Safe Haskell | None |
---|---|

Language | Haskell2010 |

Some basic univariate power series expansions. This module is not re-exported by Math.Combinat.

Note: the "`convolveWithXXX`

" functions are much faster than the equivalent
`(XXX `convolve`)`

!

TODO: better names for these functions.

- unitSeries :: Num a => [a]
- zeroSeries :: Num a => [a]
- constSeries :: Num a => a -> [a]
- idSeries :: Num a => [a]
- powerTerm :: Num a => Int -> [a]
- addSeries :: Num a => [a] -> [a] -> [a]
- sumSeries :: Num a => [[a]] -> [a]
- subSeries :: Num a => [a] -> [a] -> [a]
- negateSeries :: Num a => [a] -> [a]
- scaleSeries :: Num a => a -> [a] -> [a]
- mulSeries :: Num a => [a] -> [a] -> [a]
- productOfSeries :: Num a => [[a]] -> [a]
- convolve :: Num a => [a] -> [a] -> [a]
- convolveMany :: Num a => [[a]] -> [a]
- reciprocalSeries :: (Eq a, Fractional a) => [a] -> [a]
- integralReciprocalSeries :: (Eq a, Num a) => [a] -> [a]
- composeSeries :: (Eq a, Num a) => [a] -> [a] -> [a]
- substitute :: (Eq a, Num a) => [a] -> [a] -> [a]
- lagrangeCoeff :: Partition -> Integer
- integralLagrangeInversion :: (Eq a, Num a) => [a] -> [a]
- lagrangeInversion :: (Eq a, Fractional a) => [a] -> [a]
- expSeries :: Fractional a => [a]
- cosSeries :: Fractional a => [a]
- sinSeries :: Fractional a => [a]
- coshSeries :: Fractional a => [a]
- sinhSeries :: Fractional a => [a]
- log1Series :: Fractional a => [a]
- dyckSeries :: Num a => [a]
- coinSeries :: [Int] -> [Integer]
- coinSeries' :: Num a => [(a, Int)] -> [a]
- convolveWithCoinSeries :: [Int] -> [Integer] -> [Integer]
- convolveWithCoinSeries' :: Num a => [(a, Int)] -> [a] -> [a]
- productPSeries :: [[Int]] -> [Integer]
- productPSeries' :: Num a => [[(a, Int)]] -> [a]
- convolveWithProductPSeries :: [[Int]] -> [Integer] -> [Integer]
- convolveWithProductPSeries' :: Num a => [[(a, Int)]] -> [a] -> [a]
- pseries :: [Int] -> [Integer]
- convolveWithPSeries :: [Int] -> [Integer] -> [Integer]
- pseries' :: Num a => [(a, Int)] -> [a]
- convolveWithPSeries' :: Num a => [(a, Int)] -> [a] -> [a]
- signedPSeries :: [(Sign, Int)] -> [Integer]
- convolveWithSignedPSeries :: [(Sign, Int)] -> [Integer] -> [Integer]

# Trivial series

unitSeries :: Num a => [a] Source #

The series [1,0,0,0,0,...], which is the neutral element for the convolution.

zeroSeries :: Num a => [a] Source #

Constant zero series

constSeries :: Num a => a -> [a] Source #

Power series representing a constant function

# Basic operations on power series

negateSeries :: Num a => [a] -> [a] Source #

scaleSeries :: Num a => a -> [a] -> [a] Source #

productOfSeries :: Num a => [[a]] -> [a] Source #

# Convolution (product)

convolve :: Num a => [a] -> [a] -> [a] Source #

Convolution of series (that is, multiplication of power series). The result is always an infinite list. Warning: This is slow!

convolveMany :: Num a => [[a]] -> [a] Source #

Convolution (= product) of many series. Still slow!

# Reciprocals of general power series

reciprocalSeries :: (Eq a, Fractional a) => [a] -> [a] Source #

Given a power series, we iteratively compute its multiplicative inverse

integralReciprocalSeries :: (Eq a, Num a) => [a] -> [a] Source #

Given a power series starting with `1`

, we can compute its multiplicative inverse
without divisions.

# Composition of formal power series

composeSeries :: (Eq a, Num a) => [a] -> [a] -> [a] Source #

`g `composeSeries` f`

is the power series expansion of `g(f(x))`

.
This is a synonym for `flip substitute`

.

