f11 Chapter Contents
f11 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_sparse_sym_sol (f11jec)

## 1  Purpose

nag_sparse_sym_sol (f11jec) solves a real sparse symmetric system of linear equations, represented in symmetric coordinate storage format, using a conjugate gradient or Lanczos method, without preconditioning, with Jacobi or with SSOR preconditioning.

## 2  Specification

 #include #include
 void nag_sparse_sym_sol (Nag_SparseSym_Method method, Nag_SparseSym_PrecType precon, Integer n, Integer nnz, const double a[], const Integer irow[], const Integer icol[], double omega, const double b[], double tol, Integer maxitn, double x[], double *rnorm, Integer *itn, Nag_Sparse_Comm *comm, NagError *fail)

## 3  Description

nag_sparse_sym_sol (f11jec) solves a real sparse symmetric linear system of equations:
 $Ax = b ,$
using a preconditioned conjugate gradient method (see Barrett et al. (1994)), or a preconditioned Lanczos method based on the algorithm SYMMLQ (Paige and Saunders (1975)). The conjugate gradient method is more efficient if $A$ is positive definite, but may fail to converge for indefinite matrices. In this case the Lanczos method should be used instead. For further details see Barrett et al. (1994).
The function allows the following choices for the preconditioner:
• no preconditioning;
• Jacobi preconditioning (see Young (1971);
• symmetric successive-over-relaxation (SSOR) preconditioning (see Young (1971)).
For incomplete Cholesky (IC) preconditioning see nag_sparse_sym_chol_sol (f11jcc).
The matrix $A$ is represented in symmetric coordinate storage (SCS) format (see the f11 Chapter Introduction) in the arrays a, irow and icol. The array a holds the nonzero entries in the lower triangular part of the matrix, while irow and icol hold the corresponding row and column indices.

## 4  References

Barrett R, Berry M, Chan T F, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C and Van der Vorst H (1994) Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods SIAM, Philadelphia
Paige C C and Saunders M A (1975) Solution of sparse indefinite systems of linear equations SIAM J. Numer. Anal. 12 617–629
Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York

