# NAG FL Interfacef11dbf (real_​gen_​precon_​ilu_​solve)

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## 1Purpose

f11dbf solves a system of linear equations involving the incomplete $LU$ preconditioning matrix generated by f11daf.

## 2Specification

Fortran Interface
 Subroutine f11dbf ( n, a, la, irow, icol, istr, y, x,
 Integer, Intent (In) :: n, la, irow(la), icol(la), istr(n+1), idiag(n) Integer, Intent (Inout) :: ipivp(n), ipivq(n), ifail Real (Kind=nag_wp), Intent (In) :: a(la), y(n) Real (Kind=nag_wp), Intent (Out) :: x(n) Character (1), Intent (In) :: trans, check
#include <nag.h>
 void f11dbf_ (const char *trans, const Integer *n, const double a[], const Integer *la, const Integer irow[], const Integer icol[], Integer ipivp[], Integer ipivq[], const Integer istr[], const Integer idiag[], const char *check, const double y[], double x[], Integer *ifail, const Charlen length_trans, const Charlen length_check)
The routine may be called by the names f11dbf or nagf_sparse_real_gen_precon_ilu_solve.

## 3Description

f11dbf 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 f11daf.
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 f11dbf through the matrix
 $C=L+D-1+U-2I$
which is an n by n sparse matrix, stored in CS format, as returned by f11daf. The permutation matrices $P$ and $Q$ are returned from f11daf via the arrays ipivp and ipivq.
It is envisaged that a common use of f11dbf will be to carry out the preconditioning step required in the application of f11bef to sparse linear systems. f11dbf is used for this purpose by the Black Box routine f11dcf.
f11dbf may also be used in combination with f11daf to solve a sparse system of linear equations directly (see Section 9.5 in f11daf).

None.

## 5Arguments

1: $\mathbf{trans}$Character(1) Input
On entry: specifies whether or not the matrix $M$ is transposed.
${\mathbf{trans}}=\text{'N'}$
$Mx=y$ is solved.
${\mathbf{trans}}=\text{'T'}$
${M}^{\mathrm{T}}x=y$ is solved.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'T'}$.
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 f11daf.
Constraint: ${\mathbf{n}}\ge 1$.
3: $\mathbf{a}\left({\mathbf{la}}\right)$Real (Kind=nag_wp) array Input
On entry: the values returned in the array a by a previous call to f11daf.
4: $\mathbf{la}$Integer Input
On entry: the dimension of the arrays a, irow and icol as declared in the (sub)program from which f11dbf is called. This must be the same value returned by the preceding call to f11daf.
5: $\mathbf{irow}\left({\mathbf{la}}\right)$Integer array Input
6: $\mathbf{icol}\left({\mathbf{la}}\right)$Integer array Input
7: $\mathbf{ipivp}\left({\mathbf{n}}\right)$Integer array Input
8: $\mathbf{ipivq}\left({\mathbf{n}}\right)$Integer array Input
9: $\mathbf{istr}\left({\mathbf{n}}+1\right)$Integer array Input
10: $\mathbf{idiag}\left({\mathbf{n}}\right)$Integer array Input
On entry: the values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to f11daf.
11: $\mathbf{check}$Character(1) Input
On entry: specifies whether or not the CS representation of the matrix $M$ should be checked.
${\mathbf{check}}=\text{'C'}$
Checks are carried on the values of n, irow, icol, ipivp, ipivq, istr and idiag.
${\mathbf{check}}=\text{'N'}$
None of these checks are carried out.
Constraint: ${\mathbf{check}}=\text{'C'}$ or $\text{'N'}$.
12: $\mathbf{y}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the right-hand side vector $y$.
13: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: the solution vector $x$.
14: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{check}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{check}}=\text{'C'}$ or $\text{'N'}$.
On entry, ${\mathbf{trans}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'T'}$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 1$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{a}}\left(i\right)$ is out of order: $i=⟨\mathit{\text{value}}⟩$.
On entry, $i=⟨\mathit{\text{value}}⟩$, ${\mathbf{icol}}\left(i\right)=⟨\mathit{\text{value}}⟩$, and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{icol}}\left(i\right)\ge 1$ and ${\mathbf{icol}}\left(i\right)\le {\mathbf{n}}$.
On entry, $i=⟨\mathit{\text{value}}⟩$, ${\mathbf{ipivp}}\left(i\right)=⟨\mathit{\text{value}}⟩$, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ipivp}}\left(i\right)\ge 1$ and ${\mathbf{ipivp}}\left(i\right)\le {\mathbf{n}}$.
On entry, $i=⟨\mathit{\text{value}}⟩$, ${\mathbf{ipivq}}\left(i\right)=⟨\mathit{\text{value}}⟩$, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ipivq}}\left(i\right)\ge 1$ and ${\mathbf{ipivq}}\left(i\right)\le {\mathbf{n}}$.
On entry, $i=⟨\mathit{\text{value}}⟩$, ${\mathbf{irow}}\left(i\right)=⟨\mathit{\text{value}}⟩$, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{irow}}\left(i\right)\ge 1$ and ${\mathbf{irow}}\left(i\right)\le {\mathbf{n}}$.
On entry, ${\mathbf{idiag}}\left(i\right)$ appears to be incorrect: $i=⟨\mathit{\text{value}}⟩$.
On entry, ${\mathbf{ipivp}}\left(i\right)$ is a repeated value: $i=⟨\mathit{\text{value}}⟩$.
On entry, ${\mathbf{ipivq}}\left(i\right)$ is a repeated value: $i=⟨\mathit{\text{value}}⟩$.
On entry, istr appears to be invalid.
On entry, ${\mathbf{istr}}\left(i\right)$ is inconsistent with irow: $i=⟨\mathit{\text{value}}⟩$.
On entry, the location (${\mathbf{irow}}\left(i\right),{\mathbf{icol}}\left(i\right)$) is a duplicate: $i=⟨\mathit{\text{value}}⟩$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dbf and f11daf.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

If ${\mathbf{trans}}=\text{'N'}$ 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 ${\mathbf{trans}}=\text{'T'}$.

## 8Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f11dbf is not threaded in any implementation.

### 9.1Timing

The time taken for a call to f11dbf is proportional to the value of nnzc returned from f11daf.

### 9.2Use of check

It is expected that a common use of f11dbf will be to carry out the preconditioning step required in the application of f11bef to sparse linear systems. In this situation f11dbf 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 ${\mathbf{check}}=\text{'C'}$ for the first of such calls, and for all subsequent calls set ${\mathbf{check}}=\text{'N'}$.

## 10Example

This example reads in a sparse nonsymmetric matrix $A$ and a vector $y$. It then calls f11daf, with ${\mathbf{lfill}}=-1$ and ${\mathbf{dtol}}=0.0$, to compute the complete $LU$ decomposition
 $A=PLDUQ.$
Finally it calls f11dbf to solve the system
 $PLDUQx=y.$

### 10.1Program Text

Program Text (f11dbfe.f90)

### 10.2Program Data

Program Data (f11dbfe.d)

### 10.3Program Results

Program Results (f11dbfe.r)