library.sparse Submodule

Module Summary

Interfaces for the NAG Mark 30.3 sparse Chapter.

sparse - Large Scale Linear Systems

This module provides functions for the solution of large sparse systems of simultaneous linear equations. These include iterative methods for real nonsymmetric and symmetric, complex non-Hermitian and Hermitian linear systems and direct methods for general real linear systems. Further direct methods are currently available in submodule matop and submodule linsys.

See Also

naginterfaces.library.examples.sparse :

This subpackage contains examples for the sparse module. See also the Examples subsection.

Functionality Index

Basic functions for complex Hermitian linear systems

diagnostic function: complex_herm_basic_diag()

reverse communication CG or SYMMLQ solver function: complex_herm_basic_solver()

setup function: complex_herm_basic_setup()

Basic functions for complex non-Hermitian linear systems

diagnostic function: complex_gen_basic_diag()

reverse communication RGMRES, CGS, Bi-CGSTAB$\left(\ell \right)$ or TFQMR solver function: complex_gen_basic_solver()

setup function: complex_gen_basic_setup()

Basic functions for real nonsymmetric linear systems

diagnostic function: real_gen_basic_diag()

reverse communication RGMRES, CGS, Bi-CGSTAB$\left(\ell \right)$ or TFQMR solver function: real_gen_basic_solver()

setup function: real_gen_basic_setup()

Basic functions for real symmetric linear systems

diagnostic function: real_symm_basic_diag()

reverse communication CG or SYMMLQ solver: real_symm_basic_solver()

setup function: real_symm_basic_setup()

Black Box functions for complex Hermitian linear systems

CG or SYMMLQ solver

with incomplete Cholesky preconditioning: complex_herm_solve_ilu()

with no preconditioning, Jacobi or SSOR preconditioning: complex_herm_solve_jacssor()

Black Box functions for complex non-Hermitian linear systems

RGMRES, CGS, Bi-CGSTAB$\left(\ell \right)$ or TFQMR solver

with block Jacobi or additive Schwarz preconditioning: complex_gen_solve_bdilu()

with incomplete $LU$ preconditioning: complex_gen_solve_ilu()

with no preconditioning, Jacobi, or SSOR preconditioning: complex_gen_solve_jacssor()

Black Box functions for real nonsymmetric linear systems

RGMRES, CGS, Bi-CGSTAB$\left(\ell \right)$ or TFQMR solver

with block Jacobi or additive Schwarz preconditioning: real_gen_solve_bdilu()

with incomplete $LU$ preconditioning: real_gen_solve_ilu()

with no preconditioning, Jacobi, or SSOR preconditioning: real_gen_solve_jacssor()

Black Box functions for real symmetric linear systems

CG or SYMMLQ solver

with incomplete Cholesky preconditioning: real_symm_solve_ichol()

with no preconditioning, Jacobi, or SSOR preconditioning: real_symm_solve_jacssor()

Direct methods for real sparse nonsymmetric linear systems in CCS format

apply iterative refinement to the solution and compute error estimates, after factorizing the matrix of coefficients: direct_real_gen_refine()

condition number estimation, after factorizing the matrix of coefficients: direct_real_gen_cond()

$LU$ factorization

diagnostic: direct_real_gen_diag()

factorize: direct_real_gen_lu()

setup: direct_real_gen_setup()

solution of simultaneous linear equations, after factorizing the matrix of coefficients: direct_real_gen_solve()

utility

compute a norm or the element of largest absolute value: direct_real_gen_norm()

matrix-matrix multiplier: direct_real_gen_matmul()

Utility function for complex Hermitian linear systems

incomplete Cholesky factorization: complex_herm_precon_ichol()

matrix-vector multiplier for complex Hermitian matrices in SCS format: complex_herm_matvec()

solver for linear systems involving preconditioning matrix from complex_herm_precon_ichol(): complex_herm_precon_ilu_solve()

solver for linear systems involving SSOR preconditioning matrix: complex_herm_precon_ssor_solve()

sort function for complex Hermitian matrices in SCS format: complex_herm_sort()

Utility function for complex linear systems

solver for linear systems involving iterated Jacobi method: complex_gen_precon_jacobi()

Utility function for complex non-Hermitian linear systems

incomplete $LU$ factorization: complex_gen_precon_ilu()

incomplete $LU$ factorization of local or overlapping diagonal blocks: complex_gen_precon_bdilu()

matrix-vector multiplier for complex non-Hermitian matrices in CS format: complex_gen_matvec()

solver for linear systems involving preconditioning matrix from complex_gen_precon_ilu(): complex_gen_precon_ilu_solve()

solver for linear systems involving SSOR preconditioning matrix: complex_gen_precon_ssor_solve()

sort function for complex non-Hermitian matrices in CS format: complex_gen_sort()

Utility function for real linear systems

solver for linear systems involving iterated Jacobi method: real_gen_precon_jacobi()

Utility function for real nonsymmetric linear systems

incomplete $LU$ factorization: real_gen_precon_ilu()

incomplete $LU$ factorization of local or overlapping diagonal blocks: real_gen_precon_bdilu()

matrix-vector multiplier for real nonsymmetric matrices in CS format: real_gen_matvec()

solver for linear systems involving preconditioning matrix from real_gen_precon_ilu(): real_gen_precon_ilu_solve()

solver for linear systems involving SSOR preconditioning matrix: real_gen_precon_ssor_solve()

sort function for real nonsymmetric matrices in CS format: real_gen_sort()

sort function for real rectangular matrices in CS or CCS format: real_rect_sort()

Utility function for real symmetric linear systems

incomplete Cholesky factorization: real_symm_precon_ichol()

matrix-vector multiplier for real symmetric matrices in SCS format: real_symm_matvec()

solver for linear systems involving preconditioning matrix from real_symm_precon_ichol(): real_symm_precon_ichol_solve()

solver for linear systems involving SSOR preconditioning matrix: real_symm_precon_ssor_solve()

sort function for real symmetric matrices in SCS format: real_symm_sort()

Utility function for real symmetric linear systems, compute bandwidth-reducing reverse Cuthill–McKee permutation: sym_rcm()

For full information please refer to the NAG Library document

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/f11/f11intro.html

Examples

naginterfaces.library.examples.sparse.real_gen_basic_solver_ex.main()

Example for naginterfaces.library.sparse.real_gen_basic_solver().

This example solves an 8×8 nonsymmetric system of simultaneous linear equations using the bi-conjugate gradient stabilized method of order l=1, where the coefficient matrix A has a random sparsity pattern. An incomplete LU preconditioner is used.

>>> main()
naginterfaces.library.sparse.real_gen_basic_solver Python Example Results.
 Monitoring at iteration no.    1
 residual no rm:     1.4059e+02

 Monitoring at iteration no.    2
 residual no rm:     3.2742e+01

 Final results
 Number of iterations for convergence:       3
 Residual norm:                               ...
 Right-hand side of termination criterion:    5.5900e-06
 1-norm of matrix A:                          1.1000e+01

   Solution vector   Residual vector
      1.0000e+00      ...
      2.0000e+00      ...
      3.0000e+00      ...
      4.0000e+00      ...
      5.0000e+00      ...
      6.0000e+00      ...
      7.0000e+00      ...
      8.0000e+00      ...