# NAG Library Function Document

## 1Purpose

nag_sparse_herm_precon_ssor_solve (f11jrc) solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a complex sparse Hermitian matrix, represented in symmetric coordinate storage format.

## 2Specification

 #include #include
 void nag_sparse_herm_precon_ssor_solve (Integer n, Integer nnz, const Complex a[], const Integer irow[], const Integer icol[], const double rdiag[], double omega, Nag_SparseSym_CheckData check, const Complex y[], Complex x[], NagError *fail)

## 3Description

nag_sparse_herm_precon_ssor_solve (f11jrc) solves a system of equations
 $Mx=y$
involving the preconditioning matrix
 $M=1ω2-ω D+ω L D-1 D+ω LH$
corresponding to symmetric successive-over-relaxation (SSOR) (see Young (1971)) on a linear system $Ax=b$, where $A$ is a sparse complex Hermitian matrix stored in symmetric coordinate storage (SCS) format (see Section 2.1.2 in the f11 Chapter Introduction).
In the definition of $M$ given above $D$ is the diagonal part of $A$, $L$ is the strictly lower triangular part of $A$ and $\omega$ is a user-defined relaxation parameter. Note that since $A$ is Hermitian the matrix $D$ is necessarily real.

## 4References

Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York

## 5Arguments

1:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
2:    $\mathbf{nnz}$IntegerInput
On entry: the number of nonzero elements in the lower triangular part of the matrix $A$.
Constraint: $1\le {\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
3:    $\mathbf{a}\left[{\mathbf{nnz}}\right]$const ComplexInput
On entry: the nonzero elements in the lower triangular part of the matrix $A$, ordered by increasing row index, and by increasing column index within each row. Multiple entries for the same row and column indices are not permitted. The function nag_sparse_herm_sort (f11zpc) may be used to order the elements in this way.
4:    $\mathbf{irow}\left[{\mathbf{nnz}}\right]$const IntegerInput
5:    $\mathbf{icol}\left[{\mathbf{nnz}}\right]$const IntegerInput
On entry: the row and column indices of the nonzero elements supplied in array a.
Constraints:
irow and icol must satisfy the following constraints (which may be imposed by a call to nag_sparse_herm_sort (f11zpc)):
• $1\le {\mathbf{irow}}\left[\mathit{i}\right]\le {\mathbf{n}}$ and $1\le {\mathbf{icol}}\left[\mathit{i}\right]\le {\mathbf{irow}}\left[\mathit{i}\right]$, for $\mathit{i}=0,1,\dots ,{\mathbf{nnz}}-1$;
• ${\mathbf{irow}}\left[\mathit{i}-1\right]<{\mathbf{irow}}\left[\mathit{i}\right]$ or ${\mathbf{irow}}\left[\mathit{i}-1\right]={\mathbf{irow}}\left[\mathit{i}\right]$ and ${\mathbf{icol}}\left[\mathit{i}-1\right]<{\mathbf{icol}}\left[\mathit{i}\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{nnz}}-1$.
6:    $\mathbf{rdiag}\left[{\mathbf{n}}\right]$const doubleInput
On entry: the elements of the diagonal matrix ${D}^{-1}$, where $D$ is the diagonal part of $A$. Note that since $A$ is Hermitian the elements of ${D}^{-1}$ are necessarily real.
7:    $\mathbf{omega}$doubleInput
On entry: the relaxation parameter $\omega$.
Constraint: $0.0<{\mathbf{omega}}<2.0$.
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, nnz, irow, icol and omega.
${\mathbf{check}}=\mathrm{Nag_SparseSym_NoCheck}$
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

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: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{nnz}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nnz}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{nnz}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
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_SCS
On entry, $\mathit{I}=〈\mathit{\text{value}}〉$, ${\mathbf{icol}}\left[\mathit{I}-1\right]=〈\mathit{\text{value}}〉$, ${\mathbf{irow}}\left[\mathit{I}-1\right]=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{icol}}\left[i-1\right]\le {\mathbf{irow}}\left[i-1\right]$.
On entry, $\mathit{I}=〈\mathit{\text{value}}〉$, ${\mathbf{irow}}\left[\mathit{I}-1\right]=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{irow}}\left[\mathit{i}-1\right]\le {\mathbf{n}}$.
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: $\mathit{I}=〈\mathit{\text{value}}〉$. Consider calling nag_sparse_herm_sort (f11zpc) to reorder and sum or remove duplicates.
NE_REAL
On entry, ${\mathbf{omega}}=〈\mathit{\text{value}}〉$.
Constraint: $0.0<{\mathbf{omega}}<2.0$.
NE_ZERO_DIAG_ELEM
The matrix $A$ has no diagonal entry in row $〈\mathit{\text{value}}〉$.

## 7Accuracy

The computed solution $x$ is the exact solution of a perturbed system of equations $\left(M+\delta M\right)x=y$, where
 $δM≤cnεD+ωLD-1D+ωLT,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision.

## 8Parallelism and Performance

nag_sparse_herm_precon_ssor_solve (f11jrc) is not threaded in any implementation.

### 9.1Timing

The time taken for a call to nag_sparse_herm_precon_ssor_solve (f11jrc) is proportional to nnz.

## 10Example

This example program solves the preconditioning equation $Mx=y$ for a $9$ by $9$ sparse complex Hermitian matrix $A$, given in symmetric coordinate storage (SCS) format.

### 10.1Program Text

Program Text (f11jrce.c)

### 10.2Program Data

Program Data (f11jrce.d)

### 10.3Program Results

Program Results (f11jrce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017