Numeric.Series

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

Data type for infinite sequences.

data Sequence a

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

Constructors

Sequence s (s -> (a, s))

Instances

Functor Sequence

Defined in Numeric.Series

Methods

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

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

Applicative Sequence

Defined in Numeric.Series

Methods

pure :: a -> Sequence a

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

liftA2 :: (a -> b -> c) -> Sequence a -> Sequence b -> Sequence c

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

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

Fractional a => Fractional (Sequence a)

Elementwise operations with sequences

Defined in Numeric.Series

Methods

(/) :: Sequence a -> Sequence a -> Sequence a

recip :: Sequence a -> Sequence a

fromRational :: Rational -> Sequence a

Num a => Num (Sequence a)

Elementwise operations with sequences

Defined in Numeric.Series

Methods

(+) :: Sequence a -> Sequence a -> Sequence a

(-) :: Sequence a -> Sequence a -> Sequence a

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

negate :: Sequence a -> Sequence a

abs :: Sequence a -> Sequence a

signum :: Sequence a -> Sequence a

fromInteger :: Integer -> Sequence a

Constructors

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

enumSequenceFrom x generate sequence:

\[ a_n = x + n \]

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

enumSequenceFromStep x d generate sequence:

\[ a_n = x + nd \]

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

Analog of scanl for sequence.

Summation of series

sumSeries :: Sequence Double -> Double

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

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]

Convert series to infinite list

Evaluation of continued fractions

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

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"