How to speed up symbolic derivatives of long functions using SymPy?

Question

I am writing a program in Python to solve the Schrödinger equation using the Free ICI Method (well, SICI method right now... but Free ICI is what it will turn into). If this does not sound familiar, that is because there is very little information out there on the subject, and absolutely no sample code to work from.

This process involves iteratively arriving at a solution to the partial differential equation. In doing this, there are a lot of symbolic derivatives that need to be performed. The problem is, as the program runs, the functions that need to be differentiated continue to get larger and larger so that by the fifth iteration it takes a very large amount of time to compute the symbolic derivatives.

I need to speed this up because I'd like to be able to achieve at least 30 iterations, and I'd like to have it do that before I retire.

I've gone through and removed unnecessary repeats of calculations (or at least the ones I know of), which has helped quite a bit. Beyond this, I have absolutely no clue how to speed things up.

Here is the code where containing the function that is computing the derivatives (the inf_integrate function is just the composite Simpson’s method, as it is way faster than using SymPy’s integrate, and doesn’t raise errors due to oscillatory functions):

from sympy import *


def inf_integrate(fun, n, a, b):
    f = lambdify(r, fun)
    h = (b-a)/n
    XI0 = f(a) + f(b)
    XI1 = 0
    XI2 = 0

    for i in range(1, n):
        X = a + i*h

        if i % 2 == 0:
            XI2 = XI2 + f(X)
        else:
            XI1 = XI1 + f(X)

    XI = h*(XI0 + 2*XI2 + 4*XI1)/3

    return XI


r = symbols('r')

def H(fun):
    return (-1/2)*diff(fun, r, 2) - (1/r)*diff(fun, r) - (1/r)*fun

E1 = symbols('E1')
low = 10**(-5)
high = 40
n = 5000

g = Lambda(r, r)


psi0 = Lambda(r, exp(-1.5*r))

I1 = inf_integrate(4*pi*(r**2)*psi0(r)*H(psi0(r)), n, low, high)
I2 = inf_integrate(4*pi*(r**2)*psi0(r)*psi0(r), n, low, high)

E0 = I1/I2
print(E0)

for x in range(10):

    f1 = Lambda(r, psi0(r))
    f2 = Lambda(r, g(r)*(H(psi0(r)) - E0*psi0(r)))
    Hf1 = Lambda(r, H(f1(r)))
    Hf2 = Lambda(r, H(f2(r)))

    H11 = inf_integrate(4*pi*(r**2)*f1(r)*Hf1(r), n, low, high)
    H12 = inf_integrate(4*pi*(r**2)*f1(r)*Hf2(r), n, low, high)
    H21 = inf_integrate(4*pi*(r**2)*f2(r)*Hf1(r), n, low, high)
    H22 = inf_integrate(4*pi*(r**2)*f2(r)*Hf2(r), n, low, high)

    S11 = inf_integrate(4*pi*(r**2)*f1(r)*f1(r), n, low, high)
    S12 = inf_integrate(4*pi*(r**2)*f1(r)*f2(r), n, low, high)
    S21 = S12
    S22 = inf_integrate(4*pi*(r**2)*f2(r)*f2(r), n, low, high)

    eqn = Lambda(E1, (H11 - E1*S11)*(H22 - E1*S22) - (H12 - E1*S12)*(H21 - E1*S21))

    roots = solve(eqn(E1), E1)

    E0 = roots[0]

    C = -(H11 - E0*S11)/(H12 - E0*S12)

    psi0 = Lambda(r, f1(r) + C*f2(r))

    print(E0)

The program is working and converges to exactly what the expected result is, but it is way too slow. Any help on speeding this up is very much appreciated.

Answer 1

There are several things you can do here:

1. If you profile your code, you will notice that you spend most time in the integration function inf_integrate, mostly because you are using manual Python loops. This can be amended by turning the argument into a vectorised function and using SciPy’s integration routines (which are compiled and thus fast).

2. When you are using nested symbolic expressions, it may be worthwhile checking whether an occasional explicit simplification can help to keep the exploding complexity in check. This appears to be the case here.

3. All the Lamda functions you defined are not needed. You can simplify work with expressions. I haven’t checked whether this actually affects the runtime, but it certainly helps with the next step (since SymEngine does not have Lambda yet).

4. Use SymEngine instead of SymPy. SymPy (as of now) is purely Python-based and hence slow. SymEngine is its compiled core in the making and can be considerably faster. It has almost all the functionalities you need.

5. With every step, you solve an equation that does not change its nature: It’s always the same quadratic equation, only the coefficients change. By solving this once in general, you save a lot of time, in particular by SymPy not having to deal with complicated coefficients.

Taking all together, I arrive at the following:

from symengine import *
import sympy
from scipy.integrate import trapz
import numpy as np

r, E1 = symbols('r, E1')
H11, H22, H12, H21 = symbols("H11, H22, H12, H21")
S11, S22, S12, S21 = symbols("S11, S22, S12, S21")
low = 1e-5
high = 40
n = 5000

quadratic_expression = (H11-E1*S11)*(H22-E1*S22)-(H12-E1*S12)*(H21-E1*S21)
general_solution = sympify( sympy.solve(quadratic_expression,E1)[0] )
def solve_quadratic(**kwargs):
    return general_solution.subs(kwargs)

sampling_points = np.linspace(low,high,n)
def inf_integrate(fun):
    f = lambdify([r],[fun])
    values = f(sampling_points)
    return trapz(values,sampling_points)

def H(fun):
    return -fun.diff(r,2)/2 - fun.diff(r)/r - fun/r

psi0 = exp(-3*r/2)
I1 = inf_integrate(4*pi*(r**2)*psi0*H(psi0))
I2 = inf_integrate(4*pi*(r**2)*psi0**2)
E0 = I1/I2
print(E0)

for x in range(30):
    f1 = psi0
    f2 = r * (H(psi0)-E0*psi0)
    Hf1 = H(f1)
    Hf2 = H(f2)

    H11 = inf_integrate( 4*pi*(r**2)*f1*Hf1 )
    H12 = inf_integrate( 4*pi*(r**2)*f1*Hf2 )
    H21 = inf_integrate( 4*pi*(r**2)*f2*Hf1 )
    H22 = inf_integrate( 4*pi*(r**2)*f2*Hf2 )

    S11 = inf_integrate( 4*pi*(r**2)*f1**2 )
    S12 = inf_integrate( 4*pi*(r**2)*f1*f2 )
    S21 = S12
    S22 = inf_integrate( 4*pi*(r**2)*f2**2 )

    E0 = solve_quadratic(
            H11=H11, H22=H22, H12=H12, H21=H21,
            S11=S11, S22=S22, S12=S12, S21=S21,
        )
    print(E0)

    C = -( H11 - E0*S11 )/( H12 - E0*S12 )
    psi0 = (f1 + C*f2).simplify()

This converges to −½ in a few seconds on my machine.