Error in calculating integral for 2D interpolation. Comparing numpy arrays

Question

My optimization task deals with calculation of the following integral and finding the best values of xl and xu:

Iterations take to

Answer 1

I can't reproduce your error unless I omit the np.vectorize decorator. Setting xl/xu values that coincide does give me a ZeroDivisionError though.

Anyway, there's nothing stopping you from checking the values of xu vs xl in your higher-level function. That way you can skip integration entirely for nonsensical data points and return np.nan early:

import numpy as np
import mpmath
import scipy.integrate as integrate

def k_integrand(x, xl, xu):    
    return ((x**2)*mpmath.exp(x))/((xu - xl)*(mpmath.exp(x)-1)**2)

@np.vectorize   
def K(xl, xu):
    if xu <= xl:
        # don't even try to integrate
        return np.nan
    y, err = integrate.quad(k_integrand, xl, xu, args = (xl, xu))
    return y

grid_xl = np.linspace(0.1,1,10)        # shape (10,) ~ (1,10)
grid_xu = np.linspace(0.5,4,8)[:,None] # shape (8,1)

With these definitions I get (following np.set_printoptions(linewidth=200) for easier comparison:

In [35]: K(grid_xl, grid_xu)
Out[35]: 
array([[0.99145351, 0.98925197, 0.98650808, 0.98322919,        nan,        nan,        nan,        nan,        nan,        nan],
       [0.97006703, 0.96656815, 0.96254363, 0.95800307, 0.95295785, 0.94742104, 0.94140733, 0.93493293, 0.9280154 ,        nan],
       [0.93730403, 0.93263063, 0.92745487, 0.92178832, 0.91564423, 0.90903747, 0.90198439, 0.89450271, 0.88661141, 0.87833062],
       [0.89565597, 0.88996696, 0.88380385, 0.87717991, 0.87010995, 0.8626103 , 0.85469862, 0.84639383, 0.83771595, 0.82868601],
       [0.84794429, 0.8414176 , 0.83444842, 0.82705134, 0.81924245, 0.81103915, 0.8024601 , 0.79352503, 0.7842547 , 0.77467065],
       [0.79692339, 0.78974   , 0.78214742, 0.77416128, 0.76579857, 0.75707746, 0.74801726, 0.73863822, 0.72896144, 0.71900874],
       [0.7449893 , 0.73732055, 0.7292762 , 0.72087263, 0.71212741, 0.70305921, 0.69368768, 0.68403329, 0.67411725, 0.66396132],
       [0.69402415, 0.68602325, 0.67767956, 0.66900991, 0.66003222, 0.65076537, 0.6412291 , 0.63144388, 0.62143077, 0.61121128]])

You can see that the values perfectly agree with your linked image.

Now, I've got bad news and good news. The bad news is that while np.vectorize provides syntactical sugar around calling your scalar integration function with array inputs, it won't actually give you speed-up compared to a native for loop. The good news is that you can replace the calls to mpmath.exp with calls to np.exp and you'll end up with the same result much faster:

def k_integrand_np(x, xl, xu):    
    return ((x**2)*np.exp(x))/((xu - xl)*(np.exp(x)-1)**2)

@np.vectorize   
def K_np(xl, xu):
    if xu <= xl:
        # don't even try to integrate
        return np.nan
    y, err = integrate.quad(k_integrand_np, xl, xu, args = (xl, xu))
    return y

With these definitions

In [14]: res_mpmath = K(grid_xl, grid_xu)
    ...: res_np = K_np(grid_xl, grid_xu)
    ...: inds = ~np.isnan(res_mpmath)
    ...: 

In [15]: np.array_equal(res_mpmath[inds], res_np[inds])
Out[15]: True

In [16]: %timeit K(grid_xl, grid_xu)
107 ms ± 521 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)

In [17]: %timeit K_np(grid_xl, grid_xu)
7.26 ms ± 157 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)

So the two methods give the same result (exactly!), but the numpy version is almost 15 times faster.