\( x \approx f(A)b \) More...

Collaboration diagram for Matrix-functions:

Classes
class	dg::mat::UniversalLanczos< ContainerType >
	Tridiagonalize \(A\) and approximate \(f(A)b \approx \|b\|_W V f(T) e_1\) via Lanczos algorithm. A is self-adjoint in the weights \( W\). More...

struct	dg::mat::MatrixSqrt< ContainerType >
	Fast computation of \( \vec x = A^{\pm 1/2}\vec b\) for self-adjoint positive definite \(A\). More...

struct	dg::mat::MatrixFunction< ContainerType >
	Computation of \( \vec x = f(A)\vec b\) for self-adjoint positive definite \( A\). More...

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 \( W\). More...

class	dg::mat::MCGFuncEigen< Container >
	EXPERIMENTAL Shortcut for \(x \approx f(A) b \) solve via exploiting first a Krylov projection achieved by the M-CG method and a matrix function computation via Eigen-decomposition. More...

struct	dg::mat::SqrtCauchyInt< Container >
	Cauchy integral \( \sqrt{A} b= A\frac{ 2 K' \sqrt{m}}{\pi N} \sum_{j=1}^{N} ( w_j^2 I + A)^{-1} c_j d_j b \) More...

struct	dg::mat::DirectSqrtCauchy< Container >
	Shortcut for \(b \approx \sqrt{A} x \) solve directly via sqrt Cauchy combined with PCG inversions. More...

struct	dg::mat::InvSqrtODE< Container >
	Right hand side of the square root ODE. More...

Functions
template<class value_type , class ExplicitRHS >
auto	dg::mat::make_directODESolve (ExplicitRHS &&ode, std::string tableau, value_type epsTimerel, value_type epsTimeabs, unsigned &number, value_type t0=0., value_type t1=1.)
	Operator that integrates an ODE from 0 to 1 with an adaptive ERK class as timestepper More...

template<class Matrix , class Preconditioner , class Container >
InvSqrtODE< Container >	dg::mat::make_inv_sqrtodeCG (Matrix &A, const Preconditioner &P, const Container &weights, dg::get_value_type< Container > epsCG)
	Right hand side of the square root ODE. More...

template<class MatrixType >
auto	dg::mat::make_expode (MatrixType &A)
	Right hand side of the exponential ODE. More...

template<class MatrixType >
auto	dg::mat::make_besselI0ode (MatrixType &A)
	Right hand side of the (zeroth order) modified Bessel function ODE, rewritten as a system of coupled first order ODEs: More...

Detailed Description

\( x \approx f(A)b \)

approximation

Function Documentation

◆ make_besselI0ode()

template<class MatrixType >

auto dg::mat::make_besselI0ode ( MatrixType & A )

Right hand side of the (zeroth order) modified Bessel function ODE, rewritten as a system of coupled first order ODEs:

\[ \dot{z_0}= z_1 \]

\[ \dot{z_1}= A^2 z_0 - t^{-1} z_1 \]

where \( A\) is the matrix and

\[z=(y,\dot{y})\]

Note: Solution of ODE: \( y(1) = I_0(A) y(0)\) for initial condition \( z(0) = (y(0),0)^T \)

◆ make_directODESolve()

template<class value_type , class ExplicitRHS >

auto dg::mat::make_directODESolve	(	ExplicitRHS &&	ode,
		std::string	tableau,
		value_type	epsTimerel,
		value_type	epsTimeabs,
		unsigned &	number,
		value_type	t0 = `0.`,
		value_type	t1 = `1.`
	)

Operator that integrates an ODE from 0 to 1 with an adaptive ERK class as timestepper

The intended use is to integrate Matrix equations

dg::Helmholtz<Geometry, Matrix, Container> A( g, alpha, dg::centered);
auto sqrt_ode = dg::mat::make_sqrtode( A, 1., A.weights(), eps);
unsigned number = 0;
auto sqrtA = dg::mat::make_directODESolve( sqrt_ode, "Dormand-Prince-7-4-5",1e-5, 1e-7, number);
dg::apply ( sqrtA , x , b);

The call \( b = f(x) \) corresponds to integrating \( \dot y = F(y)\) with \( y(0 ) = x\) to \( b = y(1)\)

Parameters

ode	The differential equation to integrate (forwarded to `dg::Adaptive<dg::ERKStep>`)
tableau	The tableau for `dg::ERKStep`
epsTimerel	relative accuracy of adaptive ODE solver
epsTimeabs	absolute accuracy of adaptive ODE solver
number	Is linked to the lambda. Contains the number of steps the adaptive timestepper used to completion
t0	Change starting time to `t0`
t1	Change end time to `t1`

Returns: Operator that integrates the ode from t0 to t1

See also: dg::make_sqrtode dg::make_expode, dg::make_besselI0ode

◆ make_expode()

template<class MatrixType >

auto dg::mat::make_expode ( MatrixType & A )

Right hand side of the exponential ODE.

\[ \dot{y}= A y \]

where \( A\) is the matrix

Note: Solution of ODE: \( y(1) = \exp{A} y(0)\)

◆ make_inv_sqrtodeCG()

template<class Matrix , class Preconditioner , class Container >

InvSqrtODE< Container > dg::mat::make_inv_sqrtodeCG	(	Matrix &	A,
		const Preconditioner &	P,
		const Container &	weights,
		dg::get_value_type< Container >	epsCG
	)

Right hand side of the square root ODE.

\[ \dot{y}= \left[tI + (1-t) A\right]^{-1} (I - A)/2 y \]

where \( A\) is the matrix and the inverse is computed via a dg::PCG solver

This class is based on the approach of the paper Numerical approximation of the product of the square root of a matrix with a vector by E. J. Allen et al

Note: Solution of ODE: \( y(1) = 1/\sqrt{A} y(0)\)