f11 Chapter Contents
f11 Chapter Introduction
NAG C Library Manual

NAG Library Function Documentnag_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 #include
 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:
 $Ax = b ,$
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 M-matrix 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:     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:     nIntegerInput
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:     nnzIntegerInput
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: $1\le {\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
4:     a[la]const doubleInput
On entry: the values returned in array a by a previous call to nag_sparse_sym_chol_fac (f11jac).
5:     laIntegerInput
On entry: this must be the same value as returned by a previous call to nag_sparse_sym_chol_fac (f11jac).
Constraint: ${\mathbf{la}}\ge 2×{\mathbf{nnz}}$.
6:     irow[la]const IntegerInput
7:     icol[la]const IntegerInput
8:     ipiv[n]const IntegerInput
9:     istr[${\mathbf{n}}+1$]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:   b[n]const doubleInput
On entry: the right-hand side vector $b$.
11:   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: ${\mathbf{tol}}<1.0$.
12:   maxitnIntegerInput
On entry: the maximum number of iterations allowed.
Constraint: ${\mathbf{maxitn}}\ge 1$.
13:   x[n]doubleInput/Output
On entry: an initial approximation to the solution vector $x$.
On exit: an improved approximation to the solution vector $x$.
14:   rnormdouble *Output
On exit: the final value of the residual norm ${‖{r}_{k}‖}_{\infty }$, where $k$ is the output value of itn.
15:   itnInteger *Output
On exit: the number of iterations carried out.
16:   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.
17:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_2_INT_ARG_LT
On entry, ${\mathbf{la}}=〈\mathit{\text{value}}〉$ while ${\mathbf{nnz}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{la}}\ge 2×{\mathbf{nnz}}$.
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.
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, ${\mathbf{nnz}}=〈\mathit{\text{value}}〉$, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
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_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: ${\mathbf{tol}}=〈\mathit{\text{value}}〉$.

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_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 two-dimensional elliptic partial differential equations, and to random-valued randomly structured symmetric positive definite linear systems, can be found in Salvini and Shaw (1995).

9  Example

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

9.1  Program Text

Program Text (f11jcce.c)

9.2  Program Data

Program Data (f11jcce.d)

9.3  Program Results

Program Results (f11jcce.r)