

In [1]:

    
import numpy as np
import sympy as sp
import pandas as pd
import math

import interpolate_exp_cos as p1
import plot_sin_eps as p2
import integrate_exp as p3
import diff_functions as p4

import matplotlib.pyplot as plt

# Needed only in Jupyter to render properly in-notebook
%matplotlib inline

Homework 5

Chinmai Raman

3/18/2016

B.1 Interpolating a discrete function

Part 1:



In [3]:

    
p1.f(-.45)









    Out[3]:





-0.77671500104510183



In [2]:

    
p1.S_k(-1, 1, -.45, 10)









    Out[2]:





-0.93597357483441579

Part 2:

Value of function f(x) = $e^(-x^2) * cos(2\pi x)$ at x = -0.45 :

q = 2



In [2]:

    
p1.f(-.45)









    Out[2]:





-0.77671500104510183



In [3]:

    
p1.S_k(-1, 1, -.45, 2)









    Out[3]:





1.284454251472851

Error in approximation at q = 2:



In [4]:

    
p1.error(-1, 1, -.45, 2)









    Out[4]:





2.0611692525179528

q = 4



In [5]:

    
p1.S_k(-1, 1, -.45, 4)









    Out[5]:





2.6009207047642646

Error in approximation at q = 4:



In [6]:

    
p1.error(-1, 1, -.45, 4)









    Out[6]:





3.3776357058093662

q = 8



In [9]:

    
p1.S_k(-1, 1, -.45, 8)









    Out[9]:





-0.79999999999999982

Error in approximation at q = 8:



In [8]:

    
p1.error(-1, 1, -.45, 8)









    Out[8]:





0.023284998954897995

q= 16



In [10]:

    
p1.S_k(-1, 1, -.45, 16)









    Out[10]:





-0.98295202860868525

Error in approximation at q = 16:



In [13]:

    
p1.error(-1, 1, -.45, 16)









    Out[13]:





0.20623702756358342

B.2 Studying a function for different parameter values

$f(x) = sin\dfrac{1}{x+eps}$ in [0,1]



In [2]:

    
p2.sin_graph(0.2, 10)



In [2]:

    
p2.multigraph(0.2, 10)

How large n needs to be in order for the difference between the max of the function $f(x) = sin\dfrac{1}{x+eps}$ in [0,1] using n nodes and n+10 nodes to be less than 0.1:



In [2]:

    
p2.choose_n(0.2)









    Out[2]:





3



In [3]:

    
p2.choose_n(0.1)









    Out[3]:





3



In [4]:

    
p2.choose_n(0.05)









    Out[4]:





1

n = 200 * eps (for an epsilon of 0.2). Increasing n further does not change the plot so that it is visible on the screen.



In [8]:

    
p2.multigraph(0.2, 40)

B.6 Using the trapezoid method to approximate integrals



In [2]:

    
p3.graph()

The plot is symmetric over the y-axis. Therefore the integral of the function from -$\infty$ to $\infty$ will be the same as twice the integral of the function from 0 to $\infty$



In [3]:

    
p3.T(100, 10)









    Out[3]:





1.7370047738874057



In [5]:

    
p3.trap(100, 10) * 2









    Out[5]:





1.7370047738874057

$$\int_{-L}^{L} {e}^{-x^2} dx = 2\int_{0}^{L} {e}^{-x^2} dx$$



In [4]:

    
p3.table()



In [5]:

    
p3.error()

The error decreases as n increases and as L increases

B.8 Plotting functions and their derivatives

$$f'(x)={\dfrac{1}{x+\dfrac{1}{100}}}$$



In [2]:

    
p4.graph(p4.diff(p4.f, 1/1000, 1, 100))



In [2]:

    
p4.multigraph(p4.diff(p4.f, 1/1000, 1, 100), p4.fprime)

$$g'(x)-10{e}^{10x}sin({e}^{10x})$$



In [3]:

    
p4.graph(p4.diff(p4.g, 1/1000, 1, 100))



In [5]:

    
p4.multigraph(p4.diff(p4.g, 1/1000, 1, 100), p4.gprime)

$$h'(x)=(ln x)x^x+xx^{x-1}$$



In [4]:

    
p4.graph(p4.diff(p4.h, 1/1000, 1, 100))



In [6]:

    
p4.multigraph(p4.diff(p4.h, 1/1000, 1, 100), p4.hprime)

For the functions f(x) and h(x), an n of 100 seems to be sufficient as to make the graphs of the analytical and discrete derivatives visibly indistinguishable. For g(x), however, it is unlikely that any value of n will make the graphs indistinguishable upon superimposition, as the slope of g(x) as x -> 1 approaches very high values.



In [ ]:

	L = 2	L = 4	L = 6	L = 8	L = 10
n = 100	1.729607	1.737005	1.737005	1.737005	1.737005
n = 200	1.746886	1.754729	1.754729	1.754729	1.754729
n = 300	1.752645	1.760637	1.760637	1.760637	1.760637
n = 400	1.755525	1.763592	1.763592	1.763592	1.763592
n = 500	1.757252	1.765364	1.765364	1.765364	1.765364

	L = 2	L = 4	L = 6	L = 8	L = 10
n = 100	-0.042847	-0.035449	-0.035449	-0.035449	-0.035449
n = 200	-0.025568	-0.017725	-0.017725	-0.017725	-0.017725
n = 300	-0.019808	-0.011816	-0.011816	-0.011816	-0.011816
n = 400	-0.016929	-0.008862	-0.008862	-0.008862	-0.008862
n = 500	-0.015201	-0.007090	-0.007090	-0.007090	-0.007090