# NAG FL Interfacef11jbf (real_​symm_​precon_​ichol_​solve)

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## 1Purpose

f11jbf solves a system of linear equations involving the incomplete Cholesky preconditioning matrix generated by f11jaf.

## 2Specification

Fortran Interface
 Subroutine f11jbf ( n, a, la, irow, icol, ipiv, istr, y, x,
 Integer, Intent (In) :: n, la, irow(la), icol(la), istr(n+1) Integer, Intent (Inout) :: ipiv(n), ifail Real (Kind=nag_wp), Intent (In) :: a(la), y(n) Real (Kind=nag_wp), Intent (Out) :: x(n) Character (1), Intent (In) :: check
C Header Interface
#include <nag.h>
 void f11jbf_ (const Integer *n, const double a[], const Integer *la, const Integer irow[], const Integer icol[], Integer ipiv[], const Integer istr[], const char *check, const double y[], double x[], Integer *ifail, const Charlen length_check)
The routine may be called by the names f11jbf or nagf_sparse_real_symm_precon_ichol_solve.

## 3Description

f11jbf solves a system of linear equations
 $Mx=y$
involving the preconditioning matrix $M=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, corresponding to an incomplete Cholesky decomposition of a sparse symmetric matrix stored in symmetric coordinate storage (SCS) format (see Section 2.1.2 in the F11 Chapter Introduction), as generated by f11jaf.
In the above decomposition $L$ is a lower triangular sparse matrix with unit diagonal, $D$ is a diagonal matrix and $P$ is a permutation matrix. $L$ and $D$ are supplied to f11jbf through the matrix
 $C=L+D-1-I$
which is a lower triangular n by n sparse matrix, stored in SCS format, as returned by f11jaf. The permutation matrix $P$ is returned from f11jaf via the array ipiv.
It is envisaged that a common use of f11jbf will be to carry out the preconditioning step required in the application of f11gef to sparse symmetric linear systems. f11jbf is used for this purpose by the Black Box routine f11jcf.
f11jbf may also be used in combination with f11jaf to solve a sparse symmetric positive definite system of linear equations directly (see Section 9.4 in f11jaf).

None.

## 5Arguments

1: $\mathbf{n}$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 f11jaf.
Constraint: ${\mathbf{n}}\ge 1$.
2: $\mathbf{a}\left({\mathbf{la}}\right)$Real (Kind=nag_wp) array Input
On entry: the values returned in the array a by a previous call to f11jaf.
3: $\mathbf{la}$Integer Input
On entry: the dimension of the arrays a, irow and icol as declared in the (sub)program from which f11jbf is called. This must be the same value returned by the preceding call to f11jaf.
4: $\mathbf{irow}\left({\mathbf{la}}\right)$Integer array Input
5: $\mathbf{icol}\left({\mathbf{la}}\right)$Integer array Input
6: $\mathbf{ipiv}\left({\mathbf{n}}\right)$Integer array Input
7: $\mathbf{istr}\left({\mathbf{n}}+1\right)$Integer array Input
On entry: the values returned in arrays irow, icol, ipiv and istr by a previous call to f11jaf.
8: $\mathbf{check}$Character(1) Input
On entry: specifies whether or not the input data should be checked.
${\mathbf{check}}=\text{'C'}$
Checks are carried out on the values of n, irow, icol, ipiv and istr.
${\mathbf{check}}=\text{'N'}$
No checks are carried out.
See also Section 9.2.
Constraint: ${\mathbf{check}}=\text{'C'}$ or $\text{'N'}$.
9: $\mathbf{y}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Input
On entry: the right-hand side vector $y$.
10: $\mathbf{x}\left({\mathbf{n}}\right)$Real (Kind=nag_wp) array Output
On exit: the solution vector $x$.
11: $\mathbf{ifail}$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 $\text{'N'}$.
${\mathbf{ifail}}=2$
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, $\mathit{I}=⟨\mathit{\text{value}}⟩$, ${\mathbf{icol}}\left(\mathit{I}\right)=⟨\mathit{\text{value}}⟩$ and ${\mathbf{irow}}\left(\mathit{I}\right)=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{icol}}\left(\mathit{I}\right)\ge 1$ and ${\mathbf{icol}}\left(\mathit{I}\right)\le {\mathbf{irow}}\left(\mathit{I}\right)$.
On entry, $i=⟨\mathit{\text{value}}⟩$, ${\mathbf{ipiv}}\left(i\right)=⟨\mathit{\text{value}}⟩$, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ipiv}}\left(i\right)\ge 1$ and ${\mathbf{ipiv}}\left(i\right)\le {\mathbf{n}}$.
On entry, $i=⟨\mathit{\text{value}}⟩$, ${\mathbf{irow}}\left(i\right)=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{irow}}\left(i\right)\ge 1$ and ${\mathbf{irow}}\left(i\right)\le {\mathbf{n}}$.
On entry, ${\mathbf{ipiv}}\left(i\right)$ 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 (${\mathbf{irow}}\left(\mathit{I}\right),{\mathbf{icol}}\left(\mathit{I}\right)$) is a duplicate: $\mathit{I}=⟨\mathit{\text{value}}⟩$.
Check that a, irow, icol, ipiv and istr have not been corrupted between calls to f11jaf and f11jbf.
${\mathbf{ifail}}=-99$
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

The computed solution $x$ is the exact solution of a perturbed system of equations $\left(M+\delta M\right)x=y$, where
 $|δM|≤c(n)εP|L||D||LT|PT,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f11jbf is not threaded in any implementation.

## 9Further Comments

### 9.1Timing

The time taken for a call to f11jbf is proportional to the value of nnzc returned from f11jaf.

### 9.2Use of check

It is expected that a common use of f11jbf will be to carry out the preconditioning step required in the application of f11gef to sparse symmetric linear systems. In this situation f11jbf 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 symmetric positive definite sparse matrix $A$ and a vector $y$. It then calls f11jaf, with ${\mathbf{lfill}}=-1$ and ${\mathbf{dtol}}=0.0$, to compute the complete Cholesky decomposition of $A$:
 $A=PLDLTPT.$
Then it calls f11jbf to solve the system
 $PLDLTPTx=y.$
It then repeats the exercise for the same matrix permuted with the bandwidth-reducing Reverse Cuthill–McKee permutation, calculated with f11yef.

### 10.1Program Text

Program Text (f11jbfe.f90)

### 10.2Program Data

Program Data (f11jbfe.d)

### 10.3Program Results

Program Results (f11jbfe.r)