naginterfaces.library.sparse.complex_herm_solve_ilu¶
- naginterfaces.library.sparse.complex_herm_solve_ilu(method, nnz, a, irow, icol, ipiv, istr, b, tol, maxitn, x)[source]¶
complex_herm_solve_ilu
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.For full information please refer to the NAG Library document for f11jq
https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/f11/f11jqf.html
- Parameters
- methodstr
Specifies the iterative method to be used.
Conjugate gradient method.
Lanczos method (SYMMLQ).
- nnzint
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
complex_herm_precon_ichol()
.- acomplex, array-like, shape
The values returned in the array by a previous call to
complex_herm_precon_ichol()
.- irowint, array-like, shape
The values returned in arrays , , and by a previous call to
complex_herm_precon_ichol()
.- icolint, array-like, shape
The values returned in arrays , , and by a previous call to
complex_herm_precon_ichol()
.- ipivint, array-like, shape
The values returned in arrays , , and by a previous call to
complex_herm_precon_ichol()
.- istrint, array-like, shape
The values returned in arrays , , and by a previous call to
complex_herm_precon_ichol()
.- bcomplex, array-like, shape
The right-hand side vector .
- tolfloat
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.
- maxitnint
The maximum number of iterations allowed.
- xcomplex, array-like, shape
An initial approximation to the solution vector .
- Returns
- xcomplex, ndarray, shape
An improved approximation to the solution vector .
- rnormfloat
The final value of the residual norm , where is the output value of .
- itnint
The number of iterations carried out.
- Raises
- NagValueError
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: or .
- (errno )
On entry, and .
Constraint: .
- (errno )
On entry, and .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, .
Constraint: .
- (errno )
On entry, the location () is a duplicate: .
- (errno )
On entry, is out of order: .
- (errno )
On entry, , , .
Constraint: and .
- (errno )
On entry, , and .
Constraint: and .
- (errno )
On entry, appears to be invalid.
- (errno )
The solution has not converged after iterations.
- (errno )
The preconditioner appears not to be positive definite. The computation cannot continue.
- (errno )
The matrix of the coefficients appears not to be positive definite. The computation cannot continue.
- (errno )
A serious error, code , has occurred in an internal call. Check all function calls and array sizes. Seek expert help.
- Warns
- NagAlgorithmicWarning
- (errno )
The required accuracy could not be obtained. However a reasonable accuracy has been achieved.
- Notes
complex_herm_solve_ilu
solves a complex sparse Hermitian linear system of equationsusing 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).
complex_herm_solve_ilu
uses the incomplete Cholesky factorization determined bycomplex_herm_precon_ichol()
as the preconditioning matrix. A call tocomplex_herm_solve_ilu
must always be preceded by a call tocomplex_herm_precon_ichol()
. Alternative preconditioners for the same storage scheme are available by callingcomplex_herm_solve_jacssor()
.The matrix and the preconditioning matrix are represented in symmetric coordinate storage (SCS) format (see the F11 Introduction) in the arrays , and , as returned from
complex_herm_precon_ichol()
. The array holds the nonzero entries in the lower triangular parts of these matrices, while and hold the corresponding row and column indices.
- 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