# NAG CL Interfacef11dbc (real_​gen_​precon_​ilu_​solve)

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## 1Purpose

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

## 2Specification

 #include
 void f11dbc (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)
The function may be called by the names: f11dbc, nag_sparse_real_gen_precon_ilu_solve or nag_sparse_nsym_precon_ilu_solve.

## 3Description

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 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 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 f11dac. The permutation matrices $P$ and $Q$ are returned from f11dac via the arrays ipivp and ipivq.
It is envisaged that a common use of f11dbc will be to carry out the preconditioning step required in the application of f11bec to sparse linear systems. f11dbc is used for this purpose by the Black Box function f11dcc.
f11dbc may also be used in combination with f11dac to solve a sparse system of linear equations directly (see Section 9.5 in f11dac).

None.

## 5Arguments

1: $\mathbf{trans}$Nag_TransType Input
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: $\mathbf{n}$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 f11dac.
Constraint: ${\mathbf{n}}\ge 1$.
3: $\mathbf{a}\left[{\mathbf{la}}\right]$const double Input
On entry: the values returned in the array a by a previous call to f11dac.
4: $\mathbf{la}$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 f11dac.
5: $\mathbf{irow}\left[{\mathbf{la}}\right]$const Integer Input
6: $\mathbf{icol}\left[{\mathbf{la}}\right]$const Integer Input
7: $\mathbf{ipivp}\left[{\mathbf{n}}\right]$const Integer Input
8: $\mathbf{ipivq}\left[{\mathbf{n}}\right]$const Integer Input
9: $\mathbf{istr}\left[{\mathbf{n}}+1\right]$const Integer Input
10: $\mathbf{idiag}\left[{\mathbf{n}}\right]$const Integer Input
On entry: the values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to f11dac.
11: $\mathbf{check}$Nag_SparseNsym_CheckData Input
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.
See also Section 9.2.
Constraint: ${\mathbf{check}}=\mathrm{Nag_SparseNsym_Check}$ or $\mathrm{Nag_SparseNsym_NoCheck}$.
12: $\mathbf{y}\left[{\mathbf{n}}\right]$const double Input
On entry: the right-hand side vector $y$.
13: $\mathbf{x}\left[{\mathbf{n}}\right]$double Output
On exit: the solution vector $x$.
14: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dbc and f11dac.
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 $⟨\mathit{\text{value}}⟩$ 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 7.5 in the Introduction to the NAG Library CL Interface 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 8 in the Introduction to the NAG Library CL Interface 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
 $|δM|≤c(n)εP|L||D||U|Q,$
$c\left(n\right)$ 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

f11dbc is not threaded in any implementation.

## 9Further Comments

### 9.1Timing

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

### 9.2Use of check

It is expected that a common use of f11dbc will be to carry out the preconditioning step required in the application of f11bec to sparse linear systems. In this situation 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 ${\mathbf{check}}=\mathrm{Nag_SparseNsym_NoCheck}$.

## 10Example

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

### 10.1Program Text

Program Text (f11dbce.c)

### 10.2Program Data

Program Data (f11dbce.d)

### 10.3Program Results

Program Results (f11dbce.r)