# NAG C Library Function Document

## 1Purpose

nag_sparse_sym_precon_ssor_solve (f11jdc) solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a real sparse symmetric matrix, represented in symmetric coordinate storage format.

## 2Specification

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

## 3Description

nag_sparse_sym_precon_ssor_solve (f11jdc) solves a system of equations
 $Mx=y$
involving the preconditioning matrix
 $M=1ω2-ω D+ω L D-1 D+ω LT$
corresponding to symmetric successive-over-relaxation (SSOR) (see Young (1971)) on a linear system $Ax=b$, where $A$ is a sparse symmetric 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.
It is envisaged that a common use of nag_sparse_sym_precon_ssor_solve (f11jdc) will be to carry out the preconditioning step required in the application of nag_sparse_sym_basic_solver (f11gec) to sparse linear systems. For an illustration of this use of nag_sparse_sym_precon_ssor_solve (f11jdc) see the example program given in Section 10.1. nag_sparse_sym_precon_ssor_solve (f11jdc) is also used for this purpose by the Black Box function nag_sparse_sym_sol (f11jec).

## 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 $A$.
Constraint: $1\le {\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
3:    $\mathbf{a}\left[{\mathbf{nnz}}\right]$const doubleInput
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_sym_sort (f11zbc) 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 these constraints (which may be imposed by a call to nag_sparse_sym_sort (f11zbc)):
• $1\le {\mathbf{irow}}\left[\mathit{i}\right]\le {\mathbf{n}}$ 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 $A$.
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 doubleInput
On entry: the right-hand side vector $y$.
10:  $\mathbf{x}\left[{\mathbf{n}}\right]$doubleOutput
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

Consider calling nag_sparse_sym_sort (f11zbc) to reorder and sum or remove duplicates.
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, $\mathit{I}=〈\mathit{\text{value}}〉$, ${\mathbf{icol}}\left[\mathit{I}-1\right]=〈\mathit{\text{value}}〉$ 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_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: $\mathit{I}=〈\mathit{\text{value}}〉$.
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_sym_precon_ssor_solve (f11jdc) is not threaded in any implementation.

### 9.1Timing

The time taken for a call to nag_sparse_sym_precon_ssor_solve (f11jdc) is proportional to nnz.

### 9.2Use of check

It is expected that a common use of nag_sparse_sym_precon_ssor_solve (f11jdc) will be to carry out the preconditioning step required in the application of nag_sparse_sym_basic_solver (f11gec) to sparse symmetric linear systems. In this situation nag_sparse_sym_precon_ssor_solve (f11jdc) 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 solves a sparse symmetric linear system of equations
 $Ax=b,$
using the conjugate-gradient (CG) method with SSOR preconditioning.
The CG algorithm itself is implemented by the reverse communication function nag_sparse_sym_basic_solver (f11gec), which returns repeatedly to the calling program with various values of the argument irevcm. This argument indicates the action to be taken by the calling program.
• If ${\mathbf{irevcm}}=1$, a matrix-vector product $v=Au$ is required. This is implemented by a call to nag_sparse_sym_matvec (f11xec).
• If ${\mathbf{irevcm}}=2$, a solution of the preconditioning equation $Mv=u$ is required. This is achieved by a call to nag_sparse_sym_precon_ssor_solve (f11jdc).
• If ${\mathbf{irevcm}}=4$, nag_sparse_sym_basic_solver (f11gec) has completed its tasks. Either the iteration has terminated, or an error condition has arisen.
For further details see the function document for nag_sparse_sym_basic_solver (f11gec).

### 10.1Program Text

Program Text (f11jdce.c)

### 10.2Program Data

Program Data (f11jdce.d)

### 10.3Program Results

Program Results (f11jdce.r)