We require that the constant term of `f`

is zero.

substitute :: (Eq a, Num a) => [a] -> [a] -> [a] Source #

`substitute f g`

is the power series corresponding to `g(f(x))`

.
Equivalently, this is the composition of univariate functions (in the "wrong" order).

Note: for this to be meaningful in general (not depending on convergence properties),
we need that the constant term of `f`

is zero.

# Lagrange inversions

lagrangeCoeff :: Partition -> Integer Source #

Coefficients of the Lagrange inversion

integralLagrangeInversion :: (Eq a, Num a) => [a] -> [a] Source #

We expect the input series to match `(0:1:_)`

. The following is true for the result (at least with exact arithmetic):

substitute f (integralLagrangeInversion f) == (0 : 1 : repeat 0) substitute (integralLagrangeInversion f) f == (0 : 1 : repeat 0)

lagrangeInversion :: (Eq a, Fractional a) => [a] -> [a] Source #

We expect the input series to match `(0:a1:_)`

. with a1 nonzero The following is true for the result (at least with exact arithmetic):

substitute f (lagrangeInversion f) == (0 : 1 : repeat 0) substitute (lagrangeInversion f) f == (0 : 1 : repeat 0)

# Power series expansions of elementary functions

expSeries :: Fractional a => [a] Source #

Power series expansion of `exp(x)`

cosSeries :: Fractional a => [a] Source #

Power series expansion of `cos(x)`

sinSeries :: Fractional a => [a] Source #

Power series expansion of `sin(x)`

coshSeries :: Fractional a => [a] Source #

Power series expansion of `cosh(x)`

sinhSeries :: Fractional a => [a] Source #

Power series expansion of `sinh(x)`

log1Series :: Fractional a => [a] Source #

Power series expansion of `log(1+x)`

dyckSeries :: Num a => [a] Source #

Power series expansion of `(1-Sqrt[1-4x])/(2x)`

(the coefficients are the Catalan numbers)

# "Coin" series

coinSeries :: [Int] -> [Integer] Source #

Power series expansion of

1 / ( (1-x^k_1) * (1-x^k_2) * ... * (1-x^k_n) )

Example:

`(coinSeries [2,3,5])!!k`

is the number of ways
to pay `k`

dollars with coins of two, three and five dollars.

TODO: better name?

coinSeries' :: Num a => [(a, Int)] -> [a] Source #

Generalization of the above to include coefficients: expansion of

1 / ( (1-a_1*x^k_1) * (1-a_2*x^k_2) * ... * (1-a_n*x^k_n) )

convolveWithCoinSeries' :: Num a => [(a, Int)] -> [a] -> [a] Source #

# Reciprocals of products of polynomials

productPSeries :: [[Int]] -> [Integer] Source #

Convolution of many `pseries`

, that is, the expansion of the reciprocal
of a product of polynomials

productPSeries' :: Num a => [[(a, Int)]] -> [a] Source #

The same, with coefficients.

convolveWithProductPSeries' :: Num a => [[(a, Int)]] -> [a] -> [a] Source #

This is the most general function in this module; all the others are special cases of this one.

# Reciprocals of polynomials

pseries :: [Int] -> [Integer] Source #

The power series expansion of

1 / (1 - x^k_1 - x^k_2 - x^k_3 - ... - x^k_n)

convolveWithPSeries :: [Int] -> [Integer] -> [Integer] Source #

Convolve with (the expansion of)

1 / (1 - x^k_1 - x^k_2 - x^k_3 - ... - x^k_n)

pseries' :: Num a => [(a, Int)] -> [a] Source #

The expansion of

1 / (1 - a_1*x^k_1 - a_2*x^k_2 - a_3*x^k_3 - ... - a_n*x^k_n)

convolveWithPSeries' :: Num a => [(a, Int)] -> [a] -> [a] Source #

Convolve with (the expansion of)

1 / (1 - a_1*x^k_1 - a_2*x^k_2 - a_3*x^k_3 - ... - a_n*x^k_n)

convolveWithSignedPSeries :: [(Sign, Int)] -> [Integer] -> [Integer] Source #

Convolve with (the expansion of)

1 / (1 +- x^k_1 +- x^k_2 +- x^k_3 +- ... +- x^k_n)

Should be faster than using `convolveWithPSeries'`

.
Note: `Plus`

corresponds to the coefficient `-1`

in `pseries'`

(since
there is a minus sign in the definition there)!