Richardson <: SumHelper

Desciption

Richardson method helper. This struct is initialized with a range of integers dom and a list of exponents exponents, which are used internally. More exponents can lead to better convergence at the cost of noise. There are two methods available for the fitting of internal weights: :bender, :rohringer. See C. Bender, A. Orszag 99, p. 375; G. Rohringer, A. Toschi 2016 for the derivation Due to the different fit methods, method=:bender will not use

Usage

Fields

• indices : AbstractVector{Int} of indices of partial sum array which have been fitted
• weights : Matrix{Float64} fit weights

Shanks <: SumHelper

Desciption

Shanks method helper.

Usage

Arguments

Examples

build_M_matrix(dom::AbstractArray{Int,1}, exponents::AbstractArray{Int,1})

Helper function that build the matrix $M$ used to fit data obtained for indices dom to a $\sum_{n\in \text{exponents}} c_n/i^n$ tail.

dom specifies the number of terms for the partial sum, that are to be fitted. exponents specifies the exponents of $\sum_{x \in \text{dom}} \sum_{p \in \text{exponents}} 1/i^p$ that the partial sums should be fitted to.

The coefficients $c_n$ are obtained by solving $M c = b$. $b$ can be constructed from the data using build_weights_rohringer.

build_weights_bender(dom::AbstractArray{Int,1}, exponents::AbstractArray{Int,1})

Build weight matrix in closed form. See C. Bender, A. Orszag 99, p. 375. Fit coefficients can be obtained by multiplying w with data: $a_k = W_{kj} g_j$

build_weights_rohringer(dom::AbstractArray{Int,1}, exponents::AbstractArray{Int,1})

Build weight matrix i.e. $W = M^{-1} R$ with M from build_M_matrix and $R_{kj} = \frac{1}{j^k}$. Fit coefficients can be obtained by multiplying w with data: $a_k = W_{kj} g_j$

build_weights_wynn(arr::AbstractArray{T1,1}) where T1 <: Number

esum(arr::AbstractArray{T1,1}, type::T2) where {T1 <: Number, T2 <: SumHelper}

Desciption

Computes improved estimated for infinite series using the summands of the series arr, as input. The method can be chosen by setting type. In cases where the cumulative sum can not be naively computed, one can also specify a function csum_f to obtain the 1 dimensional array of partial sums.

Usage

esum(arr, r)

Arguments

• arr : AbstractVector of summands.
• type : Instance SumHelper for construction of improved estimation of the limit.
• csum_f : Optional. Function which constructs partial sum array from arr.

See also

esum_c, Richardson, Shanks.

Examples

using SeriesAcceleration

r = Richardson(1:5,0:3)
arr = S1_100 = 1 ./ (1:100) .^ 2
limit = esum(arr, r)

esum_c(carr::AbstractArray{T1,1}, type::T2) where {T1 <: Number, T2 <: SumHelper}

Desciption

Computes improved estimated for infinite series using the cumulative sum carr as input. The method can be chosen by setting type. See also Richardson, Shanks. For technical reasons. the algorithm uses partial sums internally. This method therefore provides a faster and more flexible interface, than the esum method, which will construct the partial sums itself.

Usage

esum_c(arr, r)

Arguments

• carr : AbstractVector of partial sum's up to each index.
• type : Instance SumHelper for construction of improved estimation of the limit.

See also

esum, Richardson, Shanks.

Examples

using SeriesAcceleration

r = Richardson(1:5,0:3)
arr = cumsum(S1_100 = 1 ./ (1:100) .^ 2)
limit = esum_c(arr, r)

npartial_sum(type::T2)

Desciption

Returns the number of partial sum terms needed for this sum helper.

rateOfConv(arr:AbstractArray{T, 1})

Estimates rate of convergence $\alpha_n = \frac{\log |(x_{n+1} - x_n)/(x_n - x_{n-1})|}{\log |(x_n - x_{n-1})/(x_{n-1}-x_{n-2})|}$ from array. trace can be set to true, to obtain the roc for all partial sums.