# NAG C Library Function Document

## 1Purpose

nag_sparse_nherm_precon_ilu_solve (f11dpc) solves a system of complex linear equations involving the incomplete $LU$ preconditioning matrix generated by nag_sparse_nherm_fac (f11dnc).

## 2Specification

 #include #include
 void nag_sparse_nherm_precon_ilu_solve (Nag_TransType trans, Integer n, const Complex 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 Complex y[], Complex x[], NagError *fail)

## 3Description

nag_sparse_nherm_precon_ilu_solve (f11dpc) solves a system of complex 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 complex sparse matrix stored in coordinate storage (CS) format (see Section 2.1.1 in the f11 Chapter Introduction), as generated by nag_sparse_nherm_fac (f11dnc).
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_nherm_precon_ilu_solve (f11dpc) 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_nherm_fac (f11dnc). The permutation matrices $P$ and $Q$ are returned from nag_sparse_nherm_fac (f11dnc) via the arrays ipivp and ipivq.
It is envisaged that a common use of nag_sparse_nherm_precon_ilu_solve (f11dpc) will be to carry out the preconditioning step required in the application of nag_sparse_nherm_basic_solver (f11bsc) to sparse complex linear systems. nag_sparse_nherm_precon_ilu_solve (f11dpc) is used for this purpose by the Black Box function nag_sparse_nherm_fac_sol (f11dqc).
nag_sparse_nherm_precon_ilu_solve (f11dpc) may also be used in combination with nag_sparse_nherm_fac (f11dnc) to solve a sparse system of complex linear equations directly (see Section 9.5 in nag_sparse_nherm_fac (f11dnc)). This use of nag_sparse_nherm_precon_ilu_solve (f11dpc) is illustrated in Section 10.

None.

## 5Arguments

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_nherm_fac (f11dnc).
Constraint: ${\mathbf{n}}\ge 1$.
3:    $\mathbf{a}\left[{\mathbf{la}}\right]$const ComplexInput
On entry: the values returned in the array a by a previous call to nag_sparse_nherm_fac (f11dnc).
4:    $\mathbf{la}$IntegerInput
On entry: the dimension of the arrays a, irow and icol. This must be the same value supplied in the preceding call to nag_sparse_nherm_fac (f11dnc).
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_nherm_fac (f11dnc).
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 ComplexInput
On entry: the right-hand side vector $y$.
13:  $\mathbf{x}\left[{\mathbf{n}}\right]$ComplexOutput
On exit: the solution vector $x$.
14:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

Check that the call to nag_sparse_nherm_precon_ilu_solve (f11dpc) has been preceded by a valid call to nag_sparse_nherm_fac (f11dnc) and that the arrays a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between the two calls.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation 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 ${\mathbf{icol}}\left[i-1\right]\le {\mathbf{n}}$.
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 ${\mathbf{irow}}\left[i-1\right]\le {\mathbf{n}}$.
NE_INVALID_CS_PRECOND
On entry, ${\mathbf{idiag}}\left[i-1\right]$ appears to be incorrect: $i=〈\mathit{\text{value}}〉$.
On entry, istr appears to be invalid.
On entry, ${\mathbf{istr}}\left[i-1\right]$ is inconsistent with irow: $i=〈\mathit{\text{value}}〉$.
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 ${\mathbf{ipivp}}\left[i-1\right]\le {\mathbf{n}}$.
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 ${\mathbf{ipivq}}\left[i-1\right]\le {\mathbf{n}}$.
On entry, ${\mathbf{ipivp}}\left[i-1\right]$ is a repeated value: $i=〈\mathit{\text{value}}〉$.
On entry, ${\mathbf{ipivq}}\left[i-1\right]$ is a repeated value: $i=〈\mathit{\text{value}}〉$.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_NOT_STRICTLY_INCREASING
On entry, ${\mathbf{a}}\left[i-1\right]$ is out of order: $i=〈\mathit{\text{value}}〉$.
On entry, the location (${\mathbf{irow}}\left[i-1\right],{\mathbf{icol}}\left[i-1\right]$) is a duplicate: $i=〈\mathit{\text{value}}〉$.

## 7Accuracy

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 ${\mathbf{trans}}=\mathrm{Nag_Trans}$.

## 8Parallelism and Performance

nag_sparse_nherm_precon_ilu_solve (f11dpc) is not threaded in any implementation.

### 9.1Timing

The time taken for a call to nag_sparse_nherm_precon_ilu_solve (f11dpc) is proportional to the value of nnzc returned from nag_sparse_nherm_fac (f11dnc).

### 9.2Use of check

It is expected that a common use of nag_sparse_nherm_precon_ilu_solve (f11dpc) will be to carry out the preconditioning step required in the application of nag_sparse_nherm_basic_solver (f11bsc) to sparse complex linear systems. In this situation nag_sparse_nherm_precon_ilu_solve (f11dpc) 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 to set ${\mathbf{check}}=\mathrm{Nag_SparseNsym_NoCheck}$ for all subsequent calls.

## 10Example

This example reads in a complex sparse non-Hermitian matrix $A$ and a vector $y$. It then calls nag_sparse_nherm_fac (f11dnc), with ${\mathbf{lfill}}=-1$ and ${\mathbf{dtol}}=0.0$, to compute the complete $LU$ decomposition
 $A=PLDUQ.$
Finally it calls nag_sparse_nherm_precon_ilu_solve (f11dpc) to solve the system
 $PLDUQx=y.$

### 10.1Program Text

Program Text (f11dpce.c)

### 10.2Program Data

Program Data (f11dpce.d)

### 10.3Program Results

Program Results (f11dpce.r)