# NAG CL Interfacef11jpc (complex_​herm_​precon_​ilu_​solve)

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## 1Purpose

f11jpc solves a system of complex linear equations involving the incomplete Cholesky preconditioning matrix generated by f11jnc.

## 2Specification

 #include
 void f11jpc (Integer n, const Complex a[], Integer la, const Integer irow[], const Integer icol[], const Integer ipiv[], const Integer istr[], Nag_SparseSym_CheckData check, const Complex y[], Complex x[], NagError *fail)
The function may be called by the names: f11jpc, nag_sparse_complex_herm_precon_ilu_solve or nag_sparse_herm_precon_ichol_solve.

## 3Description

f11jpc solves a system of linear equations
 $Mx=y$
involving the preconditioning matrix $M=PLD{L}^{\mathrm{H}}{P}^{\mathrm{T}}$, corresponding to an incomplete Cholesky decomposition of a complex sparse Hermitian matrix stored in symmetric coordinate storage (SCS) format (see Section 2.1.2 in the F11 Chapter Introduction), as generated by f11jnc.
In the above decomposition $L$ is a complex lower triangular sparse matrix with unit diagonal, $D$ is a real diagonal matrix and $P$ is a permutation matrix. $L$ and $D$ are supplied to f11jpc through the matrix
 $C=L+D-1-I$
which is a lower triangular $n×n$ complex sparse matrix, stored in SCS format, as returned by f11jnc. The permutation matrix $P$ is returned from f11jnc via the array ipiv.
f11jpc may also be used in combination with f11jnc to solve a sparse complex Hermitian positive definite system of linear equations directly (see f11jnc).
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 f11jnc.
Constraint: ${\mathbf{n}}\ge 1$.
2: $\mathbf{a}\left[{\mathbf{la}}\right]$const Complex Input
On entry: the values returned in the array a by a previous call to f11jnc.
3: $\mathbf{la}$Integer Input
On entry: the dimension of the arrays a, irow and icol. This must be the same value supplied in the preceding call to f11jnc.
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 f11jnc.
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.
${\mathbf{check}}=\mathrm{Nag_SparseSym_NoCheck}$
None of these checks are carried out.
Constraint: ${\mathbf{check}}=\mathrm{Nag_SparseSym_Check}$ or $\mathrm{Nag_SparseSym_NoCheck}$.
9: $\mathbf{y}\left[{\mathbf{n}}\right]$const Complex Input
On entry: the right-hand side vector $y$.
10: $\mathbf{x}\left[{\mathbf{n}}\right]$Complex 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 f11jnc and f11jpc.
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
 $|δM|≤c(n)εP|L||D||LH|PT,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

f11jpc is not threaded in any implementation.

### 9.1Timing

The time taken for a call to f11jpc is proportional to the value of nnzc returned from f11jnc.

## 10Example

This example reads in a complex sparse Hermitian positive definite matrix $A$ and a vector $y$. It then calls f11jnc, with ${\mathbf{lfill}}=-1$ and ${\mathbf{dtol}}=0.0$, to compute the complete Cholesky decomposition of $A$:
 $A=PLDLHPT.$
Finally it calls f11jpc to solve the system
 $PLDLHPTx=y.$

### 10.1Program Text

Program Text (f11jpce.c)

### 10.2Program Data

Program Data (f11jpce.d)

### 10.3Program Results

Program Results (f11jpce.r)