如何用值数组替换 SymPy 符号？

Question

I am currently have an array that has the bounds for an integral (bounds_symbolic). I am trying to substitute the symbolic bounds with an array of values.

At a high level, what I am trying to do is solve the following integral (here it is using the variable names):

integral_variables

And here is an example:

example_integral

Here is my code thus far:

import sympy
import numpy as np
a, b, x, y = sympy.symbols("a b x y")


# Equation of the ellipse solved for y
ellipse = sympy.sqrt((b ** 2) * (1 - ((x ** 2) / (a ** 2))))

# Functions to be tested
test_functions = [(a * b * x), (((a * b) ** 2) * x), (((a * b) ** 3) * x), (((a * b) ** 4) * x), (((a * b) ** 5) * x)]


# Equating ellipse and test_functions so their intersection can be symbolically solved for
equate = [sympy.Eq(ellipse, test_functions[0]), sympy.Eq(ellipse, test_functions[1]), sympy.Eq(ellipse, test_functions[2]), sympy.Eq(ellipse, test_functions[3]), sympy.Eq(ellipse, test_functions[4])]


# Calculating the intersection points of the ellipse and the testing functions

# Array that holds the bounds of the integral solved symbolically
bounds_symbolic = []
for i in range(0, 3):
    bounds_symbolic.append(sympy.solve(equate[i], x))

# Array of a-values to plug into the bounds of the integral
a_values = np.linspace(-10, 10, 201)

# Setting b equal to a constant of 1
b = 1

integrand = []
for j in range(0, 3):
    integrand.append(ellipse - test_functions[j])

# New array with a-values substituted into the bounds

bounds_a = bounds_symbolic
for j in range(0, 201):
    bounds_a.subs(a, a_values[j])

When I run it, I get an error when I attempt to perform bounds_a.subs(a, a_values[j]) operation:

AttributeError: 'list' object has no attribute 'subs'

What I would like to have happen is that the bounds_a array has the same bounds that I solved for above, but with all of the "a" values substituted into it, as opposed to just having a symbolic "a". I would also like to be able to substitute in a "b" value of 1.

How would I go about doing this? Would it be better if I used a NumPy array instead of a list?

Thank you!

Answer 1

In an isympy session with x defined as a variable:

In [51]: x                                                                                           
Out[51]: x

I can make a list of expressions:

In [52]: alist = [2*x, 3*x, 4+x]                                                                     

and perform subs on each of them:

In [53]: [expr.subs({x:12}) for expr in alist]                                                       
Out[53]: [24, 36, 16]

and multiple subs:

In [54]: [[expr.subs({x:val}) for expr in alist] for val in [1,2,3]]                                 
Out[54]: [[2, 3, 5], [4, 6, 6], [6, 9, 7]]

---

With lambdify I can create a function that takes a numpy array as input (default is numpy mode):

In [61]: f = lambdify(x, alist)                                                                      

In [62]: print(f.__doc__)                                                                            
Created with lambdify. Signature:

func(x)

Expression:

[2*x, 3*x, x + 4]

Source code:

def _lambdifygenerated(x):
    return ([2*x, 3*x, x + 4])

I can give f a number:

In [63]: [f(i) for i in [1,2,3]]                                                                     
Out[63]: [[2, 3, 5], [4, 6, 6], [6, 9, 7]]

or array:

In [64]: f(np.array([1,2,3]))                                                                        
Out[64]: [array([2, 4, 6]), array([3, 6, 9]), array([5, 6, 7])]