threaded in any implementation.

9 Further Comments

9.1 Timing

The time taken for a call to f11jbc is proportional to the value of nnzc returned from f11jac.

9.2 Use of check

It is expected that a common use of f11jbc will be to carry out the preconditioning step required in the application of f11gec to sparse symmetric linear systems. In this situation f11jbc 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_SparseSym_Check

for the first of such calls, and to set

check = Nag_SparseSym_NoCheck

for all subsequent calls.

10 Example

This example reads in a symmetric positive definite sparse matrix

A

and a vector

y

. It then calls f11jac, with

lfill = - 1

and

dtol = 0.0

, to compute the complete Cholesky decomposition of

A

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

Then it calls f11jbc to solve the system

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

It then repeats the exercise for the same matrix permuted with the bandwidth-reducing Reverse Cuthill–McKee permutation, calculated with f11yec.

f11jb: FL CL CPP AD

NAG CL Interfacef11jbc (real_​symm_​precon_​ichol_​solve)

▸▿ Contents

1 Purpose

2 Specification