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.
The function may be called by the names: f11jcc, nag_sparse_real_symm_solve_ichol or nag_sparse_sym_chol_sol.
3Description
f11jcc solves a real sparse symmetric linear system of equations:
$$Ax=b\text{,}$$
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).
f11jcc uses the incomplete Cholesky factorization determined by f11jac as the preconditioning matrix. A call to f11jcc must always be preceded by a call to f11jac. Alternative preconditioners for the same storage scheme are available by calling 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 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.
4References
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
5Arguments
1: $\mathbf{method}$ – Nag_SparseSym_MethodInput
On entry: specifies the iterative method to be 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 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 f11jac.
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 ${\mathbf{tol}}\le 0.0$, $\tau =\mathrm{max}\phantom{\rule{0.125em}{0ex}}(\sqrt{\epsilon},\sqrt{{\mathbf{n}}}\epsilon )$ is used, where $\epsilon $ is the machine precision. Otherwise $\tau =\mathrm{max}\phantom{\rule{0.125em}{0ex}}({\mathbf{tol}},10\epsilon ,\sqrt{{\mathbf{n}}},\epsilon )$ is used.
Constraint:
${\mathbf{tol}}<1.0$.
12: $\mathbf{maxitn}$ – IntegerInput
On entry: the maximum number of iterations allowed.
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 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_2_INT_ARG_LT
On entry, ${\mathbf{la}}=\u27e8\mathit{\text{value}}\u27e9$ while ${\mathbf{nnz}}=\u27e8\mathit{\text{value}}\u27e9$. 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.
The matrix of coefficients appears not to be positive definite.
NE_INT_2
On entry, ${\mathbf{nnz}}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: $1\le {\mathbf{nnz}}\le {\mathbf{n}}\times ({\mathbf{n}}+1)/2$.
NE_INT_ARG_LT
On entry, ${\mathbf{maxitn}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{maxitn}}\ge 1$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
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 f11jcc has been preceded by a valid call to 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 f11jcc has been preceded by a valid call to 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: ${\mathbf{tol}}=\u27e8\mathit{\text{value}}\u27e9$.
7Accuracy
On successful termination, the final residual ${r}_{k}={b-Ax}_{k}$, where $k={\mathbf{itn}}$, satisfies the termination criterion
The value of the final residual norm is returned in rnorm.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
f11jcc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f11jcc 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.
9Further Comments
The time taken by f11jcc for each iteration is roughly proportional to the value of nnzc returned from the preceding call to 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 $\overline{A}={M}^{\mathrm{-1}}A$.
Some illustrations of the application of 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).
10Example
This example program solves a symmetric positive definite system of equations using the conjugate gradient method, with incomplete Cholesky preconditioning.