# NAG CL Interfacef11ddc (real_​gen_​precon_​ssor_​solve)

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## 1Purpose

f11ddc solves a system of linear equations involving the preconditioning matrix corresponding to SSOR applied to a real sparse nonsymmetric matrix, represented in coordinate storage format.

## 2Specification

 #include
 void f11ddc (Nag_TransType trans, Integer n, Integer nnz, const double a[], const Integer irow[], const Integer icol[], const double rdiag[], double omega, Nag_SparseNsym_CheckData check, const double y[], double x[], NagError *fail)
The function may be called by the names: f11ddc, nag_sparse_real_gen_precon_ssor_solve or nag_sparse_nsym_precon_ssor_solve.

## 3Description

f11ddc solves a system of linear equations
 $Mx=y, or MTx=y,$
according to the value of the argument trans, where the matrix
 $M=1ω(2-ω) (D+ωL) D-1 (D+ωU)$
corresponds to symmetric successive-over-relaxation (SSOR) (see Young (1971)) applied to a linear system $Ax=b$, where $A$ is a real sparse nonsymmetric 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 f11ddc will be to carry out the preconditioning step required in the application of f11bec to sparse linear systems. f11ddc is also used for this purpose by the Black Box function f11dec.

## 4References

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

## 5Arguments

1: $\mathbf{trans}$Nag_TransType Input
On entry: specifies whether or not the matrix $M$ is transposed.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
$Mx=y$ is solved.
${\mathbf{trans}}=\mathrm{Nag_Trans}$
${M}^{\mathrm{T}}x=y$ is solved.
Constraint: ${\mathbf{trans}}=\mathrm{Nag_NoTrans}$ or $\mathrm{Nag_Trans}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
3: $\mathbf{nnz}$Integer Input
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 double Input
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 f11zac may be used to order the elements in this way.
5: $\mathbf{irow}\left[{\mathbf{nnz}}\right]$const Integer Input
6: $\mathbf{icol}\left[{\mathbf{nnz}}\right]$const Integer Input
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 f11zac):
• $1\le {\mathbf{irow}}\left[\mathit{i}\right]\le {\mathbf{n}}$ 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 double Input
On entry: the elements of the diagonal matrix ${D}^{-1}$, where $D$ is the diagonal part of $A$.
8: $\mathbf{omega}$double Input
On entry: the relaxation parameter $\omega$.
Constraint: $0.0<{\mathbf{omega}}<2.0$.
9: $\mathbf{check}$Nag_SparseNsym_CheckData Input
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 double Input
On entry: the right-hand side vector $y$.
11: $\mathbf{x}\left[{\mathbf{n}}\right]$double Output
On exit: the solution vector $x$.
12: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

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: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{nnz}}=⟨\mathit{\text{value}}⟩$.
Constraint: $1\le {\mathbf{nnz}}\le {{\mathbf{n}}}^{2}$.
NE_INT_2
On entry, ${\mathbf{nnz}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: $1\le {\mathbf{nnz}}\le {{\mathbf{n}}}^{2}$.
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 7.5 in the Introduction to the NAG Library CL Interface 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 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: $\mathit{I}=⟨\mathit{\text{value}}⟩$.
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 $⟨\mathit{\text{value}}⟩$.
The SSOR preconditioner is not appropriate for this problem.

## 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
 $|δM|≤c(n)ε|D+ωL||D-1||D+ωU|,$
$c\left(n\right)$ 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

f11ddc is not threaded in any implementation.

### 9.1Timing

The time taken for a call to f11ddc is proportional to nnz.

### 9.2Use of check

It is expected that a common use of f11ddc will be to carry out the preconditioning step required in the application of f11bec to sparse linear systems. In this situation f11ddc 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 for all subsequent calls set ${\mathbf{check}}=\mathrm{Nag_SparseNsym_NoCheck}$.

## 10Example

This example solves a sparse linear system of equations:
 $Ax=b,$
using RGMRES with SSOR preconditioning.
The RGMRES algorithm itself is implemented by the reverse communication function f11bec, 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 f11xac.
• If ${\mathbf{irevcm}}=-1$, a transposed matrix-vector product $v={A}^{\mathrm{T}}u$ is required in the estimation of the norm of $A$. This is implemented by a call to f11xac.
• If ${\mathbf{irevcm}}=2$, a solution of the preconditioning equation $Mv=u$ is required. This is achieved by a call to f11ddc.
• If ${\mathbf{irevcm}}=4$, f11bec has completed its tasks. Either the iteration has terminated, or an error condition has arisen.
For further details see the function document for f11bec.

### 10.1Program Text

Program Text (f11ddce.c)

### 10.2Program Data

Program Data (f11ddce.d)

### 10.3Program Results

Program Results (f11ddce.r)