Python numerical integration with Simpson's rule

Question

I have started to work through this book (Computational Physics Exercise 5.4) and its exercises and I got stuck with the following question:

Write a Python func

Answer 1

Your code is a little messy. You've defined a generator function float_range but you never use it, and the rest of your code is tacked onto that function, but should be separated from it, and not indented at the same level.

You've also messed up the definition of the core function that needs to be integrated to get the Bessel function. You left out the constant m, and you mixed up the x and the t (theta). When you're evaluating the integral only the parameter t should vary - m and x are fixed.

Anyway, here's some working code:

from math import sin, cos, pi

# The core function of the Bessel integral
def f(m, x, t):
    return cos(m * t - x * sin(t))

#The number of steps used in the Simpson's integral approximation
N = 1000

def J(m, x):
    ''' Approximate Bessel function Jm(x) for integer m '''

    # lower & upper limits of the integral
    a = 0.0
    b = pi

    # step size
    h = (b - a) / N

    # Sum the values for Simpson's integration
    s = f(m, x, a) + f(m, x, b)
    for i in range(1, N):
        t = a + i * h
        if i % 2 == 1:
            s += 4.0 * f(m, x, t)
        else:
            s += 2.0 * f(m, x, t)

    # multiply by h/3 to get the integral
    # and divide by pi to get the Bessel function.
    return s * h / (3.0 * pi) 

for x in range(21):
    print(x, J(0, x), J(1, x), J(2, x))

output

0 1.0 3.59712259979e-17 5.16623780792e-17
1 0.765197686558 0.440050585745 0.114903484932
2 0.223890779141 0.576724807757 0.352834028616
3 -0.260051954902 0.339058958526 0.486091260586
4 -0.397149809864 -0.0660433280235 0.364128145852
5 -0.177596771314 -0.327579137591 0.0465651162778
6 0.150645257251 -0.276683858128 -0.24287320996
7 0.30007927052 -0.00468282348235 -0.301417220086
8 0.171650807138 0.234636346854 -0.112991720424
9 -0.0903336111829 0.245311786573 0.144847341533
10 -0.245935764451 0.0434727461689 0.254630313685
11 -0.171190300407 -0.176785298957 0.139047518779
12 0.0476893107968 -0.223447104491 -0.0849304948786
13 0.206926102377 -0.0703180521218 -0.217744264242
14 0.17107347611 0.133375154699 -0.152019882582
15 -0.0142244728268 0.205104038614 0.0415716779753
16 -0.174899073984 0.0903971756613 0.186198720941
17 -0.169854252151 -0.0976684927578 0.158363841239
18 -0.013355805722 -0.187994885488 -0.0075325148878
19 0.14662943966 -0.105701431142 -0.157755906096
20 0.167024664341 0.0668331241758 -0.160341351923

---

The accuracy of this approximation is surprisingly good. Here are some values generated using the mpmath module, which supplies various Bessel functions.

0 1.0 0.0 0.0
1 0.76519768655796655145 0.44005058574493351596 0.11490348493190048047
2 0.22389077914123566805 0.5767248077568733872 0.35283402861563771915
3 -0.26005195490193343762 0.33905895852593645893 0.48609126058589107691
4 -0.39714980986384737229 -0.066043328023549136143 0.36412814585207280421
5 -0.17759677131433830435 -0.32757913759146522204 0.046565116277752215532
6 0.15064525725099693166 -0.27668385812756560817 -0.24287320996018546772
7 0.30007927051955559665 -0.0046828234823458326991 -0.30141722008594012028
8 0.17165080713755390609 0.23463634685391462438 -0.11299172042407525
9 -0.090333611182876134336 0.24531178657332527232 0.14484734153250397263
10 -0.2459357644513483352 0.04347274616886143667 0.25463031368512062253
11 -0.17119030040719608835 -0.17678529895672150114 0.13904751877870126996
12 0.047689310796833536624 -0.22344710449062761237 -0.084930494878604805352
13 0.206926102377067811 -0.070318052121778371157 -0.21774426424195679117
14 0.17107347611045865906 0.13337515469879325311 -0.15201988258205962291
15 -0.014224472826780773234 0.20510403861352276115 0.04157167797525047472
16 -0.17489907398362918483 0.090397175661304186239 0.18619872094129220811
17 -0.16985425215118354791 -0.097668492757780650236 0.15836384123850347142
18 -0.013355805721984110885 -0.18799488548806959401 -0.0075325148878013995603
19 0.14662943965965120426 -0.1057014311424092668 -0.15775590609569428497
20 0.16702466434058315473 0.066833124175850045579 -0.16034135192299815017

I'll let you figure out the plotting part of your exercise. :)

---

Here's the mpmath code I used to generate those Bessel function values above, which are accurate to 20 significant figures:

from mpmath import mp

# set precision
mp.dps = 20

for x in range(21):
    print(x, mp.besselj(0, x), mp.besselj(1, x), mp.besselj(2, x))