#!/usr/bin/env python # coding: utf-8 # In[ ]: from sympy import * init_printing(use_latex='mathjax') x, y, z = symbols('x,y,z') n, m = symbols('n,m', integer=True) # In[ ]: get_ipython().run_line_magic('matplotlib', 'inline') # # Numeric Evaluation # # In this section we'll learn how to use our symbolic equations to drive numeric computations # ## .subs and .evalf # # The simplest (and slowest) ways to evaluate an expression numerically is with the .subs and .evalf methods # In[ ]: sin(x) # In[ ]: sin(x).subs({x: 0}) # In[ ]: acos(x).subs({x: -1}) # In[ ]: acos(x).subs({x: -1}).evalf() # In[ ]: acos(x).subs({x: -1}).evalf(n=100) # ### Exercise # # In a previous section we computed the following symbolic integral # # $$\int_y^z x^n dx$$ # In[ ]: result = integrate(x**n, (x, y, z)) result # Use .subs and a dictionary with keys n, y, z to evaluate this result when # # n == 2 # y == 0 # z == 3 # In[ ]: # Evaluate the resulting integral on the above values # ### Exercise # # This integral takes on a special form when $n = -1$. Use subs to find the expression when # # n == -1 # y == 5 # z == 100 # # Then use .evalf to evaluate this subs-ed expression as a float. # In[ ]: # Evaluate the resulting integral for the values {n: -1, y: 5, z: 100} # Then use evalf to get a numeric result # ## lambdify # # The .subs and .evalf methods are great for when you want to evaluate an expression at a single point. When you want to evaluate your expression on lots of points they quickly become slow. # # To resolve this problem SymPy can rewrite its expressions as normal Python functions using the math library, vectorized computations using the NumPy library, C or Fortran Code using code printers, or even more sophisticated systems. # # We'll talk about some of the more advanced topics later. For now, lambdify... # In[ ]: # function = lambdify(input, output) f = lambdify(x, x**2) f(3) # In[ ]: import numpy as np f = lambdify(x, x**2, 'numpy') # Use numpy backend data = np.array([1, 2, 3, 4, 5]) f(data) # ### Exercise # # Here is a radial wave function for the Carbon atom at $n=3$, $l=1$ # In[ ]: from sympy.physics.hydrogen import R_nl n = 3 l = 1 r = 6 # Carbon expr = R_nl(n, l, x, r) expr # Create a function, f, that evaluate this expression using the 'numpy' backend # In[ ]: # Create Numpy function mapping x to expr with the numpy backend # We can plot your function from $x \in [0, 5]$ with the following numpy/matplotlib code # In[ ]: from matplotlib.pyplot import plot nx = np.linspace(0, 5, 1000) plot(nx, f(nx)) # ### Exercise # # Create a numpy function that computes the derivative of our expression. Plot the result alongside the original. # In[ ]: # Compute derivative of expr with respect to x # In[ ]: # Create new fprime function using lambdify # In[ ]: # Plot results alongside f(nx)