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.
The function may be called by the names: f11jec, nag_sparse_real_symm_solve_jacssor or 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:
symmetric successive-over-relaxation (SSOR) preconditioning (see Young (1971)).
For incomplete Cholesky (IC) preconditioning see f11jcc.
The matrix 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.
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
1: – Nag_SparseSym_MethodInput
On entry: specifies the iterative method to be used.
The conjugate gradient method is used.
The Lanczos method (SYMMLQ) is used.
2: – Nag_SparseSym_PrecTypeInput
On entry: specifies the type of preconditioning to be used.
No preconditioning is used.
Symmetric successive-over-relaxation is used.
Jacobi preconditioning is used.
, or .
3: – IntegerInput
On entry: the order of the matrix .
4: – IntegerInput
On entry: the number of nonzero elements in the lower triangular part of the matrix .
5: – const doubleInput
On entry: 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 f11zbc may be used to order the elements in this way.
6: – const IntegerInput
7: – 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 f11zbc):;
and , for ;
or and , for .
8: – doubleInput
On entry: if , omega is the relaxation argument to be used in the SSOR method. Otherwise omega need not be initialized.
9: – const doubleInput
On entry: the right-hand side vector .
10: – doubleInput
On entry: the required tolerance. Let denote the approximate solution at iteration , and the corresponding residual. The algorithm is considered to have converged at iteration if:
If , is used, where is the machine precision. Otherwise is used.
11: – IntegerInput
On entry: the maximum number of iterations allowed.
12: – doubleInput/Output
On entry: an initial approximation of the solution vector .
On exit: an improved approximation to the solution vector .
13: – double *Output
On exit: the final value of the residual norm , where is the output value of itn.
14: – Integer *Output
On exit: the number of iterations carried out.
15: – 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.
16: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
The required accuracy could not be obtained. However, a reasonable accuracy has been obtained and further iterations cannot improve the result.
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 for assistance.
The required accuracy has not been obtained in maxitn iterations.
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
and , for .
Call 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 , where , satisfies the termination criterion
The value of the final residual norm is returned in rnorm.
8Parallelism and Performance
f11jec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f11jec makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The time taken by 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.