package main

import (
    "fmt"
    "math"
)

type ypFunc func(t, y float64) float64
type ypStepFunc func(t, y, dt float64) float64

// newRKStep takes a function representing a differential equation
// and returns a function that performs a single step of the forth-order
// Runge-Kutta method.
func newRK4Step(yp ypFunc) ypStepFunc {
    return func(t, y, dt float64) float64 {
        dy1 := dt * yp(t, y)
        dy2 := dt * yp(t+dt/2, y+dy1/2)
        dy3 := dt * yp(t+dt/2, y+dy2/2)
        dy4 := dt * yp(t+dt, y+dy3)
        return y + (dy1+2*(dy2+dy3)+dy4)/6
    }
}

// example differential equation
func yprime(t, y float64) float64 {
    return t * math.Sqrt(y)
}

// exact solution of example
func actual(t float64) float64 {
    t = t*t + 4
    return t * t / 16
}

func main() {
    t0, tFinal := 0, 10 // task specifies times as integers,
    dtPrint := 1        // and to print at whole numbers.
    y0 := 1.            // initial y.
    dtStep := .1        // step value.

    t, y := float64(t0), y0
    ypStep := newRK4Step(yprime)
    for t1 := t0 + dtPrint; t1 <= tFinal; t1 += dtPrint {
        printErr(t, y) // print intermediate result
        for steps := int(float64(dtPrint)/dtStep + .5); steps > 1; steps-- {
            y = ypStep(t, y, dtStep)
            t += dtStep
        }
        y = ypStep(t, y, float64(t1)-t) // adjust step to integer time
        t = float64(t1)
    }
    printErr(t, y) // print final result
}

func printErr(t, y float64) {
    fmt.Printf("y(%.1f) = %f Error: %e\n", t, y, math.Abs(actual(t)-y))
}