geometric_sequence.hh File Reference

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

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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.