geometric_sequence.hh File Reference

#include <cmath>
#include <numeric>
#include <iterator>
#include <utility>
#include "utilities/power.hh"

Go to the source code of this file.

Functions
template<class Iterator >
std::pair< typename std::iterator_traits< Iterator >::value_type, typename std::iterator_traits< Iterator >::value_type >	estimateGeometricSequence (Iterator first, Iterator last)
	Estimates parameters of geometric sequence. More...
template<class Iterator >
std::iterator_traits< Iterator >::value_type	estimateGeometricTruncationError (Iterator first, Iterator last, int k)
	Estimates errors of truncating geometrically convergent sequences. More...

Function Documentation

estimateGeometricSequence()

template<class Iterator >

std::pair< typename std::iterator_traits< Iterator >::value_type, typename std::iterator_traits< Iterator >::value_type > estimateGeometricSequence	(	Iterator	first,
		Iterator	last
	)

Estimates parameters of geometric sequence.

Given a (finite) sequence $a_i, i=0,\dots , n-1$, this function estimates the parameters $c,q$ of a geometric sequence $c q^i$, such that the least squares error $\sum_{i=0}^{n-1} (ln(cq^i) - \ln a_i)^2$ is minimal. $n$ has to be at least 2. The values $a_i$ have to be positive.

Template Parameters

Iterator

an input iterator type with a scalar floating point value type

Definition at line 34 of file geometric_sequence.hh.

Referenced by estimateGeometricTruncationError().

estimateGeometricTruncationError()

template<class Iterator >

std::iterator_traits< Iterator >::value_type estimateGeometricTruncationError	(	Iterator	first,
		Iterator	last,
		int	k
	)

Estimates errors of truncating geometrically convergent sequences.

Assume there is a geometrically convergent series $(x_i)_{i\in N}$ with data available for $i=0,\dots , n$. The differences satisfy $a_i = x_{i+1} - x_i \approx c q^i$. Then we estimate the truncation error $x_k - x_\infty \approx \sum_{i=k}^{n-1} a_i + \frac{cq^n}{1-q}.$ This value is computed here, given the $n$ difference values $a_i$.

Template Parameters

Iterator

an input iterator with scalar floating point value type.

Parameters

k	defines for which iterate $x_k$ the error is estimated. $0\le k \le n$.

Definition at line 75 of file geometric_sequence.hh.