dg::mat Namespace Reference

Classes
struct	BESSELI0
	\( f(x) = I_0 (x)\) with \(I_0\) the zeroth order modified Bessel function More...
struct	ConvertsToFunctionalButcherTableau
	Convert identifiers to their corresponding dg::mat::FunctionalButcherTableau. More...
struct	DirectSqrtCauchy
	Shortcut for \(b \approx \sqrt{A} x \) solve directly via sqrt Cauchy combined with PCG inversions. More...
struct	ExponentialERKStep
	Exponential Runge-Kutta fixed-step time-integration for \( \dot y = A y + g(t,y)\). More...
struct	ExponentialStep
	Exponential one step time-integration for \( \dot y = A y \). More...
struct	FunctionalButcherTableau
	Manage coefficients of a functional (extended) Butcher tableau. More...
struct	GAMMA0
	\( f(x) = \Gamma_0 (x) := I_0 (x) exp(x) \) with \(I_0\) the zeroth order modified Bessel function More...
struct	InvSqrtODE
	Right hand side of the square root ODE. More...
struct	MatrixFunction
	Computation of \( \vec x = f(A)\vec b\) for self-adjoint positive definite \( A\). More...
struct	MatrixSqrt
	Fast computation of \( \vec x = A^{\pm 1/2}\vec b\) for self-adjoint positive definite \(A\). More...
class	MCG
	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	MCGFuncEigen
	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...
class	PolCharge
	Various arbitary wavelength polarization charge operators of delta-f (df) and full-f (ff) More...
class	PolChargeN
	EXPERIMENTAL polarization solver class for N. More...
struct	SqrtCauchyInt
	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	TensorElliptic
	Matrix class that represents the arbitrary polarization operator. More...
class	TridiagInvD
	Compute the inverse of a general tridiagonal matrix. More...
class	TridiagInvDF
	USE THIS ONE Compute the inverse of a general tridiagonal matrix. The algorithm does not rely on the determinant. More...
class	TridiagInvHMGTI
	Compute the inverse of a general tridiagonal matrix. More...
class	UniversalLanczos
	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...

Enumerations
enum	func_tableau_identifier { EXPLICIT_EULER_1_1 , MIDPOINT_2_2 , CLASSIC_4_4 , HOCHBRUCK_3_3_4 }
	Identifiers for Butcher Tableaus. More...

Functions
template<class T >
T	phi1 (T x)
	\( f(x) = (\exp(x) - 1)/x\) More...
template<class T >
T	phi2 (T x)
	\( f(x) = (\exp(x) - x - 1)/x^2\) More...
template<class T >
T	phi3 (T x)
	\( f(x) = (\exp(x) - x^2/2 -x- 1)/x^3\) More...
template<class T >
T	phi4 (T x)
	\( f(x) = (\exp(x) - x^3/6 - x^2/2 -x- 1)/x^4\) More...
template<class UnaryOp >
auto	make_FuncEigen_Te1 (UnaryOp f)
	Create a functor that uses Eigenvalue decomposition to compute \( f(T)\vec e_1 = E f(\Lambda) E^T \vec e_1 \) for symmetric tridiagonal T. More...
template<class value_type >
auto	make_SqrtCauchy_Te1 (int exp, std::array< value_type, 2 > EVs, unsigned stepsCauchy)
	Create a functor that computes \( \sqrt{T^{\pm 1}} \vec e_1\) using SqrtCauchyInt. More...
template<class value_type >
auto	make_SqrtCauchyEigen_Te1 (int exp, std::array< value_type, 2 > EVs, unsigned stepsCauchy)
	Create a functor that computes \( \sqrt{T^{\pm 1}} \vec e_1\) using either Eigen or SqrtCauchy solve based on whichever is fastest for given size. More...
template<class value_type >
auto	make_SqrtODE_Te1 (int exp, std::string tableau, value_type rtol, value_type atol, unsigned &number)
	Create a functor that computes \( \sqrt{T^{\pm 1}} \vec e_1\) using ODE solve. More...
static auto	make_Linear_Te1 (int exp)
	Create a functor that computes \( T^{\pm 1} \vec e_1\) directly. More...
template<class value_type , class ExplicitRHS >
auto	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 >	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 Matrix , class Container >
InvSqrtODE< Container >	make_inv_sqrtodeTri (const Matrix &TH, const Container &copyable)
template<class MatrixType >
auto	make_expode (MatrixType &A)
	Right hand side of the exponential ODE. More...
template<class MatrixType >
auto	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...
template<class value_type >
value_type	compute_Tinv_m1 (const cusp::dia_matrix< int, value_type, cusp::host_memory > &T, unsigned size)
	Computes the value of \( (T^{-1})_{m1} = \langle \vec e_m, T^{-1}\vec e_1\rangle\) via a Thomas algorithm. More...
template<class value_type >
void	compute_Tinv_y (const cusp::dia_matrix< int, value_type, cusp::host_memory > &T, thrust::host_vector< value_type > &x, const thrust::host_vector< value_type > &y, value_type a=1., value_type d=0.)
	Computes the value of \( x = ((aT+dI)^{-1})y \) via Thomas algorithm. More...
template<class value_type >
void	invert (const cusp::dia_matrix< int, value_type, cusp::host_memory > &T, cusp::coo_matrix< int, value_type, cusp::host_memory > &Tinv)
	Invert a tridiagonal matrix. More...
template<class value_type >
cusp::coo_matrix< int, value_type, cusp::host_memory >	invert (const cusp::dia_matrix< int, value_type, cusp::host_memory > &T)
	Invert a tridiagonal matrix. More...
template<class value_type >
std::array< value_type, 2 >	compute_extreme_EV (const cusp::dia_matrix< int, value_type, cusp::host_memory > &T)
	Compute extreme Eigenvalues of a symmetric tridiangular matrix. More...

Function Documentation

make_inv_sqrtodeTri()

template<class Matrix , class Container >

InvSqrtODE< Container > dg::mat::make_inv_sqrtodeTri	(	const Matrix &	TH,
		const Container &	copyable
	)

phi1()

template<class T >

T dg::mat::phi1

(

T

x

)

\( f(x) = (\exp(x) - 1)/x\)

Accurate evaluation close to and at 0

Template Parameters

T	value type

phi2()

template<class T >

T dg::mat::phi2

(

T

x

)

\( f(x) = (\exp(x) - x - 1)/x^2\)

Accurate evaluation close to and at 0

Template Parameters

T	value type

phi3()

template<class T >

T dg::mat::phi3

(

T

x

)

\( f(x) = (\exp(x) - x^2/2 -x- 1)/x^3\)

Accurate evaluation close to and at 0

Template Parameters

T	value type

phi4()

template<class T >

T dg::mat::phi4

(

T

x

)

\( f(x) = (\exp(x) - x^3/6 - x^2/2 -x- 1)/x^4\)

Accurate evaluation close to and at 0

Template Parameters

T	value type