Tridiagonal Matrix-functions

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

Collaboration diagram for Tridiagonal Matrix-functions:

Functions
template<class UnaryOp >
auto	dg::mat::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	dg::mat::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	dg::mat::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	dg::mat::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	dg::mat::make_Linear_Te1 (int exp)
	Create a functor that computes \( T^{\pm 1} \vec e_1\) directly. More...

Detailed Description

\( x \approx f(T)b \)

approximation

Function Documentation

make_FuncEigen_Te1()

template<class UnaryOp >

auto dg::mat::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.

Note

This is a general purpose solution. Very fast for small sizes (<40) of T, but scales badly for larger sizes. Use more specialized solutions if the number of iterations becomes high

Parameters

f	the matrix function (e.g. dg::SQRT<double> or dg::EXP<double>)

Returns

an operator to use in UniversalLanczos solve method

See also

UniversalLanczos

make_Linear_Te1()

static auto dg::mat::make_Linear_Te1 ( int  exp )

inlinestatic

Create a functor that computes \( T^{\pm 1} \vec e_1\) directly.

Parameters

exp	exponent if +1 compute \( Te_1\), if -1 compute \( T^{-1} e_1\)

Returns

an operator to use in UniversalLanczos solve method

See also

UniversalLanczos

make_SqrtCauchy_Te1()

template<class value_type >

auto dg::mat::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.

Note

The Eigenvalues can be estimated from a few lanczos iterations (which is at least more reliable than doing it semi-analytically)

dg::mat::UniversalLanczos lanczos( A.weights(), 20);
auto T = lanczos.tridiag( A, A.weights(), A.weights());
auto EVs = dg::mat::compute_extreme_EV( T);
auto make_SqrtCauchy_Te1( -1, EVs, 40);

Parameters

exp	exponent if +1 compute \( \sqrt(T)\), if -1 compute \( 1/\sqrt(T)\)
EVs	{minimum Eigenvalue of A, maximum Eigenvalue of A}
stepsCauchy	iterations of cauchy integral

Returns

an operator to use in UniversalLanczos solve method

See also

SqrtCauchyInt UniversalLanczos

make_SqrtCauchyEigen_Te1()

template<class value_type >

auto dg::mat::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.

Note

The Eigenvalues can be estimated from a few lanczos iterations (which is at least more reliable than doing it semi-analytically)

dg::mat::UniversalLanczos lanczos( A.weights(), 20);
auto T = lanczos.tridiag( A, A.weights(), A.weights());
auto EVs = dg::mat::compute_extreme_EV( T);
auto make_SqrtCauchyEigen_Te1( -1, EVs, 40);

This function uses an Eigen decomposition for small sizes of T and a SqrtCauchyInt solve for larger sizes to optimize execution times

Parameters

exp	exponent if +1 compute \( \sqrt{T}\), if -1 compute \( 1/\sqrt{T}\)
EVs	{minimum Eigenvalue of A, maximum Eigenvalue of A}
stepsCauchy	iterations of cauchy integral

Returns

an operator to use in UniversalLanczos solve method

See also

SqrtCauchyInt UniversalLanczos

make_SqrtODE_Te1()

template<class value_type >

auto dg::mat::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.

Parameters

exp	exponent if +1 compute \( \sqrt{T}\), if -1 compute \( 1/\sqrt{T}\)
tableau	Tableau of time integrator
rtol	relative tolerance of time integrator
atol	absolute tolerance of time integrator
number	links to number of steps in time integrator

Returns

an operator to use in UniversalLanczos solve method

See also

make_directODESolve UniversalLanczos