EXPERIMENTAL Tridiagonalize \(A\) and approximate \(f(A)b \approx R f(\tilde T) e_1\) via CG algorithm. A is self-adjoint in the weights \( W\). More...

Public Types
using	value_type = dg::get_value_type< ContainerType >
	value type of the ContainerType class More...

using	HCooMatrix = cusp::coo_matrix< int, value_type, cusp::host_memory >

using	HDiaMatrix = cusp::dia_matrix< int, value_type, cusp::host_memory >

using	HVec = dg::HVec

Public Member Functions
	MCG ()
	Allocate nothing, Call `construct` method before usage. More...

	MCG (const ContainerType &copyable, unsigned max_iterations)
	Allocate memory for the mcg method. More...

void	set_max (unsigned new_max)
	Set the maximum number of iterations. More...

unsigned	get_max () const
	Get the current maximum number of iterations. More...

void	set_verbose (bool verbose)
	Set or unset debugging output during iterations. More...

value_type	get_bnorm () const
	Norm of `b` from last call to `operator()` More...

template<class ... Params>
void	construct (Params &&...ps)
	Perfect forward parameters to one of the constructors. More...

unsigned	get_iter () const
	Get the number of iterations in the last call to `operator()` (size of T) More...

template<class MatrixType , class DiaMatrixType , class ContainerType0 , class ContainerType1 , class ContainerType2 >
void	Ry (MatrixType &&A, const DiaMatrixType &T, const ContainerType0 &y, ContainerType1 &x, const ContainerType2 &b)
	Compute x = R y. More...

template<class MatrixType , class ContainerType0 , class ContainerType1 >
const HDiaMatrix &	operator() (MatrixType &&A, const ContainerType0 &b, const ContainerType1 &weights, value_type eps=1e-12, value_type nrmb_correction=1., value_type res_fac=1.)
	Tridiagonalize the system \( A\vec b\) using CG. More...

HVec	make_e1 ()
	Return the vector \( \vec e_1\) with size `get_iter()` More...

Detailed Description

template<class ContainerType>
class dg::mat::MCG< ContainerType >

EXPERIMENTAL Tridiagonalize \(A\) and approximate \(f(A)b \approx R f(\tilde T) e_1\) via CG algorithm. A is self-adjoint in the weights \( W\).

Attention: This class has the exact same result as the corresponding UniversalLanczos class. Its use is for mere experimental purposes.

This class is based on the approach of the paper An iterative solution method for solving f(A)x = b, using Krylov subspace information obtained for the symmetric positive definite matrix A by H. A. Van Der Vorst

Note: The approximation relies on Projection \(x = f(A) b \approx R f(\tilde T) e_1\), where \(\tilde T\) and \(R\) are the tridiagonal and orthogonal matrix of the CG solve respectively and \(e_1\) is the normalized unit vector.

Member Typedef Documentation

◆ HCooMatrix

template<class ContainerType >

using dg::mat::MCG< ContainerType >::HCooMatrix = cusp::coo_matrix<int, value_type, cusp::host_memory>

◆ HDiaMatrix

template<class ContainerType >

using dg::mat::MCG< ContainerType >::HDiaMatrix = cusp::dia_matrix<int, value_type, cusp::host_memory>

◆ HVec

template<class ContainerType >

using dg::mat::MCG< ContainerType >::HVec = dg::HVec

◆ value_type

template<class ContainerType >

using dg::mat::MCG< ContainerType >::value_type = dg::get_value_type<ContainerType>

value type of the ContainerType class

Constructor & Destructor Documentation

◆ MCG() [1/2]

template<class ContainerType >

dg::mat::MCG< ContainerType >::MCG ( )

inline

Allocate nothing, Call construct method before usage.

◆ MCG() [2/2]

template<class ContainerType >

dg::mat::MCG< ContainerType >::MCG	(	const ContainerType &	copyable,
		unsigned	max_iterations
	)

inline

Allocate memory for the mcg method.

Parameters

copyable	A ContainerType must be copy-constructible from this
max_iterations	Maximum number of iterations to be used

Member Function Documentation

◆ construct()

template<class ContainerType >

template<class ... Params>

void dg::mat::MCG< ContainerType >::construct ( Params &&... ps )

inline

Perfect forward parameters to one of the constructors.

