NAG Library Function Document
nag_sparse_sym_chol_sol (f11jcc)
1
Purpose
nag_sparse_sym_chol_sol (f11jcc) solves a real sparse symmetric system of linear equations, represented in symmetric coordinate storage format, using a conjugate gradient or Lanczos method, with incomplete Cholesky preconditioning.
2
Specification
#include <nag.h> 
#include <nagf11.h> 
void 
nag_sparse_sym_chol_sol (Nag_SparseSym_Method method,
Integer n,
Integer nnz,
const double a[],
Integer la,
const Integer irow[],
const Integer icol[],
const Integer ipiv[],
const Integer istr[],
const double b[],
double tol,
Integer maxitn,
double x[],
double *rnorm,
Integer *itn,
Nag_Sparse_Comm *comm,
NagError *fail) 

3
Description
nag_sparse_sym_chol_sol (f11jcc) solves a real sparse symmetric linear system of equations:
using a preconditioned conjugate gradient method (
Meijerink and Van der Vorst (1977)), 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).
nag_sparse_sym_chol_sol (f11jcc) uses the incomplete Cholesky factorization determined by
nag_sparse_sym_chol_fac (f11jac) as the preconditioning matrix. A call to
nag_sparse_sym_chol_sol (f11jcc) must always be preceded by a call to
nag_sparse_sym_chol_fac (f11jac). Alternative preconditioners for the same storage scheme are available by calling
nag_sparse_sym_sol (f11jec).
The matrix
$A$, and the preconditioning matrix
$M$, are represented in symmetric coordinate storage (SCS) format (see the
f11 Chapter Introduction) in the arrays
a,
irow and
icol, as returned from
nag_sparse_sym_chol_fac (f11jac). The array
a holds the nonzero entries in the lower triangular parts of these matrices, 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
Meijerink J and Van der Vorst H (1977) An iterative solution method for linear systems of which the coefficient matrix is a symmetric Mmatrix Math. Comput. 31 148–162
Paige C C and Saunders M A (1975) Solution of sparse indefinite systems of linear equations SIAM J. Numer. Anal. 12 617–629
Salvini S A and Shaw G J (1995) An evaluation of new NAG Library solvers for large sparse symmetric linear systems NAG Technical Report TR1/95
5
Arguments
 1:
$\mathbf{method}$ – Nag_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:
$\mathbf{n}$ – IntegerInput

On entry: the order of the matrix
$A$. This
must be the same value as was supplied in the preceding call to
nag_sparse_sym_chol_fac (f11jac).
Constraint:
${\mathbf{n}}\ge 1$.
 3:
$\mathbf{nnz}$ – IntegerInput

On entry: the number of nonzero elements in the lower triangular part of the matrix
$A$. This
must be the same value as was supplied in the preceding call to
nag_sparse_sym_chol_fac (f11jac).
Constraint:
$1\le {\mathbf{nnz}}\le {\mathbf{n}}\times \left({\mathbf{n}}+1\right)/2$.
 4:
$\mathbf{a}\left[{\mathbf{la}}\right]$ – const doubleInput

On entry: the values returned in array
a by a previous call to
nag_sparse_sym_chol_fac (f11jac).
 5:
$\mathbf{la}$ – IntegerInput

On entry: the
second
dimension of the arrays
a,
irow and
icol.This
must be the same value as returned by a previous call to
nag_sparse_sym_chol_fac (f11jac).
Constraint:
${\mathbf{la}}\ge 2\times {\mathbf{nnz}}$.
 6:
$\mathbf{irow}\left[{\mathbf{la}}\right]$ – const IntegerInput
 7:
$\mathbf{icol}\left[{\mathbf{la}}\right]$ – const IntegerInput
 8:
$\mathbf{ipiv}\left[{\mathbf{n}}\right]$ – const IntegerInput
 9:
$\mathbf{istr}\left[{\mathbf{n}}+1\right]$ – const IntegerInput

