שיטות גרפיות ונומריות

function euler(f, t0, y0, h, steps) {
  let t = t0, y = y0;
  for (let i = 0; i < steps; i++) {
    y += h * f(t, y);
    t += h;
  }
  return y;
}

function rk4Step(f, t, y, h) {
  const k1 = f(t, y);
  const k2 = f(t + h/2, y + h*k1/2);
  const k3 = f(t + h/2, y + h*k2/2);
  const k4 = f(t + h, y + h*k3);
  return y + h*(k1 + 2*k2 + 2*k3 + k4)/6;
}

function rk4(f, t0, y0, h, steps) {
  let t = t0, y = y0;
  for (let i = 0; i < steps; i++) {
    y = rk4Step(f, t, y, h);
    t += h;
  }
  return y;
}

const f = (_t, y) => y;
const exact = Math.E;
console.table([0.5, 0.25, 0.125].map((h) => {
  const steps = Math.round(1/h);
  const eulerValue = euler(f, 0, 1, h, steps);
  const rk4Value = rk4(f, 0, 1, h, steps);
  return {
    h,
    euler: eulerValue,
    eulerError: Math.abs(eulerValue - exact),
    rk4: rk4Value,
    rk4Error: Math.abs(rk4Value - exact),
  };
}));

from math import e

def euler(f, t0, y0, h, steps):
    t, y = t0, y0
    for _ in range(steps):
        y += h * f(t, y)
        t += h
    return y

def rk4_step(f, t, y, h):
    k1 = f(t, y)
    k2 = f(t+h/2, y+h*k1/2)
    k3 = f(t+h/2, y+h*k2/2)
    k4 = f(t+h, y+h*k3)
    return y + h*(k1+2*k2+2*k3+k4)/6

def rk4(f, t0, y0, h, steps):
    t, y = t0, y0
    for _ in range(steps):
        y = rk4_step(f, t, y, h)
        t += h
    return y

f = lambda _t, y: y
for h in (0.5, 0.25, 0.125):
    steps = round(1/h)
    euler_value = euler(f, 0, 1, h, steps)
    rk4_value = rk4(f, 0, 1, h, steps)
    print({
        "h": h,
        "euler": euler_value,
        "euler_error": abs(euler_value-e),
        "rk4": rk4_value,
        "rk4_error": abs(rk4_value-e),
    })