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Python implementation of Matlab Code - Finite Difference Method

Given this Matlab Code created by my teacher:
function [] = explicitWave(T,L,N,J)
% Explicit method for the wave eq.
% T: Length time-interval
% L: Length x-interval
% N: Number of time-intervals
% J: Number of x-intervals
k=T/N;
h=L/J;
r=(k*k)/(h*h);
k/h
x=linspace(0,L,J+1); % number of points = number of intervals + 1
uOldOld=f(x); % solution two time-steps backwards. Initial condition
disp(uOldOld)
uOld=zeros(1,length(x)); % solution at previuos time-step
uNext=zeros(1,length(x));
% First time-step
for j=2:J
uOld(j)=(1-r)*f(x(j))+r/2*(f(x(j+1))+f(x(j-1)))+k*g(x(j));
end
% Remaining time-steps
for n=0:N-1
for j=2:J
uNext(j)=2*(1-r)*uOld(j)+r*(uOld(j+1)+uOld(j-1))-uOldOld(j);
end
uOldOld=uOld;
uOld=uNext;
end
plot(x,uNext,'r')
end
I tried to implement this in Python by using this code:
import numpy as np
import matplotlib.pyplot as plt
def explicit_wave(f, g, T, L, N, J):
"""
:param T: Length of Time Interval
:param L: Length of X-interval
:param N: Number of time intervals
:param J: Number of X-intervals
:return:
"""
k = T/N
h = L/J
r = (k**2) / (h**2)
x = np.linspace(0, L, J+1)
Uoldold = f(x)
Uold = np.zeros(len(x))
Unext = np.zeros(len(x))
for j in range(1, J):
Uold[j] = (1-r)*f(x[j]) + (r/2)*(f(x[j+1]) + f(x[j-1])) + k*g(x[j])
for n in range(N-1):
for j in range(1, J):
Unext[j] = 2*(1-r) * Uold[j]+r*(Uold[j+1]+Uold[j-1]) - Uoldold[j]
Uoldold = Uold
Uold = Unext
plt.plot(x, Unext)
plt.show()
return Unext, x
However when I run the code with the same inputs, I get different results when plotting them. My inputs:
g = lambda x: -np.sin(2*np.pi*x)
f = lambda x: 2*np.sin(np.pi*x)
T = 8.0
L = 1.0
J = 60
N = 480
Python plot result compared to exact result. The x-es represent the actual solution, and the red line is the function:
Matlab plot result , x-es represent the exact solution and the red line is the function:
Could you see any obvious errors I might have made when translating this code?
In case anyone needs the exact solution:
exact = lambda x,t: 2*np.sin(np.pi*x)*np.cos(np.pi*t) - (1/(2*np.pi))*np.sin(2*np.pi*x)*np.sin(2*np.pi*t)

I found the error through debugging. The main problem here is the code:
Uoldold = Uold
Uold = Unext
So in Python when you define a new variable as equal to an older variable, they become references to each other (i.e dependent on each other). Let me illustrate this as an example consisting of lists:
a = [1,2,3,4]
b = a
b[1] = 10
print(a)
>> [1, 10, 3, 4]
So the solution here was to use .copy()
Resulting in this:
Uoldold = Uold.copy()
Uold = Unext.copy()

Python: How to get Cumulative distribution function for continuous data values?

I have a set of data values, and I want to get the CDF (cumulative distribution function) for that data set.
Since this is a continuous variable, we can't use binning approach as mentioned in (How to get cumulative distribution function correctly for my data in python?). So I came up with following approach.
import scipy.stats as st
def trapezoidal_2(ag, a, b, n):
h = np.float(b - a) / n
s = 0.0
s += ag(a)[0]/2.0
for i in range(1, n):
s += ag(a + i*h)[0]
s += ag(b)[0]/2.0
return s * h
def get_cdf(data):
a = np.array(data)
ag = st.gaussian_kde(a)
cdf = [0]
x = []
k = 0
max_data = max(data)
while (k < max_data):
x.append(k)
k = k + 1
sum_integral = 0
for i in range(1, len(x)):
sum_integral = sum_integral + (trapezoidal_2(ag, x[i - 1], x[i], 2))
cdf.append(sum_integral)
return x, cdf
This is how I use this method.
b = 1
data = st.pareto.rvs(b, size=10000)
data = list(data) x_cdf, y_cdf = get_cdf(data)
Ideally I should get a value close to 1 at the end of y_cdf list. But I get a value close to 0.57.
What is going wrong here? Is my approach correct?
Thanks.

