September 2017 Update
This commit is contained in:
Ingy döt Net
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14570 changed files with 153136 additions and 63871 deletions
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@ -1,70 +1,89 @@
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approx f xs ws = sum [w * f x | (x,w) <- zip xs ws]
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approx
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:: Fractional a
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=> (a1 -> a) -> [a1] -> [a] -> a
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approx f xs ws =
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sum
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[ w * f x
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| (x, w) <- zip xs ws ]
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integrateOpen :: Fractional a => a -> [a] -> (a -> a) -> a -> a -> Int -> a
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integrateOpen v vs f a b n = approx f xs ws * h / v where
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m = fromIntegral (length vs) * n
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h = (b-a) / fromIntegral m
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ws = concat $ replicate n vs
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c = a + h/2
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xs = [c + h * fromIntegral i | i <- [0..m-1]]
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integrateOpen
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:: Fractional a
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=> a -> [a] -> (a -> a) -> a -> a -> Int -> a
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integrateOpen v vs f a b n = approx f xs ws * h / v
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where
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m = fromIntegral (length vs) * n
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h = (b - a) / fromIntegral m
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ws = concat $ replicate n vs
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c = a + h / 2
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xs =
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[ c + h * fromIntegral i
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| i <- [0 .. m - 1] ]
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integrateClosed :: Fractional a => a -> [a] -> (a -> a) -> a -> a -> Int -> a
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integrateClosed v vs f a b n = approx f xs ws * h / v where
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m = fromIntegral (length vs - 1) * n
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h = (b-a) / fromIntegral m
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ws = overlap n vs
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xs = [a + h * fromIntegral i | i <- [0..m]]
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integrateClosed
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:: Fractional a
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=> a -> [a] -> (a -> a) -> a -> a -> Int -> a
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integrateClosed v vs f a b n = approx f xs ws * h / v
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where
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m = fromIntegral (length vs - 1) * n
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h = (b - a) / fromIntegral m
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ws = overlap n vs
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xs =
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[ a + h * fromIntegral i
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| i <- [0 .. m] ]
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overlap :: Num a => Int -> [a] -> [a]
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overlap n [] = []
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overlap n (x:xs) = x : inter n xs where
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inter 1 ys = ys
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inter n [] = x : inter (n-1) xs
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inter n [y] = (x+y) : inter (n-1) xs
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inter n (y:ys) = y : inter n ys
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intLeftRect = integrateClosed 1 [1,0]
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intMidRect = integrateOpen 1 [1]
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intRightRect = integrateClosed 1 [0,1]
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intTrapezium = integrateClosed 2 [1,1]
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intSimpson = integrateClosed 3 [1,4,1]
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overlap
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:: Num a
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=> Int -> [a] -> [a]
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overlap n [] = []
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overlap n (x:xs) = x : inter n xs
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where
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inter 1 ys = ys
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inter n [] = x : inter (n - 1) xs
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inter n [y] = (x + y) : inter (n - 1) xs
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inter n (y:ys) = y : inter n ys
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uncurry4 :: (t1 -> t2 -> t3 -> t4 -> t) -> (t1, t2, t3, t4) -> t
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uncurry4 f ~(a, b, c, d) = f a b c d
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main = do
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let m1 = "rectangular left: "
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let m2 = "rectangular middle: "
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let m3 = "rectangular right: "
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let m4 = "trapezium: "
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let m5 = "simpson: "
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-- TEST ----------------------------------------------------------------------
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ms
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:: Fractional a
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=> [(String, (a -> a) -> a -> a -> Int -> a)]
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ms =
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[ ("rectangular left", integrateClosed 1 [1, 0])
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, ("rectangular middle", integrateOpen 1 [1])
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, ("rectangular right", integrateClosed 1 [0, 1])
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, ("trapezium", integrateClosed 2 [1, 1])
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, ("simpson", integrateClosed 3 [1, 4, 1])
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]
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let arg1 = ((\x -> x ^ 3), 0, 1, 100)
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putStrLn $ m1 ++ (show $ uncurry4 intLeftRect arg1)
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putStrLn $ m2 ++ (show $ uncurry4 intMidRect arg1)
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putStrLn $ m3 ++ (show $ uncurry4 intRightRect arg1)
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putStrLn $ m4 ++ (show $ uncurry4 intTrapezium arg1)
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putStrLn $ m5 ++ (show $ uncurry4 intSimpson arg1)
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putStrLn ""
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integrations
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:: (Fractional a, Num t, Num t1, Num t2)
