Approximate inverse Chebyshev Polynomial Preconditioner \( A^{-1} = \frac{c_0}{2} I + \sum_{k=1}^{r}c_kT_k( Z)\). More...

Public Types
using	container_type = ContainerType

using	value_type = get_value_type< ContainerType >
	value type of the ContainerType class More...

Public Member Functions
	ModifiedChebyshevPreconditioner (Matrix op, const ContainerType &copyable, value_type ev_min, value_type ev_max, unsigned degree)
	Construct the k-th Chebyshev Polynomial approximate. More...

template<class ContainerType0 , class ContainerType1 >
void	symv (const ContainerType0 &x, ContainerType1 &y)

Detailed Description

template<class Matrix, class ContainerType>
struct dg::ModifiedChebyshevPreconditioner< Matrix, ContainerType >

Approximate inverse Chebyshev Polynomial Preconditioner \( A^{-1} = \frac{c_0}{2} I + \sum_{k=1}^{r}c_kT_k( Z)\).

This is the polynomial preconditioner as proposed by Dag and Semlyen, A New Preconditioned Conjugate Gradient Power Flow, IEEE Transactions on power Systems, 18, (2003) We have \( c_k = \sqrt{\lambda_\min\lambda_\max}^{-1} (\sqrt{\lambda_\min/\lambda_\max}-1)^k / (\sqrt{\lambda_\min/\lambda_\max }+ 1)^k\) and \( Z = 2 ( A - (\lambda_\max + \lambda_\min)I/2)/(\lambda_\max-\lambda_\min)\)

They propose to use \( \lambda_\min = \lambda_\max / (5r)\) where r is the degree of the polynomial

Note: This class can be used as a Preconditioner in the CG algorithm. The CG algorithm forms an approximation to the solution in the form \( x_{k+1} = x_0 + P_k(A) r_0\) where \( P_k\) is a polynomial of degree k, which is optimal in minimizing the A-norm. Thus a polynomial preconditioner cannot decrease the number of matrix-vector multiplications needed to achieve a certain accuracy. However, since polynomial preconditioners do not use scalar products they may offset the increased overhead if the dot product becomes a bottleneck for performance or scalability.

Template Parameters

Matrix	Preferably a reference type
ContainerType

Member Typedef Documentation

template<class Matrix , class ContainerType >

using dg::ModifiedChebyshevPreconditioner< Matrix, ContainerType >::container_type = ContainerType

template<class Matrix , class ContainerType >

using dg::ModifiedChebyshevPreconditioner< Matrix, ContainerType >::value_type = get_value_type<ContainerType>

value type of the ContainerType class

template<class Matrix , class ContainerType >

dg::ModifiedChebyshevPreconditioner< Matrix, ContainerType >::ModifiedChebyshevPreconditioner	(	Matrix	op,
		const ContainerType &	copyable,
		value_type	ev_min,
		value_type	ev_max,
		unsigned	degree
	)

inline

Construct the k-th Chebyshev Polynomial approximate.

Parameters

op	The Matrix (copied, so maybe choose a reference type for shallow copying) will be called as `dg::blas2::symv( op, x, y)`
copyable	A ContainerType must be copy-constructible from this
ev_min	an estimate of the minimum Eigenvalue (It is important to get a good value here. The authors propose to use \( \lambda_\min = \lambda_\max / (5r)\) where `r` is the `degree`
ev_max	an estimate of the maximum Eigenvalue of \( A\) (must be larger than `min_ev`) Use `EVE` to get this value
degree	degree k of the Polynomial (5 should be a good number)

template<class Matrix , class ContainerType >

template<class ContainerType0 , class ContainerType1 >

void dg::ModifiedChebyshevPreconditioner< Matrix, ContainerType >::symv	(	const ContainerType0 &	x,
		ContainerType1 &	y
	)

inline

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