# NAG FL Interfacef11dpf (complex_​gen_​precon_​ilu_​solve)

## 1Purpose

f11dpf solves a system of complex linear equations involving the incomplete $LU$ preconditioning matrix generated by f11dnf.

## 2Specification

Fortran Interface
 Subroutine f11dpf ( 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 Complex (Kind=nag_wp), Intent (In) :: a(la), y(n) Complex (Kind=nag_wp), Intent (Out) :: x(n) Character (1), Intent (In) :: trans, check
#include <nag.h>
 void f11dpf_ (const char *trans, const Integer *n, const Complex a[], const Integer *la, const Integer irow[], const Integer icol[], Integer ipivp[], Integer ipivq[], const Integer istr[], const Integer idiag[], const char *check, const Complex y[], Complex x[], Integer *ifail, const Charlen length_trans, const Charlen length_check)
The routine may be called by the names f11dpf or nagf_sparse_complex_gen_precon_ilu_solve.

## 3Description

f11dpf 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 f11dnf.
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 f11dpf through the matrix
 $C=L+D-1+U-2I$
which is an n by n sparse matrix, stored in CS format, as returned by f11dnf. The permutation matrices $P$ and $Q$ are returned from f11dnf via the arrays ipivp and ipivq.
It is envisaged that a common use of f11dpf will be to carry out the preconditioning step required in the application of f11bsf to sparse complex linear systems. f11dpf is used for this purpose by the Black Box routine f11dqf.
f11dpf may also be used in combination with f11dnf to solve a sparse system of complex linear equations directly (see Section 9.5 in f11dnf). This use of f11dpf is illustrated in Section 10.
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 f11dnf.
Constraint: ${\mathbf{n}}\ge 1$.
3: $\mathbf{a}\left({\mathbf{la}}\right)$Complex (Kind=nag_wp) array Input
On entry: the values returned in the array a by a previous call to f11dnf.
4: $\mathbf{la}$Integer Input
On entry: the dimension of the arrays a, irow and icol as declared in the (sub)program from which f11dpf is called. This must be the same value supplied in the preceding call to f11dnf.
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 f11dnf.
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: $\mathbf{y}\left({\mathbf{n}}\right)$Complex (Kind=nag_wp) array Input
On entry: the right-hand side vector $y$.
13: $\mathbf{x}\left({\mathbf{n}}\right)$Complex (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 the call to f11dpf has been preceded by a valid call to f11dnf and that the arrays a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between the two calls.
${\mathbf{ifail}}=-99$
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
 $δM≤cnεPLDUQ,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision. An equivalent result holds when ${\mathbf{trans}}=\text{'T'}$.

## 8Parallelism and Performance

f11dpf is not threaded in any implementation.

### 9.1Timing

The time taken for a call to f11dpf is proportional to the value of nnzc returned from f11dnf.

### 9.2Use of check

It is expected that a common use of f11dpf will be to carry out the preconditioning step required in the application of f11bsf to sparse complex linear systems. In this situation f11dpf 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 to set ${\mathbf{check}}=\text{'N'}$ for all subsequent calls.

## 10Example

This example reads in a complex sparse non-Hermitian matrix $A$ and a vector $y$. It then calls f11dnf, with ${\mathbf{lfill}}=-1$ and ${\mathbf{dtol}}=0.0$, to compute the complete $LU$ decomposition
 $A=PLDUQ.$
Finally it calls f11dpf to solve the system
 $PLDUQx=y.$

### 10.1Program Text

Program Text (f11dpfe.f90)

### 10.2Program Data

Program Data (f11dpfe.d)

### 10.3Program Results

Program Results (f11dpfe.r)