Task: Numerical integration




Go: package main

import (
    "fmt"
    "math"
)


type spec struct {
    lower, upper float64               
    n            int                   
    exact        float64               
    fs           string                
    f            func(float64) float64 
}


var data = []spec{
    spec{0, 1, 100, .25, "x^3", func(x float64) float64 { return x * x * x }},
    spec{1, 100, 1000, float64(math.Log(100)), "1/x",
        func(x float64) float64 { return 1 / x }},
    spec{0, 5000, 5e5, 12.5e6, "x", func(x float64) float64 { return x }},
    spec{0, 6000, 6e6, 18e6, "x", func(x float64) float64 { return x }},
}


type method struct {
    name      string
    integrate func(spec) float64
}


var methods = []method{
    method{"Rectangular (left)    ", rectLeft},
    method{"Rectangular (right)   ", rectRight},
    method{"Rectangular (midpoint)", rectMid},
    method{"Trapezium             ", trap},
    method{"Simpson's             ", simpson},
}

func rectLeft(t spec) float64 {
    var a adder
    r := t.upper - t.lower
    nf := float64(t.n)
    x0 := t.lower
    for i := 0; i < t.n; i++ {
        x1 := t.lower + float64(i+1)*r/nf
        
        
        a.add(t.f(x0) * (x1 - x0))
        x0 = x1
    }
    return a.total()
}

func rectRight(t spec) float64 {
    var a adder
    r := t.upper - t.lower
    nf := float64(t.n)
    x0 := t.lower
    for i := 0; i < t.n; i++ {
        x1 := t.lower + float64(i+1)*r/nf
        a.add(t.f(x1) * (x1 - x0))
        x0 = x1
    }
    return a.total()
}

func rectMid(t spec) float64 {
    var a adder
    r := t.upper - t.lower
    nf := float64(t.n)
    
    
    
    
    
    
    x0 := t.lower - .5*r/nf
    for i := 0; i < t.n; i++ {
        x1 := t.lower + (float64(i)+.5)*r/nf
        a.add(t.f(x1) * (x1 - x0))
        x0 = x1
    }
    return a.total()
}

func trap(t spec) float64 {
    var a adder
    r := t.upper - t.lower
    nf := float64(t.n)
    x0 := t.lower
    f0 := t.f(x0)
    for i := 0; i < t.n; i++ {
        x1 := t.lower + float64(i+1)*r/nf
        f1 := t.f(x1)
        a.add((f0 + f1) * .5 * (x1 - x0))
        x0, f0 = x1, f1
    }
    return a.total()
}

func simpson(t spec) float64 {
    var a adder
    r := t.upper - t.lower
    nf := float64(t.n)
    
    
    dx0 := r / nf
    a.add(t.f(t.lower) * dx0)
    a.add(t.f(t.lower+dx0*.5) * dx0 * 4)
    x0 := t.lower + dx0
    for i := 1; i < t.n; i++ {
        x1 := t.lower + float64(i+1)*r/nf
        xmid := (x0 + x1) * .5
        dx := x1 - x0
        a.add(t.f(x0) * dx * 2)
        a.add(t.f(xmid) * dx * 4)
        x0 = x1
    }
    a.add(t.f(t.upper) * dx0)
    return a.total() / 6
}

func sum(v []float64) float64 {
    var a adder
    for _, e := range v {
        a.add(e)
    }
    return a.total()
}

type adder struct {
    sum, e float64
}

func (a *adder) total() float64 {
    return a.sum + a.e
}

func (a *adder) add(x float64) {
    sum := a.sum + x
    e := sum - a.sum
    a.e += a.sum - (sum - e) + (x - e)
    a.sum = sum
}

func main() {
    for _, t := range data {
        fmt.Println("Test case: f(x) =", t.fs)
        fmt.Println("Integration from", t.lower, "to", t.upper,
            "in", t.n, "parts")
        fmt.Printf("Exact result            %.7e     Error\n", t.exact)
        for _, m := range methods {
            a := m.integrate(t)
            e := a - t.exact
            if e < 0 {
                e = -e
            }
            fmt.Printf("%s  %.7e  %.7e\n", m.name, a, e)
        }
        fmt.Println("")
    }
}




Python: from fractions import Fraction

def left_rect(f,x,h):
  return f(x)
 
def mid_rect(f,x,h):
  return f(x + h/2)
 
def right_rect(f,x,h):
  return f(x+h)
 
def trapezium(f,x,h):
  return (f(x) + f(x+h))/2.0
 
def simpson(f,x,h):
  return (f(x) + 4*f(x + h/2) + f(x+h))/6.0
 
def cube(x):
  return x*x*x
 
def reciprocal(x):
  return 1/x
 
def identity(x):
  return x
 
def integrate( f, a, b, steps, meth):
   h = (b-a)/steps
   ival = h * sum(meth(f, a+i*h, h) for i in range(steps))
   return ival  


for a, b, steps, func in ((0., 1., 100, cube), (1., 100., 1000, reciprocal)):
    for rule in (left_rect, mid_rect, right_rect, trapezium, simpson):
        print('%s integrated using %s\n  from %r to %r (%i steps) = %r' %
              (func.__name__, rule.__name__, a, b, steps,
               integrate( func, a, b, steps, rule)))
    a, b = Fraction.from_float(a), Fraction.from_float(b)
    for rule in (left_rect, mid_rect, right_rect, trapezium, simpson):
        print('%s integrated using %s\n  from %r to %r (%i steps and fractions) = %r' %
              (func.__name__, rule.__name__, a, b, steps,
               float(integrate( func, a, b, steps, rule))))


for a, b, steps, func in ((0., 5000., 5000000, identity),
                          (0., 6000., 6000000, identity)):
    for rule in (left_rect, mid_rect, right_rect, trapezium, simpson):
        print('%s integrated using %s\n  from %r to %r (%i steps) = %r' %
              (func.__name__, rule.__name__, a, b, steps,
               integrate( func, a, b, steps, rule)))
    a, b = Fraction.from_float(a), Fraction.from_float(b)
    for rule in (left_rect, mid_rect, right_rect, trapezium, simpson):
        print('%s integrated using %s\n  from %r to %r (%i steps and fractions) = %r' %
              (func.__name__, rule.__name__, a, b, steps,
               float(integrate( func, a, b, steps, rule))))