math-functions-0.2.1.0: Special functions and Chebyshev polynomials

Copyright(c) 2016 Alexey Khudyakov
LicenseBSD3
Maintaineralexey.skladnoy@gmail.com, bos@serpentine.com
Stabilityexperimental
Portabilityportable
Safe HaskellSafe
LanguageHaskell2010

Numeric.Series

Contents

Description

Functions for working with infinite sequences. In particular summation of series and evaluation of continued fractions.

Synopsis

Data type for infinite sequences.

data Sequence a Source #

Infinite series. It's represented as opaque state and step function.

Constructors

Sequence s (s -> (a, s)) 

Instances

Functor Sequence Source # 

Methods

fmap :: (a -> b) -> Sequence a -> Sequence b #

(<$) :: a -> Sequence b -> Sequence a #

Applicative Sequence Source # 

Methods

pure :: a -> Sequence a #

(<*>) :: Sequence (a -> b) -> Sequence a -> Sequence b #

(*>) :: Sequence a -> Sequence b -> Sequence b #

(<*) :: Sequence a -> Sequence b -> Sequence a #

Fractional a => Fractional (Sequence a) Source #

Elementwise operations with sequences

Num a => Num (Sequence a) Source #

Elementwise operations with sequences

Constructors

enumSequenceFrom :: Num a => a -> Sequence a Source #

enumSequenceFrom x generate sequence:

\[ a_n = x + n \]

enumSequenceFromStep :: Num a => a -> a -> Sequence a Source #

enumSequenceFromStep x d generate sequence:

\[ a_n = x + nd \]

scanSequence :: (b -> a -> b) -> b -> Sequence a -> Sequence b Source #

Analog of scanl for sequence.

Summation of series

sumSeries :: Sequence Double -> Double Source #

Calculate sum of series

\[ \sum_{i=0}^\infty a_i \]

Summation is stopped when

\[ a_{n+1} < \varepsilon\sum_{i=0}^n a_i \]

where ε is machine precision (m_epsilon)

sumPowerSeries :: Double -> Sequence Double -> Double Source #

Calculate sum of series

\[ \sum_{i=0}^\infty x^ia_i \]

Calculation is stopped when next value in series is less than ε·sum.

sequenceToList :: Sequence a -> [a] Source #

Convert series to infinite list

Evaluation of continued fractions

evalContFractionB :: Sequence (Double, Double) -> Double Source #

Evaluate continued fraction using modified Lentz algorithm. Sequence contain pairs (a[i],b[i]) which form following expression:

\[ b_0 + \frac{a_1}{b_1+\frac{a_2}{b_2+\frac{a_3}{b_3 + \cdots}}} \]

Modified Lentz algorithm is described in Numerical recipes 5.2 "Evaluation of Continued Fractions"