On entry: the values returned in the arrays
irow,
icol,
ipiv and
istr by a previous call to
nag_sparse_sym_chol_fac (f11jac).
 10:
$\mathbf{b}\left[{\mathbf{n}}\right]$ – const doubleInput

On entry: the righthand side vector $b$.
 11:
$\mathbf{tol}$ – doubleInput

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:
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:
${\mathbf{tol}}<1.0$.
 12:
$\mathbf{maxitn}$ – IntegerInput

On entry: the maximum number of iterations allowed.
Constraint:
${\mathbf{maxitn}}\ge 1$.
 13:
$\mathbf{x}\left[{\mathbf{n}}\right]$ – doubleInput/Output

On entry: an initial approximation to the solution vector $x$.
On exit: an improved approximation to the solution vector $x$.
 14:
$\mathbf{rnorm}$ – double *Output

On exit: the final value of the residual norm
${\Vert {r}_{k}\Vert}_{\infty}$, where
$k$ is the output value of
itn.
 15:
$\mathbf{itn}$ – Integer *Output

On exit: the number of iterations carried out.
 16:
$\mathbf{comm}$ – Nag_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.
 17:
$\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
 NE_2_INT_ARG_LT

On entry, ${\mathbf{la}}=\u2329\mathit{\text{value}}\u232a$ while ${\mathbf{nnz}}=\u2329\mathit{\text{value}}\u232a$. These arguments must satisfy ${\mathbf{la}}\ge 2\times {\mathbf{nnz}}$.
 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.
 NE_BAD_PARAM

On entry, argument
method had an illegal value.
 NE_COEFF_NOT_POS_DEF

The matrix of coefficients appears not to be positive definite.
 NE_INT_2

On entry, ${\mathbf{nnz}}=\u2329\mathit{\text{value}}\u232a$, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: $1\le {\mathbf{nnz}}\le {\mathbf{n}}\times \left({\mathbf{n}}+1\right)/2$.
 NE_INT_ARG_LT

On entry, ${\mathbf{maxitn}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{maxitn}}\ge 1$.
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
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_INVALID_SCS

The SCS representation of the matrix
$A$ is invalid. Check that the call to
nag_sparse_sym_chol_sol (f11jcc) has been preceded by a valid call to
nag_sparse_sym_chol_fac (f11jac), and that the arrays
a,
irow and
icol have not been corrupted between the two calls.
 NE_INVALID_SCS_PRECOND

The SCS representation of the preconditioning matrix
$M$ is invalid. Check that the call to
nag_sparse_sym_chol_sol (f11jcc) has been preceded by a valid call to
nag_sparse_sym_chol_fac (f11jac), and that the arrays
a,
irow,
icol,
ipiv and
istr have not been corrupted between the two calls.
 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_ARG_GE

On entry,
tol must not be greater than or equal to 1.0:
${\mathbf{tol}}=\u2329\mathit{\text{value}}\u232a$.
7
Accuracy
On successful termination, the final residual
${r}_{k}={bAx}_{k}$, where
$k={\mathbf{itn}}$, satisfies the termination criterion
The value of the final residual norm is returned in
rnorm.
8
Parallelism and Performance
nag_sparse_sym_chol_sol (f11jcc) is not threaded in any implementation.
The time taken by
nag_sparse_sym_chol_sol (f11jcc) for each iteration is roughly proportional to the value of
nnzc returned from the preceding call to
nag_sparse_sym_chol_fac (f11jac). 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 $\stackrel{}{A}={M}^{1}A$.
Some illustrations of the application of
nag_sparse_sym_chol_sol (f11jcc) to linear systems arising from the discretization of twodimensional elliptic partial differential equations, and to randomvalued randomly structured symmetric positive definite linear systems, can be found in
Salvini and Shaw (1995).
10
Example
This example program solves a symmetric positive definite system of equations using the conjugate gradient method, with incomplete Cholesky preconditioning.
10.1
Program Text
Program Text (f11jcce.c)
10.2
Program Data
Program Data (f11jcce.d)
10.3
Program Results
Program Results (f11jcce.r)