NAG Library Function Document
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.
||nag_sparse_sym_sol (Nag_SparseSym_Method method,
const double a,
const Integer irow,
const Integer icol,
const double b,
nag_sparse_sym_sol (f11jec) solves a real sparse symmetric linear system of equations:
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
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)
is represented in symmetric coordinate storage (SCS) format (see the f11 Chapter Introduction
) in the arrays a
. The array a
holds the nonzero entries in the lower triangular part of the matrix, while irow
hold the corresponding row and column indices.
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
method – Nag_SparseSym_MethodInput
: specifies the iterative method to be used.
- The conjugate gradient method is used.
- The Lanczos method (SYMMLQ) is used.
precon – Nag_SparseSym_PrecTypeInput
: specifies the type of preconditioning to be used.
- No preconditioning is used.
- Symmetric successive-over-relaxation is used.
- Jacobi preconditioning is used.
, or .
n – IntegerInput
On entry: the order of the matrix .
nnz – IntegerInput
On entry: the number of nonzero elements in the lower triangular part of the matrix .
a[nnz] – const doubleInput
: the nonzero elements of the lower triangular part of the matrix
, 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.
irow[nnz] – const IntegerInput
icol[nnz] – const IntegerInput
On entry: the row and column indices of the nonzero elements supplied in .
- irow and icol must satisfy the following constraints (which may be imposed by a call to nag_sparse_sym_sort (f11zbc)):;
- and , for ;
- or and , for .
omega – doubleInput
is the relaxation argument
to be used in the SSOR method. Otherwise omega
need not be initialized.
b[n] – const doubleInput
On entry: the right-hand side vector .
tol – doubleInput
: the required tolerance. Let
denote the approximate solution at iteration
the corresponding residual. The algorithm is considered to have converged at iteration
is used, where
is the machine precision
maxitn – IntegerInput
On entry: the maximum number of iterations allowed.
x[n] – doubleInput/Output
On entry: an initial approximation of the solution vector .
On exit: an improved approximation to the solution vector .
rnorm – double *Output
: the final value of the residual norm
is the output value of itn
itn – Integer *Output
On exit: the number of iterations carried out.
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.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
The required accuracy could not be obtained. However, a reasonable accuracy has been obtained and further iterations cannot improve the result.
Dynamic memory allocation failed.
On entry, argument method
had an illegal value.
On entry, argument precon
had an illegal value.
The matrix of coefficients appears not to be positive definite (conjugate gradient method only).
On entry, , .
On entry, .
On entry, .
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
The required accuracy has not been obtained in maxitn
The preconditioner appears not to be positive definite.
On entry, .
On entry, tol
must not be greater than or equal to 1.0:
A nonzero element has been supplied which does not lie in the lower triangular part of the matrix
, is out of order, or has duplicate row and column indices, i.e., one or more of the following constraints has been violated:
- and , for
- , or
- and , for .
Call nag_sparse_sym_sort (f11zbc)
to reorder and sum or remove duplicates.
The matrix has a zero diagonal element. Jacobi and SSOR preconditioners are not appropriate for this problem.
On successful termination, the final residual
, satisfies the termination criterion
The value of the final residual norm is returned in rnorm
8 Parallelism and Performance
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 .
This example program solves a symmetric positive definite system of equations using the conjugate gradient method, with SSOR preconditioning.
10.1 Program Text
Program Text (f11jece.c)
10.2 Program Data
Program Data (f11jece.d)
10.3 Program Results
Program Results (f11jece.r)