| Safe Haskell | None |
|---|---|
| Language | Haskell2010 |
Math.Combinat.Numbers.Series
Description
Some basic 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]
- convolve :: Num a => [a] -> [a] -> [a]
- convolveMany :: Num a => [[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]
- data Sign
- signValue :: Num a => Sign -> a
- signedPSeries :: [(Sign, Int)] -> [Integer]
- convolveWithSignedPSeries :: [(Sign, Int)] -> [Integer] -> [Integer]
Documentation
unitSeries :: Num a => [a] Source
The series [1,0,0,0,0,...], which is the neutral element for the convolution.
convolve :: Num a => [a] -> [a] -> [a] Source
Convolution of series. The result is always an infinite list. Warning: This is slow!
convolveMany :: Num a => [[a]] -> [a] Source
Convolution of many series. Still slow!
"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 :: [Int] -> [Integer] -> [Integer] Source
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 :: [[Int]] -> [Integer] -> [Integer] Source
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)
signedPSeries :: [(Sign, Int)] -> [Integer] Source
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)!