f11 Chapter Contents
f11 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_sparse_nsym_precon_ilu_solve (f11dbc)

## 1  Purpose

nag_sparse_nsym_precon_ilu_solve (f11dbc) solves a system of linear equations involving the incomplete $LU$ preconditioning matrix generated by nag_sparse_nsym_fac (f11dac).

## 2  Specification

 #include #include
 void nag_sparse_nsym_precon_ilu_solve (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)

## 3  Description

nag_sparse_nsym_precon_ilu_solve (f11dbc) solves a system of linear equations
 $Mx=y, or MTx=y,$
according to the value of the argument trans, where the matrix $M=PLDUQ$, corresponds to an incomplete $LU$ decomposition of a sparse matrix stored in coordinate storage (CS) format (see Section 2.1.1 in the f11 Chapter Introduction), as generated by nag_sparse_nsym_fac (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 nag_sparse_nsym_precon_ilu_solve (f11dbc) through the matrix
 $C=L+D-1+U-2I$
which is an n by n sparse matrix, stored in CS format, as returned by nag_sparse_nsym_fac (f11dac). The permutation matrices $P$ and $Q$ are returned from nag_sparse_nsym_fac (f11dac) via the arrays ipivp and ipivq.
It is envisaged that a common use of nag_sparse_nsym_precon_ilu_solve (f11dbc) will be to carry out the preconditioning step required in the application of nag_sparse_nsym_basic_solver (f11bec) to sparse linear systems. nag_sparse_nsym_precon_ilu_solve (f11dbc) is used for this purpose by the Black Box function nag_sparse_nsym_fac_sol (f11dcc).
nag_sparse_nsym_precon_ilu_solve (f11dbc) may also be used in combination with nag_sparse_nsym_fac (f11dac) to solve a sparse system of linear equations directly (see Section 9.5 in nag_sparse_nsym_fac (f11dac)). This use of nag_sparse_nsym_precon_ilu_solve (f11dbc) is demonstrated in Section 10.
None.

