Python数值积分与Simpson规则
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我已经开始研究这本书(Computational Physics Exercise 5.4)及其练习,并且遇到了以下问题：
编写一个Python函数J(m,x),该函数使用N = 1000点的Simpson规则计算Jm(x)的值.在程序中使用函数在单个图形上绘制Bessel函数J0,J1和J2作为x从x = 0到x = 20的函数的图.
我创建了以下代码来评估问题的第一部分,但不确定是否正确：
def f(x, t):
return 1 / pi * (math.cos(x - t * math.sin(x)))
def float_range(a, b, c):
while a < b:
yield a
a += c
N = 1000
a = 0.0
b = 20.0
h = (b - a) / N
c = 0.0
d = pi
h2 = (d - c) / N
s = 0.5 * f(a, 1) + 0.5 * f(b, 1)
s / 3
S1 = 0
S2 = 0
for k in range(1, N):
for j in range(0, N):
if k%2 == 0:
S1 += 2 / 3 * f(a + k * h, c + k * h2)
else:
S2 += 4 / 3 * f(a + k * h, c + k * h2)
s += S1 + S2
print(h * s)
有人可以帮我解决这个问题吗,我以前从未使用过Bessel函数吗？
解决方法:
您的代码有点混乱.您已经定义了生成器函数float_range,但从未使用过,其余代码都附加在该函数上,但应与该函数分开,并且不要缩进同一级别.
您还弄乱了需要集成以获得Bessel函数的核心函数的定义.您忽略了常数m,然后将x和t(θ)混合在一起.当您评估积分时,仅参数t应该变化-m和x是固定的.
无论如何,这是一些工作代码：
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))
输出
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
这种近似的准确性令人惊讶地良好.以下是使用mpmath模块生成的一些值,该模块提供了各种Bessel功能.
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
我会让您弄清楚练习的绘图部分.
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