Template Parameters

Params deduced by the compiler

Parameters

ps	parameters forwarded to constructors

◆ get_bnorm()

template<class ContainerType >

value_type dg::mat::MCG< ContainerType >::get_bnorm ( ) const

inline

Norm of b from last call to operator()

Returns: bnorm

◆ get_iter()

template<class ContainerType >

unsigned dg::mat::MCG< ContainerType >::get_iter ( ) const

inline

Get the number of iterations in the last call to operator() (size of T)

Returns: the number of iterations in the last call to operator()

◆ get_max()

template<class ContainerType >

unsigned dg::mat::MCG< ContainerType >::get_max ( ) const

inline

Get the current maximum number of iterations.

Returns: the current maximum

◆ make_e1()

template<class ContainerType >

HVec dg::mat::MCG< ContainerType >::make_e1 ( )

inline

Return the vector \( \vec e_1\) with size get_iter()

Parameters

iter size

Returns: e_1

◆ operator()()

template<class ContainerType >

template<class MatrixType , class ContainerType0 , class ContainerType1 >

const HDiaMatrix & dg::mat::MCG< ContainerType >::operator()	(	MatrixType &&	A,
		const ContainerType0 &	b,
		const ContainerType1 &	weights,
		value_type	eps = `1e-12`,
		value_type	nrmb_correction = `1.`,
		value_type	res_fac = `1.`
	)

inline

Tridiagonalize the system \( A\vec b\) using CG.

Parameters

A	A self-adjoint, positive definit matrix
b	The right hand side vector.
weights	Weights that define the scalar product in which `A` is self-adjoint and in which the error norm is computed.
eps	relative accuracy of residual
nrmb_correction	the absolute error `C` in units of `eps` to be respected
res_fac	factor that is multiplied to the norm of the residual. Used to account for specific matrix function and operator in the convergence criterium

Returns: The tridiagonal matrix \( T\) with size get_iter()

Note: So far only ordinary convergence criterium of CG method, in particular for \( A x = b \). If used for matrix function computation, \( f(A) x = b \), the parameter eps should be multiplied with appropriate factors to account for the different convergence criteria. The Matrix R and T of the tridiagonalization are further used for computing matrix functions. The x vector must be initialized with 0 if used for tridiagonalization.

◆ Ry()

template<class ContainerType >

template<class MatrixType , class DiaMatrixType , class ContainerType0 , class ContainerType1 , class ContainerType2 >

void dg::mat::MCG< ContainerType >::Ry	(	MatrixType &&	A,
		const DiaMatrixType &	T,
		const ContainerType0 &	y,
		ContainerType1 &	x,
		const ContainerType2 &	b
	)

inline

Compute x = R y.

Parameters

A	A self-adjoint, positive definit matrix
T	T non-symmetric tridiagonal Matrix from MCG tridiagonalization
y	(host) vector with v.size() = iter. Must have size of `T.num_rows`. Typically \( T^{(-1)} e_1 \) or \( f(T^{(-1)}) e_1 \)
x	Contains the matrix approximation \(x = Ry \) (output)
b	The right hand side vector.

◆ set_max()

template<class ContainerType >

void dg::mat::MCG< ContainerType >::set_max ( unsigned new_max )

inline

Set the maximum number of iterations.

Parameters

new_max New maximum number

◆ set_verbose()

template<class ContainerType >

void dg::mat::MCG< ContainerType >::set_verbose ( bool verbose )

inline

Set or unset debugging output during iterations.

Parameters

verbose If true, additional output will be written to std::cout during solution

The documentation for this class was generated from the following file:

mcg.h

Public Types

Public Member Functions

Detailed Description

Member Typedef Documentation

◆ HCooMatrix

◆ HDiaMatrix

◆ HVec

◆ value_type

Constructor & Destructor Documentation

◆ MCG() [1/2]

◆ MCG() [2/2]

Member Function Documentation

◆ construct()

◆ get_bnorm()

◆ get_iter()

◆ get_max()

◆ make_e1()

◆ operator()()

◆ Ry()

◆ set_max()

◆ set_verbose()