# NAG Library Function Document

## 1Purpose

nag_sparse_nherm_precon_ssor_solve (f11drc) solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a complex sparse non-Hermitian matrix, represented in coordinate storage format.

## 2Specification

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

## 3Description

nag_sparse_nherm_precon_ssor_solve (f11drc) solves a system of linear equations
 $Mx=y, or MHx=y,$
according to the value of the argument trans, where the matrix
 $M=1ω2-ω D+ω L D-1 D+ω U$
corresponds to symmetric successive-over-relaxation (SSOR) Young (1971) applied to a linear system $Ax=b$, where $A$ is a complex sparse non-Hermitian matrix stored in coordinate storage (CS) format (see Section 2.1.1 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 $A$, $U$ is the strictly upper triangular part of $A$, and $\omega$ is a user-defined relaxation parameter.
It is envisaged that a common use of nag_sparse_nherm_precon_ssor_solve (f11drc) will be to carry out the preconditioning step required in the application of nag_sparse_nherm_basic_solver (f11bsc) to sparse linear systems. For an illustration of this use of nag_sparse_nherm_precon_ssor_solve (f11drc) see the example program given in Section 10. nag_sparse_nherm_precon_ssor_solve (f11drc) is also used for this purpose by the Black Box function nag_sparse_nherm_sol (f11dsc).

## 4References

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

## 5Arguments

1:    $\mathbf{trans}$Nag_TransTypeInput
On entry: specifies whether or not the matrix $M$ is transposed.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$Mx=y$ is solved.
${\mathbf{trans}}=\mathrm{Nag_Trans}$
${M}^{\mathrm{H}}x=y$ is solved.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ or $\mathrm{Nag_Trans}$.
2:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
3:    $\mathbf{nnz}$IntegerInput
On entry: the number of nonzero elements in the matrix $A$.
Constraint: $1\le {\mathbf{nnz}}\le {{\mathbf{n}}}^{2}$.
4:    $\mathbf{a}\left[{\mathbf{nnz}}\right]$const ComplexInput
On entry: the nonzero elements in 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_nherm_sort (f11znc) may be used to order the elements in this way.
5:    $\mathbf{irow}\left[{\mathbf{nnz}}\right]$const IntegerInput
6:    $\mathbf{icol}\left[{\mathbf{nnz}}\right]$const IntegerInput
On entry: the row and column indices of the nonzero elements supplied in a.
Constraints:
irow and icol must satisfy the following constraints (which may be imposed by a call to nag_sparse_nherm_sort (f11znc)):
• $1\le {\mathbf{irow}}\left[\mathit{i}\right]\le {\mathbf{n}}$ and $1\le {\mathbf{icol}}\left[\mathit{i}\right]\le {\mathbf{n}}$, for $\mathit{i}=0,1,\dots ,{\mathbf{nnz}}-1$;
• either ${\mathbf{irow}}\left[\mathit{i}-1\right]<{\mathbf{irow}}\left[\mathit{i}\right]$ or both ${\mathbf{irow}}\left[\mathit{i}-1\right]={\mathbf{irow}}\left[\mathit{i}\right]$ and ${\mathbf{icol}}\left[\mathit{i}-1\right]<{\mathbf{icol}}\left[\mathit{i}\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{nnz}}-1$.
7:    $\mathbf{rdiag}\left[{\mathbf{n}}\right]$const ComplexInput
On entry: the elements of the diagonal matrix ${D}^{-1}$, where $D$ is the diagonal part of $A$.
8:    $\mathbf{omega}$doubleInput
On entry: the relaxation parameter $\omega$.
Constraint: $0.0<{\mathbf{omega}}<2.0$.
9:    $\mathbf{check}$Nag_SparseNsym_CheckDataInput
On entry: specifies whether or not the CS representation of the matrix $M$ should be checked.
${\mathbf{check}}=\mathrm{Nag_SparseNsym_Check}$
Checks are carried on the values of n, nnz, irow, icol and omega.
${\mathbf{check}}=\mathrm{Nag_SparseNsym_NoCheck}$
None of these checks are carried out.
Constraint: ${\mathbf{check}}=\mathrm{Nag_SparseNsym_Check}$ or $\mathrm{Nag_SparseNsym_NoCheck}$.
10:  $\mathbf{y}\left[{\mathbf{n}}\right]$const ComplexInput
On entry: the right-hand side vector $y$.
11:  $\mathbf{x}\left[{\mathbf{n}}\right]$ComplexOutput
On exit: the solution vector $x$.
12:  $\mathbf{fail}$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: ${\mathbf{nnz}}\le {{\mathbf{n}}}^{2}$.
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_CS
On entry, $i=〈\mathit{\text{value}}〉$, ${\mathbf{icol}}\left[i-1\right]=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{icol}}\left[i-1\right]\ge 1$ and ${\mathbf{icol}}\left[i-1\right]\le {\mathbf{n}}$.
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

If ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ the computed solution $x$ is the exact solution of a perturbed system of equations $\left(M+\delta M\right)x=y$, where
 $δM≤cnεD+ωLD-1D+ωU,$
$c\left(n\right)$ is a modest linear function of $n$, and $\epsilon$ is the machine precision. An equivalent result holds when ${\mathbf{trans}}=\mathrm{Nag_Trans}$.

## 8Parallelism and Performance

nag_sparse_nherm_precon_ssor_solve (f11drc) is not threaded in any implementation.

### 9.1Timing

The time taken for a call to nag_sparse_nherm_precon_ssor_solve (f11drc) is proportional to nnz.

### 9.2Use of check

It is expected that a common use of nag_sparse_nherm_precon_ssor_solve (f11drc) will be to carry out the preconditioning step required in the application of nag_sparse_nherm_basic_solver (f11bsc) to sparse linear systems. In this situation nag_sparse_nherm_precon_ssor_solve (f11drc) 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 ${\mathbf{check}}=\mathrm{Nag_SparseNsym_Check}$ for the first of such calls, and ${\mathbf{check}}=\mathrm{Nag_SparseNsym_NoCheck}$ for all subsequent calls.

## 10Example

This example solves a complex sparse linear system of equations
 $Ax=b,$
using RGMRES with SSOR preconditioning.
The RGMRES algorithm itself is implemented by the reverse communication function nag_sparse_nherm_basic_solver (f11bsc), 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_nherm_matvec (f11xnc).
• If ${\mathbf{irevcm}}=-1$, a conjugate transposed matrix-vector product $v={A}^{\mathrm{H}}u$ is required in the estimation of the norm of $A$. This is implemented by a call to nag_sparse_nherm_matvec (f11xnc).
• If ${\mathbf{irevcm}}=2$, a solution of the preconditioning equation $Mv=u$ is required. This is achieved by a call to nag_sparse_nherm_precon_ssor_solve (f11drc).
• If ${\mathbf{irevcm}}=4$, nag_sparse_nherm_basic_solver (f11bsc) 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_nherm_basic_solver (f11bsc).

### 10.1Program Text

Program Text (f11drce.c)

### 10.2Program Data

Program Data (f11drce.d)

### 10.3Program Results

Program Results (f11drce.r)

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