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=> [(String, (a -> a, t, t1, t2))]
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integrations =
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[ ("x^3", ((^ 3), 0, 1, 100))
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, ("1/x", ((1 /), 1, 100, 1000))
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, ("x", (id, 0, 5000, 500000))
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, ("x", (id, 0, 6000, 600000))
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]
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let arg2 = ((\x -> 1 / x), 1, 100, 1000)
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putStrLn $ m1 ++ (show $ uncurry4 intLeftRect arg2)
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putStrLn $ m2 ++ (show $ uncurry4 intMidRect arg2)
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putStrLn $ m3 ++ (show $ uncurry4 intRightRect arg2)
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putStrLn $ m4 ++ (show $ uncurry4 intTrapezium arg2)
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putStrLn $ m5 ++ (show $ uncurry4 intSimpson arg2)
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putStrLn ""
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let arg3 = ((\x -> x), 0, 5000, 5000000)
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putStrLn $ m1 ++ (show $ uncurry4 intLeftRect arg3)
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putStrLn $ m2 ++ (show $ uncurry4 intMidRect arg3)
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putStrLn $ m3 ++ (show $ uncurry4 intRightRect arg3)
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putStrLn $ m4 ++ (show $ uncurry4 intTrapezium arg3)
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putStrLn $ m5 ++ (show $ uncurry4 intSimpson arg3)
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putStrLn ""
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let arg4 = ((\x -> x), 0, 6000, 6000000)
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putStrLn $ m1 ++ (show $ uncurry4 intLeftRect arg4)
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putStrLn $ m2 ++ (show $ uncurry4 intMidRect arg4)
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putStrLn $ m3 ++ (show $ uncurry4 intRightRect arg4)
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putStrLn $ m4 ++ (show $ uncurry4 intTrapezium arg4)
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putStrLn $ m5 ++ (show $ uncurry4 intSimpson arg4)
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main :: IO ()
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main =
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mapM_
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(\(s, e@(_, a, b, n)) -> do
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putStrLn
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(concat
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[ indent 20 ("f(x) = " ++ s)
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, show [a, b]
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, " ("
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, show n
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, " approximations)"
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])
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mapM_
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(\(s, integration) ->
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putStrLn (indent 20 (s ++ ":") ++ show (uncurry4 integration e)))
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ms
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putStrLn [])
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integrations
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where
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indent n = take n . (++ replicate n ' ')
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@ -0,0 +1,26 @@
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// version 1.1.2
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typealias Func = (Double) -> Double
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fun integrate(a: Double, b: Double, n: Int, f: Func) {
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val h = (b - a) / n
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val sum = DoubleArray(5)
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for (i in 0 until n) {
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val x = a + i * h
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sum[0] += f(x)
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sum[1] += f(x + h / 2.0)
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sum[2] += f(x + h)
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sum[3] += (f(x) + f(x + h)) / 2.0
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sum[4] += (f(x) + 4.0 * f(x + h / 2.0) + f(x + h)) / 6.0
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}
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val methods = listOf("LeftRect ", "MidRect ", "RightRect", "Trapezium", "Simpson ")
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for (i in 0..4) println("${methods[i]} = ${"%f".format(sum[i] * h)}")
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println()
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}
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fun main(args: Array<String>) {
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integrate(0.0, 1.0, 100) { it * it * it }
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integrate(1.0, 100.0, 1_000) { 1.0 / it }
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integrate(0.0, 5000.0, 5_000_000) { it }
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integrate(0.0, 6000.0, 6_000_000) { it }
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}
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24
Task/Numerical-integration/Zkl/numerical-integration.zkl
Normal file
24
Task/Numerical-integration/Zkl/numerical-integration.zkl
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@ -0,0 +1,24 @@
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fcn integrate(F,f,a,b,steps){
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h:=(b - a) / steps;
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h*(0).reduce(steps,'wrap(s,i){ F(f, h*i + a, h) + s },0.0);
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}
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fcn rectangularLeft(f,x) { f(x) }
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fcn rectangularMiddle(f,x,h){ f(x+h/2) }
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fcn rectangularRight(f,x,h) { f(x+h) }
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fcn trapezium(f,x,h) { (f(x) + f(x+h))/2 }
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fcn simpson(f,x,h) { (f(x) + 4.0*f(x+h/2) + f(x+h))/6 }
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args:=T( T(fcn(x){ x.pow(3) }, 0.0, 1.0, 10),
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T(fcn(x){ 1.0 / x }, 1.0, 100.0, 1000),
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T(fcn(x){ x }, 0.0, 5000.0, 0d5_000_000),
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T(fcn(x){ x }, 0.0, 6000.0, 0d6_000_000) );
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fs:=T(rectangularLeft,rectangularMiddle,rectangularRight,
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trapezium,simpson);
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names:=fs.pump(List,"name",'+(":"),"%-18s".fmt);
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foreach a in (args){
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names.zipWith('wrap(nm,f){
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"%s %f".fmt(nm,integrate(f,a.xplode())).println() }, fs);
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println();
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}
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