f11 Chapter Contents
f11 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_sparse_sym_precon_ichol_solve (f11jbc)

## 1  Purpose

nag_sparse_sym_precon_ichol_solve (f11jbc) solves a system of linear equations involving the incomplete Cholesky preconditioning matrix generated by nag_sparse_sym_chol_fac (f11jac).

## 2  Specification

 #include #include
 void nag_sparse_sym_precon_ichol_solve (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)

## 3  Description

nag_sparse_sym_precon_ichol_solve (f11jbc) solves a system of linear equations
 $Mx=y$
involving the preconditioning matrix $M=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, 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 nag_sparse_sym_chol_fac (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 nag_sparse_sym_precon_ichol_solve (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 nag_sparse_sym_chol_fac (f11jac). The permutation matrix $P$ is returned from nag_sparse_sym_chol_fac (f11jac) via the array ipiv.
It is envisaged that a common use of nag_sparse_sym_precon_ichol_solve (f11jbc) will be to carry out the preconditioning step required in the application of nag_sparse_sym_basic_solver (f11gec) to sparse symmetric linear systems. nag_sparse_sym_precon_ichol_solve (f11jbc) is used for this purpose by the Black Box function nag_sparse_sym_chol_sol (f11jcc).
nag_sparse_sym_precon_ichol_solve (f11jbc) may also be used in combination with nag_sparse_sym_chol_fac (f11jac) to solve a sparse symmetric positive definite system of linear equations directly (see Section 8.3 in nag_sparse_sym_chol_fac (f11jac)). This use of nag_sparse_sym_precon_ichol_solve (f11jbc) is demonstrated in Section 9.

None.

## 5  Arguments

1:     nIntegerInput
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_sym_chol_fac (f11jac).
Constraint: ${\mathbf{n}}\ge 1$.
2:     a[la]const doubleInput
On entry: the values returned in the array a by a previous call to nag_sparse_sym_chol_fac (f11jac).
3:     laIntegerInput
On entry: the dimension of the arrays a, irow and icol. This must be the same value returned by the preceding call to nag_sparse_sym_chol_fac (f11jac).
4:     irow[la]const IntegerInput
5:     icol[la]const IntegerInput
6:     ipiv[n]const IntegerInput
7:     istr[${\mathbf{n}}+1$]const IntegerInput
On entry: the values returned in arrays irow, icol, ipiv and istr by a previous call to nag_sparse_sym_chol_fac (f11jac).
8:     checkNag_SparseSym_CheckDataInput
On entry: specifies whether or not the input data should be checked.
${\mathbf{check}}=\mathrm{Nag_SparseSym_Check}$
Checks are carried out on the values of n, irow, icol, ipiv and istr.
${\mathbf{check}}=\mathrm{Nag_SparseSym_NoCheck}$
No checks are carried out.
Constraint: ${\mathbf{check}}=\mathrm{Nag_SparseSym_Check}$ or $\mathrm{Nag_SparseSym_NoCheck}$.
9:     y[n]const doubleInput
On entry: the right-hand side vector $y$.
10:   x[n]doubleOutput
On exit: the solution vector $x$.
11:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
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.
NE_INVALID_ROWCOL_PIVOT
Check that a, irow, icol, ipiv and istr have not been corrupted between calls to nag_sparse_sym_chol_fac (f11jac) and nag_sparse_sym_precon_ichol_solve (f11jbc).
NE_INVALID_SCS
Check that a, irow, icol, ipiv and istr have not been corrupted between calls to nag_sparse_sym_chol_fac (f11jac) and nag_sparse_sym_precon_ichol_solve (f11jbc).
NE_INVALID_SCS_PRECOND
Check that a, irow, icol, ipiv and istr have not been corrupted between calls to nag_sparse_sym_chol_fac (f11jac) and nag_sparse_sym_precon_ichol_solve (f11jbc).
NE_NOT_STRICTLY_INCREASING
Check that a, irow, icol, ipiv and istr have not been corrupted between calls to nag_sparse_sym_chol_fac (f11jac) and nag_sparse_sym_precon_ichol_solve (f11jbc).

## 7  Accuracy

The computed solution $x$ is the exact solution of a perturbed system of equations $\left(M+\delta M\right)x=y$, where
 $δM≤cnεPLDLTPT,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

### 8.1  Timing

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

### 8.2  Use of check

It is expected that a common use of nag_sparse_sym_precon_ichol_solve (f11jbc) will be to carry out the preconditioning step required in the application of nag_sparse_sym_basic_solver (f11gec) to sparse symmetric linear systems. In this situation nag_sparse_sym_precon_ichol_solve (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 ${\mathbf{check}}=\mathrm{Nag_SparseSym_Check}$ for the first of such calls, and to set ${\mathbf{check}}=\mathrm{Nag_SparseSym_NoCheck}$ for all subsequent calls.

## 9  Example

This example reads in a symmetric positive definite sparse matrix $A$ and a vector $y$. It then calls nag_sparse_sym_chol_fac (f11jac), with ${\mathbf{lfill}}=-1$ and ${\mathbf{dtol}}=0.0$, to compute the complete Cholesky decomposition of $A$:
 $A=PLDLTPT.$
Finally it calls nag_sparse_sym_precon_ichol_solve (f11jbc) to solve the system
 $PLDLTPTx=y.$

### 9.1  Program Text

Program Text (f11jbce.c)

### 9.2  Program Data

Program Data (f11jbce.d)

### 9.3  Program Results

Program Results (f11jbce.r)