[Haskell-cafe]
Solving differential equations in terms of power series
Henning Thielemann
iakd0 at clusterf.urz.uni-halle.de
Sat May 22 16:50:21 EDT 2004
I've attached a Haskell module demonstrating the beauty of solving
ordinary differential equations in terms of power series. A power series
is expressed by the (infinite) list of its coefficients. What do you think
about that?
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module Polynomials where
-- polynomial evaluation
horner :: Num a => a -> [a] -> a
horner x = foldr (\c sum -> c+x*sum) 0
-- Horner's scheme for arbitrary rings
horner_ring :: c -> (a -> b -> c) -> (c -> b) -> [a] -> c
horner_ring zero add scale = foldr (\c sum -> add c (scale sum)) zero
-- express horner using horner_ring
horner_ring_poly :: Num a => a -> [a] -> a
horner_ring_poly x = horner_ring 0 (+) (x*)
-- add two polynomials or series
add :: Num a => [a] -> [a] -> [a]
add [] ys = ys
add xs [] = xs
add (x:xs) (y:ys) = x+y:(add xs ys)
-- subtract two polynomials or series
sub :: Num a => [a] -> [a] -> [a]
sub [] ys = map negate ys
sub xs [] = xs
sub (x:xs) (y:ys) = x-y:(sub xs ys)
-- scale a polynomial or series by a factor
scale :: Num a => a -> [a] -> [a]
scale s = map (s*)
-- multiply a polynomial by a monomial
shift :: Num a => Int -> [a] -> [a]
shift n x =
if n>=0
then replicate n 0 ++ x
else if (head x) == 0
then shift (n+1) (tail x)
else error "shift impossible"
-- multiply two polynomials or series
mul :: Num a => [a] -> [a] -> [a]
mul [] _ = []
mul (x:xs) ys = add (scale x ys) (0:(mul xs ys))
-- divide two series, walks from low exponents to high ones
-- this behaviour is uncommon for polynomials
divSeries :: Fractional a => [a] -> [a] -> [a]
divSeries [] _ = []
divSeries (0:xs) (0:ys) = divSeries xs ys
divSeries (x:xs) (y:ys) =
let q = x/y
in q : divSeries (sub xs (scale q ys)) (y:ys)
differentiate :: (Enum a, Num a) => [a] -> [a]
differentiate x = zipWith (*) (tail x) [1..]
integrate :: (Enum a, Fractional a) => a -> [a] -> [a]
integrate c x = c : (zipWith (/) x [1..])
expSeries, sinSeries, cosSeries :: (Enum a, Fractional a) => [a]
expSeries = map (1/) (scanl (*) 1 [1..])
sinSeries = zipWith (*) (cycle [0,1,0,-1]) expSeries
cosSeries = zipWith (*) (cycle [1,0,-1,0]) expSeries
{-
Lazy evaluation allows for the solution
of differential equations in terms of power series.
Whenever you can express the highest derivative of the solution
as explicit expression of the lower derivatives
where each coefficient of the solution series
depends only on lower coefficients,
the recursive algorithm will work.
-}
{-
Example for a linear equation:
Setup a differential equation for y with
y t = (exp (-t)) * (sin t)
y' t = -(exp (-t)) * (sin t) + (exp (-t)) * (cos t)
y'' t = -2 * (exp (-t)) * (cos t)
Thus the differential equation
y'' = -2 * (y' + y)
holds.
The following function generates
a power series for (exp (-t)) * (sin t)
by solving the differential equation.
-}
solveDiffEq0, verifyDiffEq0 :: (Fractional a, Enum a) => [a]
solveDiffEq0 =
let -- the initial conditions are passed to "integrate"
y = integrate 0 y'
y' = integrate 1 y''
y'' = scale (-2) (add y' y)
in y
verifyDiffEq0 = mul (zipWith (*) (iterate ((-1)*) 1) expSeries) sinSeries
{-
We are not restricted to linear equations!
Let the solution be y with
y t = (1-t)^-1
y' t = (1-t)^-2
y'' t = 2*(1-t)^-3
then it holds
y'' = 2 * y' * y
-}
solveDiffEq1, verifyDiffEq1 :: (Fractional a, Enum a) => [a]
solveDiffEq1 =
let -- the initial conditions are passed to "integrate"
y = integrate 1 y'
y' = integrate 1 y''
y'' = scale 2 (mul y' y)
in y
verifyDiffEq1 = divSeries [1] [1,-1]
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