Using lambda functions in RK4 algorithm

Question

There are two ways of implementing the classical Runge-Kutta scheme in Python showed here. The first using lambda functions, the second without them.

Which one is going to be faster and why exactly?

Answer 1

I adapted the code in the given link, and used cProfile to compare both techniques:

import numpy as np
import cProfile as cP

def theory(t):
    return (t**2 + 4.)**2 / 16.

def f(x, y):
    return x * np.sqrt(y)

def RK4(f):
    return lambda t, y, dt: (
            lambda dy1: (
            lambda dy2: (
            lambda dy3: (
            lambda dy4: (dy1 + 2*dy2 + 2*dy3 + dy4)/6
                      )( dt * f( t + dt  , y + dy3   ) )
                      )( dt * f( t + dt/2, y + dy2/2 ) )
                      )( dt * f( t + dt/2, y + dy1/2 ) )
                      )( dt * f( t       , y         ) )


def test_RK4(dy=f, x0=0., y0=1., x1=10, n=10):
    vx = np.empty(n+1)
    vy = np.empty(n+1)
    dy = RK4(f=dy)
    dx = (x1 - x0) / float(n)
    vx[0] = x = x0
    vy[0] = y = y0
    i = 1
    while i <= n:
        vx[i], vy[i] = x + dx, y + dy(x, y, dx)
        x, y = vx[i], vy[i]
        i += 1
    return vx, vy


def rk4_step(dy, x, y, dx):
    k1 = dx * dy(x, y)
    k2 = dx * dy(x + 0.5 * dx, y + 0.5 * k1)
    k3 = dx * dy(x + 0.5 * dx, y + 0.5 * k2)
    k4 = dx * dy(x + dx, y + k3)
    return x + dx, y + (k1 + k2 + k2 + k3 + k3 + k4) / 6.


def test_rk4(dy=f, x0=0., y0=1., x1=10, n=10):
    vx = np.empty(n+1)
    vy = np.empty(n+1)
    dx = (x1 - x0) / float(n)
    vx[0] = x = x0
    vy[0] = y = y0
    i = 1
    while i <= n:
        vx[i], vy[i] = rk4_step(dy=dy, x=x, y=y, dx=dx)
        x, y = vx[i], vy[i]
        i += 1
    return vx, vy

cP.run("test_RK4(n=10000)")
cP.run("test_rk4(n=10000)")

And got:

         90006 function calls in 0.095 seconds

   Ordered by: standard name

   ncalls  tottime  percall  cumtime  percall filename:lineno(function)
        1    0.000    0.000    0.095    0.095 <string>:1(<module>)
    40000    0.036    0.000    0.036    0.000 untitled1.py:13(f)
        1    0.000    0.000    0.000    0.000 untitled1.py:16(RK4)
    10000    0.008    0.000    0.086    0.000 untitled1.py:17(<lambda>)
    10000    0.012    0.000    0.069    0.000 untitled1.py:18(<lambda>)
    10000    0.012    0.000    0.048    0.000 untitled1.py:19(<lambda>)
    10000    0.009    0.000    0.027    0.000 untitled1.py:20(<lambda>)
    10000    0.009    0.000    0.009    0.000 untitled1.py:21(<lambda>)
        1    0.009    0.009    0.095    0.095 untitled1.py:28(test_RK4)
        1    0.000    0.000    0.000    0.000 {method 'disable' of '_lsprof.Profiler' objects}
        2    0.000    0.000    0.000    0.000 {numpy.core.multiarray.empty}


         50005 function calls in 0.064 seconds

   Ordered by: standard name

   ncalls  tottime  percall  cumtime  percall filename:lineno(function)
        1    0.000    0.000    0.064    0.064 <string>:1(<module>)
    40000    0.032    0.000    0.032    0.000 untitled1.py:13(f)
    10000    0.026    0.000    0.058    0.000 untitled1.py:43(rk4_step)
        1    0.006    0.006    0.064    0.064 untitled1.py:51(test_rk4)
        1    0.000    0.000    0.000    0.000 {method 'disable' of '_lsprof.Profiler' objects}
        2    0.000    0.000    0.000    0.000 {numpy.core.multiarray.empty}

So I would say the function call overhead in the "lambda" implementation makes it slower.

Nevertheless beware that somehow I appear to have lost some precision, as the results, despite agreeing with each other, are more off than the ones in the example:

>>> vx, vy = test_rk4()
>>> vy
array([   1.        ,    1.56110667,    3.99324757, ...,  288.78174798,
        451.27952013,  675.64427775])
>>> vx, vy = test_RK4()
>>> vy
array([   1.        ,    1.56110667,    3.99324757, ...,  288.78174798,
        451.27952013,  675.64427775])

Answer 2

If you pre-process the code with the Coconut transpiler, which implements tail-call optimization, then they are completely equivalent (as fast as the faster version un-processed), so you can use whichever style is more convenient for you.

# Save berna1111's code as rk4.coco; no modifications necessary.
$ coconut --target 3 rk4.coco & python3 rk4.py
         50007 function calls in 0.055 seconds

   Ordered by: standard name

   ncalls  tottime  percall  cumtime  percall filename:lineno(function)
        1    0.000    0.000    0.097    0.097 <string>:1(<module>)
    40000    0.038    0.000    0.038    0.000 rk4.py:243(f)
        1    0.000    0.000    0.000    0.000 rk4.py:246(RK4)
    10000    0.007    0.000    0.088    0.000 rk4.py:247(<lambda>)
        1    0.010    0.010    0.097    0.097 rk4.py:250(test_RK4)
        1    0.000    0.000    0.097    0.097 {built-in method builtins.exec}
        2    0.000    0.000    0.000    0.000 {built-in method numpy.core.multiarray.empty}
        1    0.000    0.000    0.000    0.000 {method 'disable' of '_lsprof.Profiler' objects}


         50006 function calls in 0.057 seconds

   Ordered by: standard name

   ncalls  tottime  percall  cumtime  percall filename:lineno(function)
        1    0.000    0.000    0.057    0.057 <string>:1(<module>)
    40000    0.030    0.000    0.030    0.000 rk4.py:243(f)
    10000    0.019    0.000    0.049    0.000 rk4.py:265(rk4_step)
        1    0.007    0.007    0.057    0.057 rk4.py:273(test_rk4)
        1    0.000    0.000    0.057    0.057 {built-in method builtins.exec}
        2    0.000    0.000    0.000    0.000 {built-in method numpy.core.multiarray.empty}
        1    0.000    0.000    0.000    0.000 {method 'disable' of '_lsprof.Profiler' objects}

Answer 3

Both @berna1111 and @matt2000 are correct. The lambda version incurs extra over-head due to the function calls. Tail-call optimization converts the tail calls into a while loop (i.e. auto converts the lambda version to the while version) eliminating the function call overhead.

See https://stackoverflow.com/a/13592002/7421639 for why Python doesn't do this optimization automatically and you have to use a tool like Coconut to do a pre-process pass.