The value of the cdf at x is the integral of the pdf between -inf and x, but you are computing it between 0 and x. Maybe you are assuming that the pdf is 0 for x < 0 but it is not:
rs = np.random.RandomState(seed=52221829)
b = 1
data = st.pareto.rvs(b, size=10000, random_state=rs)
ag = st.gaussian_kde(data)
x = np.linspace(-100, 100)
plt.plot(x, ag.pdf(x))
So this is probably what's going wrong here: you not checking your assumptions.
Your code for computing the integral is painfully slow, there are better ways to do this with scipy but gaussian_kde provides the method integrate_box_1d to integrate the pdf. If you take the integral from -inf everything looks right.
cdf = np.vectorize(lambda x: ag.integrate_box_1d(-np.inf, x))
plt.plot(x, cdf(x))
Integrating between 0 and x you get the same you are seeing now (to the right of 0), but that's not a cdf at all:
wrong_cdf = np.vectorize(lambda x: ag.integrate_box_1d(0, x))
plt.plot(x, wrong_cdf(x))

Not sure about why your function is not working exactly but one way of calculating CDF is as follows:
def get_cdf_1(data):
# start with sorted list of data
x = [i for i in sorted(data)]
cdf = []
for xs in x:
# get the sum of the values less than each data point and store that value
# this is normalised by the sum of all values
cum_val = sum([i for i in data if i <= xs])/sum(data)
cdf.append(cum_val)
return x, cdf
There is no doubt a faster way of computing this using numpy arrays rather than appending values to a list, but this returns values in the same format as your original example.

I think it's just:
def get_cdf(data):
return sorted(data), np.linspace(0, 1, len(data))
but I might be misinterpreting the question!
when I compare this to the analytic result I get the same:
x_cdf, y_cdf = get_cdf(st.pareto.rvs(1, size=10000))
import matplotlib.pyplot as plt
plt.semilogx(x_cdf, y_cdf)
plt.semilogx(x_cdf, st.pareto.cdf(x_cdf, 1))

Python: While loop - how do I avoid division by zero?

I am writing code for summing the Fourier Series that ranges from [-n,n]. However, I'm having trouble with it iterating when it gets to n = 0. I wrote an 'if' statement inside my while loop so it can ignore it, but it seems like it isn't. Here's my code:
from __future__ import division
import numpy as np
import math
import matplotlib.pyplot as plt
#initial values
ni = -10
nf = 10
ti = -3
tf = 3
dt = 0.01
yi = 0 #initial f(t) value
j = complex(0,1)
#initialization
tarray = [ti]
yarray = [yi]
t = ti
n = ni
y = yi
cn = 1/(8*(np.pi)**3*n**3*j**3)*(j*4*np.pi*n) #part (b)
#iterating loop
while t<tf:
n = ni
y = yi
while n<nf:
if n == 0:
cn = 1/6
y += cn
n += 1
else:
y += cn*np.exp(j*np.pi*n*t)
n += 1
yarray.append(y)
t+=dt
tarray.append(t)
#converting list-array
tarray = np.array(tarray)
yarray = np.array(yarray)
#plotting
plt.plot(tarray,yarray, linewidth = 1)
plt.axis("tight")
plt.xlabel('t')
plt.ylabel('f(t) upto n partial sums')
plt.title('Fourier Series for n terms')
plt.legend()
plt.show()
I want it to iterate and create an array of y-values for n ranging from some negative number to some positive number (say for n from [-10,10]), but as soon as it hits n = 0 it seems to be plugging that in into the 'else' clause even though I want it to use what's in the 'if' clause, giving me a "ZeroDivisionError: complex division by zero". How do I fix this?
Edit: Put the entire code block here so you can see the context.

This is not the most elegant way at all but try this:
while t<tf:
n = ni
y = yi
while n<nf:
try:
1/n
cn = 1/6
y += cn
n += 1
except ZeroDivisionError:
y += cn*np.exp(j*np.pi*n*t) #1/n*np.sin(n*t)
n += 1
yarray.append(y)
t+=dt
tarray.append(t)

The coefficient cn is a function of n and should be updated in every loop. You made it constant (and even equal to 1/6 for positive n).
The inner loop could look like
y = 1/6 # starting with n = 0
for n in range(1,nf):
y -= 1/(2*np.pi*n)**2 * np.sin(np.pi*n*t) # see below
Corresponding coefficients for positive and negative n's are equal and exp(ix) - exp(-ix) = 2i sin(x), so it nicely reduces. (Double check the calculation.)

defining integral using trapezoidal rule(beginner)

