# NAG C Library Function Document

## 1Purpose

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

## 2Specification

 #include #include
 void nag_sparse_herm_precon_ichol_solve (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)

## 3Description

nag_sparse_herm_precon_ichol_solve (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 nag_sparse_herm_chol_fac (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 nag_sparse_herm_precon_ichol_solve (f11jpc) through the matrix
 $C=L+D-1-I$
which is a lower triangular $n$ by $n$ complex sparse matrix, stored in SCS format, as returned by nag_sparse_herm_chol_fac (f11jnc). The permutation matrix $P$ is returned from nag_sparse_herm_chol_fac (f11jnc) via the array ipiv.
nag_sparse_herm_precon_ichol_solve (f11jpc) may also be used in combination with nag_sparse_herm_chol_fac (f11jnc) to solve a sparse complex Hermitian positive definite system of linear equations directly (see nag_sparse_herm_chol_fac (f11jnc)). This is illustrated in Section 10.

None.

## 5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $M$. This must be the same value as was supplied in the preceding call to nag_sparse_herm_chol_fac (f11jnc).
Constraint: ${\mathbf{n}}\ge 1$.
2:    $\mathbf{a}\left[{\mathbf{la}}\right]$const ComplexInput
On entry: the values returned in the array a by a previous call to nag_sparse_herm_chol_fac (f11jnc).
3:    $\mathbf{la}$IntegerInput
On entry: the dimension of the arrays a, irow and icol. This must be the same value supplied in the preceding call to nag_sparse_herm_chol_fac (f11jnc).
4:    $\mathbf{irow}\left[{\mathbf{la}}\right]$const IntegerInput
5:    $\mathbf{icol}\left[{\mathbf{la}}\right]$const IntegerInput
6:    $\mathbf{ipiv}\left[{\mathbf{n}}\right]$const IntegerInput
7:    $\mathbf{istr}\left[{\mathbf{n}}+1\right]$const IntegerInput
On entry: the values returned in arrays irow, icol, ipiv and istr by a previous call to nag_sparse_herm_chol_fac (f11jnc).
8:    $\mathbf{check}$Nag_SparseSym_CheckDataInput
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 ComplexInput
On entry: the right-hand side vector $y$.
10:  $\mathbf{x}\left[{\mathbf{n}}\right]$ComplexOutput
On exit: the solution vector $x$.
11:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

Check that a, irow, icol, ipiv and istr have not been corrupted between calls to nag_sparse_herm_chol_fac (f11jnc) and nag_sparse_herm_precon_ichol_solve (f11jpc).
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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 2.7.6 in How to Use the NAG Library and its Documentation 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 2.7.5 in How to Use the NAG Library and its Documentation 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
 $δM≤cnεPLDLHPT,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

nag_sparse_herm_precon_ichol_solve (f11jpc) is not threaded in any implementation.

### 9.1Timing

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

## 10Example

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