Iterative solvers¶
AMGCL provides several iterative solvers, but it should be easy to use AMGCL preconditioners with a userprovided solver as well. Each solver in AMGCL is a class template. Its single template parameter specifies the backend to use. This allows to preallocate necessary resources at class construction. Obviously, the solver backend has to coincide with the preconditioner backend.
Each of the solvers in AMGCL provides two overloads for the operator()
:

std::tuple<size_t, scalar_type>
operator()
(const Matrix &A, const Precond &P, const Vec1 &rhs, Vec2 &&x) const¶ Computes the solution for the given system matrix
A
and the righthand siderhs
. Returns the number of iterations made and the achieved residual as astd::tuple
. The solution vectorx
provides initial approximation on input and holds the computed solution on output.The system matrix may differ from the matrix used during initialization. This may be used for the solution of nonstationary problems with slowly changing coefficients. There is a strong chance that a preconditioner built for a time step will act as a reasonably good preconditioner for several subsequent time steps [DeSh12].

std::tuple<size_t, scalar_type>
operator()
(const Precond &P, const Vec1 &rhs, Vec2 &&x) const¶ Computes the solution for the given righthand side
rhs
. The system matrix is the same that was used for the setup of the preconditionerP
. Returns the number of iterations made and the achieved residual as astd::tuple
. The solution vectorx
provides initial approximation on input and holds the computed solution on output.
Conjugate Gradient¶
#include
<amgcl/solver/cg.hpp>
BiConjugate Gradient Stabilized (BiCGSTAB)¶
#include
<amgcl/solver/bicgstab.hpp>

template <class Backend, class InnerProduct = detail::default_inner_product>
classbicgstab
¶ BiConjugate Gradient Stabilized (BiCGSTAB) method.
The BiConjugate Gradient Stabilized method (BiCGSTAB) was developed to solve nonsymmetric linear systems while avoiding the often irregular convergence patterns of the Conjugate Gradient [Barr94].
Public Functions

struct
params
¶ Solver parameters.

struct
BiCGSTAB(L)¶
#include
<amgcl/solver/bicgstabl.hpp>
GMRES¶
#include
<amgcl/solver/gmres.hpp>

template <class Backend, class InnerProduct = detail::default_inner_product>
classgmres
¶ Generalized Minimal Residual (GMRES) method.
The Generalized Minimal Residual method is an extension of MINRES (which is only applicable to symmetric systems) to unsymmetric systems [Barr94].
Public Functions

struct
params
¶ Solver parameters.
Public Members

template<>
unsignedM
¶ Number of iterations before restart.

template<>
preconditioner::side::typepside
¶ Preconditioning kind (left/right).

template<>
unsignedmaxiter
¶ Maximum number of iterations.

template<>
scalar_typetol
¶ Target relative residual error.

template<>
scalar_typeabstol
¶ Target absolute residual error.

template<>

struct
LGMRES¶
#include
<amgcl/solver/lgmres.hpp>

template <class Backend, class InnerProduct = detail::default_inner_product>
classlgmres
¶ “Loose” GMRES.
The LGMRES algorithm [BaJM05] is designed to avoid some problems in the convergence in restarted GMRES, and often converges in fewer iterations.
Public Functions

struct
params
¶ Solver parameters.
Public Members

template<>
unsignedM
¶ Number of inner GMRES iterations per each outer iteration.

template<>
unsignedK
¶ Number of vectors to carry between inner GMRES iterations.
According to [BaJM05], good values are in the range of 1…3. However, note that if you want to use the additional vectors to accelerate solving multiple similar problems, larger values may be beneficial.

template<>
boolalways_reset
¶ Reset augmented vectors between solves.
If the solver is used to repeatedly solve similar problems, then keeping the augmented vectors between solves may speed up subsequent solves. This flag, when set, resets the augmented vectors at the beginning of each solve.

template<>
boolstore_Av
¶ Whether LGMRES should store also A*v in addition to vectors
v
.

template<>
preconditioner::side::typepside
¶ Preconditioning kind (left/right).

template<>
size_tmaxiter
¶ Maximum number of iterations.

template<>
scalar_typetol
¶ Target relative residual error.

template<>
scalar_typeabstol
¶ Target absolute residual error.

template<>

struct
FGMRES¶
#include
<amgcl/solver/fgmres.hpp>