Inversion of tridiagonal matrices

\(T^{-1}\) More...

Collaboration diagram for Inversion of tridiagonal matrices:

Classes
class	dg::mat::TridiagInvHMGTI< real_type >
	Compute the inverse of a general tridiagonal matrix. More...
class	dg::mat::TridiagInvDF< real_type >
	USE THIS ONE Compute the inverse of a general tridiagonal matrix. The algorithm does not rely on the determinant. More...
class	dg::mat::TridiagInvD< real_type >
	Compute the inverse of a general tridiagonal matrix. More...

Functions
template<class value_type >
value_type	dg::mat::compute_Tinv_m1 (const dg::TriDiagonal< thrust::host_vector< value_type > > &T, unsigned size)
	Computes the value of \( (T^{-1})_{m1} = \langle \vec e_m, T^{-1}\vec e_1\rangle\) via a Thomas algorithm.
template<class value_type >
void	dg::mat::compute_Tinv_y (const dg::TriDiagonal< thrust::host_vector< value_type > > &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.
template<class value_type >
void	dg::mat::invert (const dg::TriDiagonal< thrust::host_vector< value_type > > &T, dg::SquareMatrix< value_type > &Tinv)
	Invert a tridiagonal matrix.
template<class value_type >
dg::SquareMatrix< value_type >	dg::mat::invert (const dg::TriDiagonal< thrust::host_vector< value_type > > &T)
	Invert a tridiagonal matrix.
template<class value_type >
std::array< value_type, 2 >	dg::mat::compute_extreme_EV (const dg::TriDiagonal< thrust::host_vector< value_type > > &T)
	Compute extreme Eigenvalues of a symmetric tridiangular matrix.

Detailed Description

\(T^{-1}\)

Function Documentation

compute_extreme_EV()

template<class value_type >

std::array< value_type, 2 > dg::mat::compute_extreme_EV

(

const dg::TriDiagonal< thrust::host_vector< value_type > > &

T

)

Compute extreme Eigenvalues of a symmetric tridiangular matrix.

dg::mat::UniversalLanczos lanczos( A.weights(), 20);
auto T = lanczos.tridiag( A, A.weights(), A.weights());
auto EV = dg::mat::compute_extreme_EV( T);
// EV[0] is the minimum, EV[1] the maximum Eigenvalue

Template Parameters

value_type

real type

Parameters

T	symmetric tridiangular matrix

Returns

{EVmin, EVmax}

compute_Tinv_m1()

template<class value_type >

value_type dg::mat::compute_Tinv_m1	(	const dg::TriDiagonal< thrust::host_vector< value_type > > &	T,
		unsigned	size )

Computes the value of \( (T^{-1})_{m1} = \langle \vec e_m, T^{-1}\vec e_1\rangle\) via a Thomas algorithm.

Note

This is extremely fast (timings can bearly be measured; for size = 100 <1e-6s)

Template Parameters

value_type

real type

Parameters

T	The tridiagonal matrix
size	the parameter m

Returns

\( (T^{-1})_{1m}\)

compute_Tinv_y()

template<class value_type >

void dg::mat::compute_Tinv_y	(	const dg::TriDiagonal< thrust::host_vector< value_type > > &	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.

Note

This is extremely fast (timings can bearly be measured; for size = 100 <1e-6s)

Template Parameters

value_type

real type

Parameters

T	The tridiagonal matrix
x	contains the solution (resized if necessary)
y	the right hand side
a	optional scaling factor of T
d	optional addition to diagonal of T

invert() [1/2]

template<class value_type >

dg::SquareMatrix< value_type > dg::mat::invert

(

const dg::TriDiagonal< thrust::host_vector< value_type > > &

T

)

Invert a tridiagonal matrix.

This is a convenience shortcut for

return dg::TridiagInvDF( size)(T);

Template Parameters

value_type

real type

Parameters

T

Returns

Tinv

invert() [2/2]

template<class value_type >

void dg::mat::invert	(	const dg::TriDiagonal< thrust::host_vector< value_type > > &	T,
		dg::SquareMatrix< value_type > &	Tinv )

Invert a tridiagonal matrix.

This is a convenience shortcut for

return dg::TridiagInvDF( size)(T, Tinv);

Template Parameters

value_type

real type

Parameters

T
Tinv	(gets resized if necessary)