# NAG CL Interfacef11jbc (real_​symm_​precon_​ichol_​solve)

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## 1Purpose

f11jbc solves a system of linear equations involving the incomplete Cholesky preconditioning matrix generated by f11jac.

## 2Specification

 #include
 void f11jbc (Integer n, const double a[], Integer la, const Integer irow[], const Integer icol[], const Integer ipiv[], const Integer istr[], Nag_SparseSym_CheckData check, const double y[], double x[], NagError *fail)
The function may be called by the names: f11jbc, nag_sparse_real_symm_precon_ichol_solve or nag_sparse_sym_precon_ichol_solve.

## 3Description

f11jbc 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 f11jac.
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 f11jbc 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 f11jac. The permutation matrix $P$ is returned from f11jac via the array ipiv.
It is envisaged that a common use of f11jbc will be to carry out the preconditioning step required in the application of f11gec to sparse symmetric linear systems. f11jbc is used for this purpose by the Black Box function f11jcc.
f11jbc may also be used in combination with f11jac to solve a sparse symmetric positive definite system of linear equations directly (see Section 9.4 in f11jac).

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 f11jac.
Constraint: ${\mathbf{n}}\ge 1$.
2: $\mathbf{a}\left[{\mathbf{la}}\right]$const double Input
On entry: the values returned in the array a by a previous call to f11jac.
3: $\mathbf{la}$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 f11jac.
4: $\mathbf{irow}\left[{\mathbf{la}}\right]$const Integer Input
5: $\mathbf{icol}\left[{\mathbf{la}}\right]$const Integer Input
6: $\mathbf{ipiv}\left[{\mathbf{n}}\right]$const Integer Input
7: $\mathbf{istr}\left[{\mathbf{n}}+1\right]$const Integer Input
On entry: the values returned in arrays irow, icol, ipiv and istr by a previous call to f11jac.
8: $\mathbf{check}$Nag_SparseSym_CheckData Input
On entry: specifies whether or not the input data should be checked.
${\mathbf{check}}=\mathrm{Nag_SparseSym_Check}$
Checks are carried out on the values of n, irow, icol, ipiv and istr.
${\mathbf{check}}=\mathrm{Nag_SparseSym_NoCheck}$
No checks are carried out.
Constraint: ${\mathbf{check}}=\mathrm{Nag_SparseSym_Check}$ or $\mathrm{Nag_SparseSym_NoCheck}$.
9: $\mathbf{y}\left[{\mathbf{n}}\right]$const double Input
On entry: the right-hand side vector $y$.
10: $\mathbf{x}\left[{\mathbf{n}}\right]$double Output
On exit: the solution vector $x$.
11: $\mathbf{fail}$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, ipiv and istr have not been corrupted between calls to f11jac and f11jbc.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface 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 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_ROWCOL_PIVOT
On entry, $i=⟨\mathit{\text{value}}⟩$, ${\mathbf{ipiv}}\left[i-1\right]=⟨\mathit{\text{value}}⟩$, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ipiv}}\left[i-1\right]\ge 1$ and ${\mathbf{ipiv}}\left[i-1\right]\le {\mathbf{n}}$.
On entry, ${\mathbf{ipiv}}\left[i-1\right]$ is a repeated value: $i=⟨\mathit{\text{value}}⟩$.
NE_INVALID_SCS
On entry, $\mathit{I}=⟨\mathit{\text{value}}⟩$, ${\mathbf{icol}}\left[\mathit{I}-1\right]=⟨\mathit{\text{value}}⟩$ and ${\mathbf{irow}}\left[\mathit{I}-1\right]=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{icol}}\left[\mathit{I}-1\right]\ge 1$ and ${\mathbf{icol}}\left[\mathit{I}-1\right]\le {\mathbf{irow}}\left[\mathit{I}-1\right]$.
On entry, $i=⟨\mathit{\text{value}}⟩$, ${\mathbf{irow}}\left[i-1\right]=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{irow}}\left[i-1\right]\ge 1$ and ${\mathbf{irow}}\left[i-1\right]\le {\mathbf{n}}$.
NE_INVALID_SCS_PRECOND
On entry, istr appears to be invalid.
On entry, ${\mathbf{istr}}\left[i-1\right]$ is inconsistent with irow: $i=⟨\mathit{\text{value}}⟩$.
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 (${\mathbf{irow}}\left[\mathit{I}-1\right],{\mathbf{icol}}\left[\mathit{I}-1\right]$) is a duplicate: $\mathit{I}=⟨\mathit{\text{value}}⟩$.

## 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

f11jbc is not threaded in any implementation.

### 9.1Timing

The time taken for a call to f11jbc is proportional to the value of nnzc returned from f11jac.

### 9.2Use of check

It is expected that a common use of f11jbc will be to carry out the preconditioning step required in the application of f11gec to sparse symmetric linear systems. In this situation f11jbc 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 ${\mathbf{check}}=\mathrm{Nag_SparseSym_Check}$ for the first of such calls, and to set ${\mathbf{check}}=\mathrm{Nag_SparseSym_NoCheck}$ for all subsequent calls.

## 10Example

This example reads in a symmetric positive definite sparse matrix $A$ and a vector $y$. It then calls f11jac, with ${\mathbf{lfill}}=-1$ and ${\mathbf{dtol}}=0.0$, to compute the complete Cholesky decomposition of $A$:
 $A=PLDLTPT.$
Then it calls f11jbc 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 f11yec.

### 10.1Program Text

Program Text (f11jbce.c)

### 10.2Program Data

Program Data (f11jbce.d)

### 10.3Program Results

Program Results (f11jbce.r)