衍生塔以及如何使用矢量空间包(haskell)

Question

我现在和Haskell合作已经有一段时间了,但我远非专家.但我发现编程的功能方法最适合我.

到目前为止,我正在研究一个项目来计算一些严重的东西,比如给定结构辐射的电流和电位.

我跟着Conal Elliott写的博客(这里是一些更多的线性地图),这是非常好的和基础的.

不幸的是,我缺少一个简单的例子:)

更确切地说,我有一条曲线

f:[0,1] in R -> R³
t -> a*e_y + 2*t*e_z

这是(0,a,2*t)处的简单直线.当我想计算f的导数,例如用于曲线的长度,我知道数学结果,这是很简单的(0,0,2),但我要如何做到这一点在Haskell,尤其是与病媒太空包？

我真的很想使用这个库,因为它的功能,这正是我所采取的方法(但我在Haskell之路上并没有那么遥远)

到目前为止我所拥有的是:

{-# LANGUAGE Rank2Types, TypeOperators, FlexibleContexts, TypeFamilies #-}
{-# OPTIONS_GHC -Wall #-}
import Numeric.GSL.Integration
import Data.VectorSpace
import Data.Basis
import Data.Cross
import Data.Derivative
import Data.LinearMap

type Vec3 s = Three s

prec :: Double
prec = 1E-9

f1 :: (Floating s, VectorSpace s, Scalar s ~ s) => s -> s
f1 = id

c1 :: Double -> Vec3 Double
c1  = \t -> linearCombo [((v 0 0 1),f1 t),(( v 0 1 0),2)]

derivC :: Double -> Vec3 (Double :> Double)
derivC t = c1 (pureD t)

它是pureD函数的实际实现,到目前为止我没有尝试过这个片段来编译.我收到以下错误:

tests.hs:26:12:
   Couldn't match expected type `Double :> Double'
               with actual type `Double'
   Expected type: Vec3 (Double :> Double)
     Actual type: Vec3 Double
   In the return type of a call of `c1'
   In the expression: c1 (pureD t)
Failed, modules loaded: none.

还有一个使用矢量空间的图形库,甚至在圆环上有一个例子,其中使用了pureD.我试图推断出这个例子,但我不知道如何将它映射到我的问题.

任何帮助将不胜感激.

提前致谢

PS:我不能发布我想要的所有链接,但我愿意提供

Answer 1

这是一个有趣的图书馆..感谢分享.虽然我还不了解库的概念,但是这段代码怎么样:

{-# LANGUAGE Rank2Types, TypeOperators, FlexibleContexts, TypeFamilies #-}
module Main where

import Data.LinearMap
import Data.Maclaurin

diff :: (Double :~> (Double,Double,Double) ) -> (Double :~> (Double,Double,Double))
diff f = \x ->  (atBasis (derivative  (f x)) ())

eval :: (Double :~> (Double,Double,Double)) -> Double -> (Double,Double,Double)
eval f x = powVal (f x)        

f :: Double :~> (Double,Double,Double)
f x = tripleD (pureD 0,pureD 1,(2*idD) x)


*Main> map (eval f) [0,0.2 .. 1]
[(0.0,1.0,0.0),(0.0,1.0,0.4),(0.0,1.0,0.8),(0.0,1.0,1.2000000000000002),
 (0.0,1.0,1.6000000000000003),(0.0,1.0,2.0000000000000004)]

*Main> map (eval (diff f)) [0,0.2 .. 1]
[(0.0,0.0,2.0),(0.0,0.0,2.0),(0.0,0.0,2.0),(0.0,0.0,2.0),(0.0,0.0,2.0), 
 (0.0,0.0,2.0)]

*Main> map (eval (diff $ diff f)) [0,0.2 .. 1]
 [(0.0,0.0,0.0),(0.0,0.0,0.0),(0.0,0.0,0.0),(0.0,0.0,0.0),(0.0,0.0,0.0),(0.0,0.0,0.0)]

也可以尝试gx = tripleD(pureD 0,idD x,(idD*idD)x)(它们似乎代表曲线(0,x,x ^ 2)).