My programm aims in defining the integral of a given function between two numbers (x1,x2),using n trapezoids.As it seems,my department's auto evaluating programm gives different answers than the ones of mine.Problem is that i cant find anything wrong in my code...
def funct(x):
val= -(1./6)*(x-1)*(x-2)*(x+2)*(x-4)
return val
x1,x2,n=input()
Dx=float(x2-x1)/n
Sum=0
i=x1+Dx
while i<x2:
val=funct(i)
Sum+=val
i+=Dx
Sum=2*Sum
val1=funct(x1)
val2=funct(x2)
S=(Dx/2)*(val1+val2+Sum)
print "%.3f" %S

Due to rounding issues, your while cycle always includes last value of x, try using exact integer arithmetic
x0, x1 = -88.787529, 83.494648
n = 1942
dx = (x1-x0)/n
s = 0
i = 1
while i < n:
# if we allow i == n, in the following row we'll have
# x0 + n*dx = x0 + n * (x1-x0) / n = x0 + x1 - x0 = x1
# but we want to exclude the last term
s = s + funct(x0+i*dx)
i = i + 1
result = (s + funct(x0)/2.0 + funct(x1)/2.0)*dx

I know this is probably some sort of homework question, but in general, don't reinvent the wheel:
import numpy
def funct(x):
return -(1./6)*(x-1)*(x-2)*(x+2)*(x-4)
x1, x2 = -88.787529, 83.494648
n = 1942
# n "panels", n+1 points
x = numpy.linspace(x1, x2, n+1)
y = funct(x)
result = numpy.trapz(y, x)

Assigning input markers to function output in python

Sorry for the seemingly elementary question. What I'm trying to implement is summarized in the following steps:
Generate input variables: x, y.
Let z = F(x,y).
Plot z's for particular combinations of x and y.
For example:
zlist = []
for _ in range(100):
x = np.random.random()*1.
y = np.random.random()*.5
if x < .5:
z = y / 2
else:
z = y * 2
zlist.append(z)
Now if I want to plot z for all the x between (0, 0.3), I presumably would need some marker on each element in zlist indicating its inputs variables. How would I attach such marker and then access it from the list when plotting?

I don't actually know anything about Numpy, so someone please comment and tell me if I'm making a fool out of myself. It seems like vanilla python behavior, though.
Rather than appending z, let's append (z,x) instead. Now zlist is a list of tuples, and you can loop through and plot by checking zlist[i][1].
zlist = []
for _ in range(100):
x = np.random.random()*1.
y = np.random.random()*.5
if x < .5:
z = y / 2
else:
z = y * 2
zlist.append((z,x))
for value in zlist:
if value[1] > 0 and value[1] < 0.3:
# Plot value[0]
# Or if you prefer list comprehensions:
# [value[0] for value in zlist if value[1] >0 and value[1] < 0.3]
# that will return a list with only the z values in zlist.

With numpy it's almost always much more efficient to perform operations on vectors and
arrays rather than on built-in Python sequence types such as lists. Here's one
way you can quickly find F(x, y) for every combination of two sets of random x
and y values without looping in Python. The result is going to be an nx-by-ny
array Z, where Z[i, j] = F(x[i], y[j]).
First of all, you can generate all of your x, y inputs as vectors:
nx = 100
ny = 200
x = np.random.random(size=nx) * 1.
y = np.random.random(size=ny) * 5.
For the result to be an nx-by-ny array, you could take these two vectors and
multiply them by ones to get two 2D nx-by-ny arrays containing the x and y
values in the rows and columns respectively. You can do this by taking advantage of numpy's
broadcasting rules:
x_arr = x[:,np.newaxis] * np.ones((nx,ny))
y_arr = y[np.newaxis,:] * np.ones((nx,ny))
The function you will apply to each x,y pair depends on the x value.
Fortunately, you can use np.where(<condition>, <do_this>, <do_that>) to apply
different operations to the values in your input depending on some condition:
Z = np.where(x_arr < 0.5, y_arr / 2., y_arr * 2.)
We can check that all the results are correct:
for i in xrange(nx):
for j in xrange(ny):
if x[i] < 0.5:
assert Z[i, j] == y[j] / 2.
else:
assert Z[i, j] == y[j] * 2
There's actually an even cleaner way to compute Z without expanding x and y into 2D arrays. Using the same broadcasting trick we used to get
x_arr and y_arr, you can pass x and y directly to np.where():
x2 = x[:,np.newaxis]
y2 = y[np.newaxis,:]
Z2 = np.where(x2 < 0.5, y2 / 2., y2 * 2.)
assert np.all(Z == Z2)