## 5  Arguments

1:    $\mathbf{trans}$Nag_TransTypeInput
On entry: specifies whether or not the matrix $M$ is transposed.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$Mx=y$ is solved.
${\mathbf{trans}}=\mathrm{Nag_Trans}$
${M}^{\mathrm{T}}x=y$ is solved.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ or $\mathrm{Nag_Trans}$.
2:    $\mathbf{n}$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_nsym_fac (f11dac).
Constraint: ${\mathbf{n}}\ge 1$.
3:    $\mathbf{a}\left[{\mathbf{la}}\right]$const doubleInput
On entry: the values returned in the array a by a previous call to nag_sparse_nsym_fac (f11dac).
4:    $\mathbf{la}$IntegerInput
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_nsym_fac (f11dac).
5:    $\mathbf{irow}\left[{\mathbf{la}}\right]$const IntegerInput
6:    $\mathbf{icol}\left[{\mathbf{la}}\right]$const IntegerInput
7:    $\mathbf{ipivp}\left[{\mathbf{n}}\right]$const IntegerInput
8:    $\mathbf{ipivq}\left[{\mathbf{n}}\right]$const IntegerInput
9:    $\mathbf{istr}\left[{\mathbf{n}}+1\right]$const IntegerInput
10:  $\mathbf{idiag}\left[{\mathbf{n}}\right]$const IntegerInput
On entry: the values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to nag_sparse_nsym_fac (f11dac).
11:  $\mathbf{check}$Nag_SparseNsym_CheckDataInput
On entry: specifies whether or not the CS representation of the matrix $M$ should be checked.
${\mathbf{check}}=\mathrm{Nag_SparseNsym_Check}$
Checks are carried on the values of n, irow, icol, ipivp, ipivq, istr and idiag.
${\mathbf{check}}=\mathrm{Nag_SparseNsym_NoCheck}$
None of these checks are carried out.
Constraint: ${\mathbf{check}}=\mathrm{Nag_SparseNsym_Check}$ or $\mathrm{Nag_SparseNsym_NoCheck}$.
12:  $\mathbf{y}\left[{\mathbf{n}}\right]$const doubleInput
On entry: the right-hand side vector $y$.
13:  $\mathbf{x}\left[{\mathbf{n}}\right]$doubleOutput
On exit: the solution vector $x$.
14:  $\mathbf{fail}$NagError *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.
See Section 3.2.1.2 in the Essential Introduction for further information.
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.
See Section 3.6.6 in the Essential Introduction for further information.
NE_INVALID_CS
On entry, $i=〈\mathit{\text{value}}〉$, ${\mathbf{icol}}\left[i-1\right]=〈\mathit{\text{value}}〉$, and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{icol}}\left[i-1\right]\ge 1$ and ${\mathbf{icol}}\left[i-1\right]\le {\mathbf{n}}$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nsym_precon_ilu_solve (f11dbc) and nag_sparse_nsym_fac (f11dac).
On entry, $i=〈\mathit{\text{value}}〉$, ${\mathbf{irow}}\left[i-1\right]=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{irow}}\left[i-1\right]\ge 1$ and ${\mathbf{irow}}\left[i-1\right]\le {\mathbf{n}}$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nsym_precon_ilu_solve (f11dbc) and nag_sparse_nsym_fac (f11dac).
NE_INVALID_CS_PRECOND
On entry, ${\mathbf{idiag}}\left[i-1\right]$ appears to be incorrect: $i=〈\mathit{\text{value}}〉$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nsym_precon_ilu_solve (f11dbc) and nag_sparse_nsym_fac (f11dac).
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_nsym_precon_ilu_solve (f11dbc) and nag_sparse_nsym_fac (f11dac).
On entry, ${\mathbf{istr}}\left[i-1\right]$ is inconsistent with irow: $i=〈\mathit{\text{value}}〉$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nsym_precon_ilu_solve (f11dbc) and nag_sparse_nsym_fac (f11dac).
NE_INVALID_ROWCOL_PIVOT
On entry, $i=〈\mathit{\text{value}}〉$, ${\mathbf{ipivp}}\left[i-1\right]=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ipivp}}\left[i-1\right]\ge 1$ and ${\mathbf{ipivp}}\left[i-1\right]\le {\mathbf{n}}$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nsym_precon_ilu_solve (f11dbc) and nag_sparse_nsym_fac (f11dac).
On entry, $i=〈\mathit{\text{value}}〉$, ${\mathbf{ipivq}}\left[i-1\right]=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ipivq}}\left[i-1\right]\ge 1$ and ${\mathbf{ipivq}}\left[i-1\right]\le {\mathbf{n}}$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nsym_precon_ilu_solve (f11dbc) and nag_sparse_nsym_fac (f11dac).
On entry, ${\mathbf{ipivp}}\left[i-1\right]$ is a repeated value: $i=〈\mathit{\text{value}}〉$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nsym_precon_ilu_solve (f11dbc) and nag_sparse_nsym_fac (f11dac).
On entry, ${\mathbf{ipivq}}\left[i-1\right]$ is a repeated value: $i=〈\mathit{\text{value}}〉$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nsym_precon_ilu_solve (f11dbc) and nag_sparse_nsym_fac (f11dac).
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_NOT_STRICTLY_INCREASING
On entry, ${\mathbf{a}}\left[i-1\right]$ is out of order: $i=〈\mathit{\text{value}}〉$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nsym_precon_ilu_solve (f11dbc) and nag_sparse_nsym_fac (f11dac).
On entry, the location (${\mathbf{irow}}\left[i-1\right],{\mathbf{icol}}\left[i-1\right]$) is a duplicate: $i=〈\mathit{\text{value}}〉$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to nag_sparse_nsym_precon_ilu_solve (f11dbc) and nag_sparse_nsym_fac (f11dac).

## 7  Accuracy

If ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ the computed solution $x$ is the exact solution of a perturbed system of equations $\left(M+\delta M\right)x=y$, where
 $δM≤cnεPLDUQ,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision. An equivalent result holds when ${\mathbf{trans}}=\mathrm{Nag_Trans}$.

## 8  Parallelism and Performance

Not applicable.

### 9.1  Timing

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

### 9.2  Use of check

It is expected that a common use of nag_sparse_nsym_precon_ilu_solve (f11dbc) will be to carry out the preconditioning step required in the application of nag_sparse_nsym_basic_solver (f11bec) to sparse linear systems. In this situation nag_sparse_nsym_precon_ilu_solve (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 ${\mathbf{check}}=\mathrm{Nag_SparseNsym_Check}$ for the first of such calls, and for all subsequent calls set ${\mathbf{check}}=\mathrm{Nag_SparseNsym_NoCheck}$.

## 10  Example

This example reads in a sparse nonsymmetric matrix $A$ and a vector $y$. It then calls nag_sparse_nsym_fac (f11dac), with ${\mathbf{lfill}}=-1$ and ${\mathbf{dtol}}=0.0$, to compute the complete $LU$ decomposition
 $A=PLDUQ.$
Finally it calls nag_sparse_nsym_precon_ilu_solve (f11dbc) to solve the system
 $PLDUQx=y.$

### 10.1  Program Text

Program Text (f11dbce.c)

### 10.2  Program Data

Program Data (f11dbce.d)

### 10.3  Program Results

Program Results (f11dbce.r)