f11jqc solves a complex sparse Hermitian 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: f11jqc, nag_sparse_complex_herm_solve_ilu or nag_sparse_herm_chol_sol.
f11jqc solves a complex sparse Hermitian linear system of equations
using a preconditioned conjugate gradient method (see Meijerink and Van der Vorst (1977)), or a preconditioned Lanczos method based on the algorithm SYMMLQ (see 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).
f11jqc uses the incomplete Cholesky factorization determined by f11jnc as the preconditioning matrix. A call to f11jqc must always be preceded by a call to f11jnc. Alternative preconditioners for the same storage scheme are available by calling f11jsc.
The matrix and the preconditioning matrix are represented in symmetric coordinate storage (SCS) format (see Section 2.1.2 in the F11 Chapter Introduction) in the arrays a, irow and icol, as returned from f11jnc. 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.
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
1: – Nag_SparseSym_MethodInput
On entry: specifies the iterative method to be used.
Conjugate gradient method.
Lanczos method (SYMMLQ).
2: – IntegerInput
On entry: , the order of the matrix . This must be the same value as was supplied in the preceding call to f11jnc.
3: – IntegerInput
On entry: the number of nonzero elements in the lower triangular part of the matrix . This must be the same value as was supplied in the preceding call to f11jnc.
4: – const ComplexInput
On entry: the values returned in the array a by a previous call to f11jnc.
5: – IntegerInput
On entry: the dimension of the arrays a, irow and icol. This must be the same value as was supplied in the preceding call to f11jnc.
The required accuracy could not be obtained. However a reasonable accuracy has been achieved.
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
The matrix of the coefficients a appears not to be positive definite. The computation cannot continue.
The solution has not converged after iterations.
On entry, .
On entry, .
On entry, .
On entry, and .
On entry, and .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
A serious error, code , has occurred in an internal call. Check all function calls and array sizes. Seek expert help.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, is out of order: .
On entry, the location () is a duplicate: .
The preconditioner appears not to be positive definite. The computation cannot continue.
On entry, .
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
f11jqc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f11jqc 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 f11jqc for each iteration is roughly proportional to the value of nnzc returned from the preceding call to f11jnc. 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 easily be determined a priori, as it can depend dramatically on the conditioning and spectrum of the preconditioned matrix of the coefficients .
This example solves a complex sparse Hermitian positive definite system of equations using the conjugate gradient method, with incomplete Cholesky preconditioning.