## 5  Arguments

1:     methodNag_SparseSym_MethodInput
On entry: specifies the iterative method to be used.
${\mathbf{method}}=\mathrm{Nag_SparseSym_CG}$
The conjugate gradient method is used.
${\mathbf{method}}=\mathrm{Nag_SparseSym_Lanczos}$
The Lanczos method (SYMMLQ) is used.
Constraint: ${\mathbf{method}}=\mathrm{Nag_SparseSym_CG}$ or $\mathrm{Nag_SparseSym_Lanczos}$.
2:     preconNag_SparseSym_PrecTypeInput
On entry: specifies the type of preconditioning to be used.
${\mathbf{precon}}=\mathrm{Nag_SparseSym_NoPrec}$
No preconditioning is used.
${\mathbf{precon}}=\mathrm{Nag_SparseSym_SSORPrec}$
Symmetric successive-over-relaxation is used.
${\mathbf{precon}}=\mathrm{Nag_SparseSym_JacPrec}$
Jacobi preconditioning is used.
Constraint: ${\mathbf{precon}}=\mathrm{Nag_SparseSym_NoPrec}$, $\mathrm{Nag_SparseSym_SSORPrec}$ or $\mathrm{Nag_SparseSym_JacPrec}$.
3:     nIntegerInput
On entry: the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
4:     nnzIntegerInput
On entry: the number of nonzero elements in the lower triangular part of the matrix $A$.
Constraint: $1\le {\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
5:     a[nnz]const doubleInput
On entry: the nonzero elements of 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.
6:     irow[nnz]const IntegerInput
7:     icol[nnz]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_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 $\mathit{i}=1,2,\dots ,{\mathbf{nnz}}-1$.
On entry: if ${\mathbf{precon}}=\mathrm{Nag_SparseSym_SSORPrec}$, omega is the relaxation argument $\omega$ to be used in the SSOR method. Otherwise omega need not be initialized.
Constraint: $0.0\le {\mathbf{omega}}\le 2.0$.
9:     b[n]const doubleInput
On entry: the right-hand side vector $b$.
10:   toldoubleInput
On entry: the required tolerance. Let ${x}_{k}$ denote the approximate solution at iteration $k$, and ${r}_{k}$ the corresponding residual. The algorithm is considered to have converged at iteration $k$ if:
 $r k ∞ ≤ τ × b ∞ + A ∞ x k ∞ .$
If ${\mathbf{tol}}\le 0.0$, $\tau =\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(\sqrt{\epsilon },\sqrt{{\mathbf{n}}},\epsilon \right)$ is used, where $\epsilon$ is the machine precision. Otherwise $\tau =\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{tol}},10\epsilon ,\sqrt{{\mathbf{n}}},\epsilon \right)$ is used.
Constraint: ${\mathbf{tol}}<1.0$.
11:   maxitnIntegerInput
On entry: the maximum number of iterations allowed.
Constraint: ${\mathbf{maxitn}}\ge 1$.
12:   x[n]doubleInput/Output
On entry: an initial approximation of the solution vector $x$.
On exit: an improved approximation to the solution vector $x$.
13:   rnormdouble *Output
On exit: the final value of the residual norm ${‖{r}_{k}‖}_{\infty }$, where $k$ is the output value of itn.
14:   itnInteger *Output
On exit: the number of iterations carried out.
15:   commNag_Sparse_Comm *Input/Output
On entry/exit: a pointer to a structure of type Nag_Sparse_Comm whose members are used by the iterative solver.
16:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ACC_LIMIT
The required accuracy could not be obtained. However, a reasonable accuracy has been obtained and further iterations cannot improve the result.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument method had an illegal value.
On entry, argument precon had an illegal value.
NE_COEFF_NOT_POS_DEF
The matrix of coefficients appears not to be positive definite (conjugate gradient method only).
NE_INT_2
On entry, ${\mathbf{nnz}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
NE_INT_ARG_LT
On entry, ${\mathbf{maxitn}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{maxitn}}\ge 1$.
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.
NE_NOT_REQ_ACC
The required accuracy has not been obtained in maxitn iterations.
NE_PRECOND_NOT_POS_DEF
The preconditioner appears not to be positive definite.
NE_REAL
On entry, ${\mathbf{omega}}=〈\mathit{\text{value}}〉$.
Constraint: $0.0\le {\mathbf{omega}}\le 2.0$.
NE_REAL_ARG_GE
On entry, tol must not be greater than or equal to 1.0: ${\mathbf{tol}}=〈\mathit{\text{value}}〉$.
NE_SYMM_MATRIX_DUP
A nonzero element has been supplied which does not lie in the lower triangular part of the matrix $A$, is out of order, or has duplicate row and column indices, i.e., one or more of the following constraints has been violated:
$1\le {\mathbf{irow}}\left[\mathit{i}\right]\le {\mathbf{n}}$ 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 $\mathit{i}=1,2,\dots ,{\mathbf{nnz}}-1$.
Call nag_sparse_sym_sort (f11zbc) to reorder and sum or remove duplicates.
NE_ZERO_DIAGONAL_ELEM
The matrix $A$ has a zero diagonal element. Jacobi and SSOR preconditioners are not appropriate for this problem.

## 7  Accuracy

On successful termination, the final residual ${r}_{k}={b-Ax}_{k}$, where $k={\mathbf{itn}}$, satisfies the termination criterion
 $r k ∞ ≤ τ × b ∞ + A ∞ x k ∞ .$
The value of the final residual norm is returned in rnorm.

The time taken by nag_sparse_sym_sol (f11jec) for each iteration is roughly proportional to nnz. One iteration with the Lanczos method (SYMMLQ) requires a slightly larger number of operations than one iteration with the conjugate gradient method.
The number of iterations required to achieve a prescribed accuracy cannot be easily determined a priori, as it can depend dramatically on the conditioning and spectrum of the preconditioned matrix of the coefficients $\stackrel{-}{A}={M}^{-1}A$.

## 9  Example

This example program solves a symmetric positive definite system of equations using the conjugate gradient method, with SSOR preconditioning.

### 9.1  Program Text

Program Text (f11jece.c)

### 9.2  Program Data

Program Data (f11jece.d)

### 9.3  Program Results

Program Results (f11jece.r)