Exercise 4-4

Maximilian Fernaldy - C2B1702

To evaluate the Maclaurin expansions, we first need to define it. The Maclaurin series is an approximation of a function that can be defined as follows:
f(x)=f(0)0!+f′(0)1!x+f′′(0)2!x2+...∴  f(x)=∑n=0∞f(n)(0)n!xn\begin{align*} f(x) &= \frac{f(0)}{0!} + \frac{f'(0)}{1!} x + \frac{f''(0)}{2!} x^2 + ... \\ \therefore\; f(x)&= \displaystyle\sum_{n = 0}^{\infty}\frac{f^{(n)}(0)}{n!} x^n \end{align*}

Since we're going to use factorials, we should also define the factorial as a function, as it's not included in the <math.h> header file:

int factorial(int number) {
    if (number < 1) { // if zero is passed, return 1 as the factorial 
        return 1;
    }
    for (int i = number-1; i > 1; i--) {
        number *= i; // multiplies number by i and assigns that new value to number
    }
    return number;
}

Note that this function does not error handle factorials of integers less than 0 (it still returns 1 as the factorial, even though factorials of negative integers are undefined), but since we are going to use it four our own purposes, it's not a problem.

Using the newly-defined function factorial, we can define the coefficients to use in the Maclaurin series.

In poly_sin3.c:

double	c[N] = {sin(0)/factorial(0), cos(0)/factorial(1), -sin(0)/factorial(2), -cos(0)/factorial(3)};

and in poly_sin5.c:

double	c[N] = {sin(0)/factorial(0), // evaluates to 0
                cos(0)/factorial(1), // evaluates to 1
                -sin(0)/factorial(2), // evaluates to 0
                -cos(0)/factorial(3), // evaluates to 1/6
                sin(0)/factorial(4), // evaluates to 1/24
                cos(0)/factorial(5)}; // evaluates to 1/120

Even though it is possible to just use the evaluated values of the coefficients, it is good practice to define where the coefficients come from, for better code readability and comprehension.

After setting the coefficients, we use the following formula for the Maclaurin series of degree nn:

f(x)=(((cn−1x+cn−2)x+cn−3)x+...+c1)x+c0f(x) = (((c_{n-1}x + c_{n-2})x + c_{n-3})x + ... + c_1)x + c_0

translating this formula to code, we have

/**** (2) processing contents ****/
for(x = -M_PI; x <= M_PI; x += M_PI/100){
    for(y = c[N-1], i = N-2; i >= 0; --i) {
        y = y * x + c[i];
    }
    printf("%f %f\n", x, y);
}

the x level for loop iterates through the linear space of the xx-axis, giving the function an xx-value to work with. The y level for loop is being used to sum the terms of the formula. We can see that at the first iteration, y = c[N-1] which corresponds to cn−1c_{n-1} gets multiplied by x and added by c[i], which corresponds to cn−2c_{n-2}. Then the whole of this corresponds to cn−1x+cn−2c_{n-1} x + c_{n-2}, and it all gets multiplied by xx and added by c[i] again in the next iteration, but now c[i] corresponds to cn−3c_{n-3}. The operation will repeat itself, changing c[i] until it is c[0], before printing the result for the specific value of xx, and continuing with the next value of xx. This will happen until the x level for loop also ends, ending the whole program.

Since we want to plot the values using matplotlib later, we should modify the program to print out the x and y values separated by a comma and space.

/**** (2) processing contents ****/
printf("x values:\n");
for(x = -M_PI; x <= M_PI; x += M_PI/100){
    printf("%f, ", x);
}
printf("\ny values:\n");
for(x = -M_PI; x <= M_PI; x += M_PI/100){
    for(y = c[N-1], i = N-2; i >= 0; --i) {
        y = y * x + c[i];
    }
    printf("%f, ", y);
}

The first for loop iterates from −π-\pi to π\pi in π100\dfrac{\pi}{100} increments and prints out the xx-values. The second calculates the yy-values and prints them out. This part of the code stays the same for whichever degree of polynomial we are using.

Compiling and running poly_sin3.c gives:

and poly_sin5.c gives:

We can use the data from the output and plug it into an array in python (see last page for python code) to plot it and see how close they are to the actual sin function.

Clearly, the 5th degree Maclaurin series approximates sin(x)sin(x) better than the 3rd degree series, as it stays close to the real function further from x=0x = 0. It's still not a viable option for xx-values higher than π\pi or lower than −π-\pi though, as we can see that it deviates quite severely after that point.

Python code for plotting 

import matplotlib.pyplot as plt 
import numpy as np
import matplotlib.font_manager as font_manager
from scipy.stats import binom

# Actual sin function
lowerlimit = -np.pi; upperlimit = np.pi

x_values = list(np.arange(lowerlimit, upperlimit, np.pi/100))

print(x_values)

y_values = [np.sin(x) for x in x_values]

# Data obtained from poly_sin3
c_program_x_range=[-3.141593, -3.110177, -3.078761, -3.047345, -3.015929, -2.984513, -2.953097, -2.921681, -2.890265, -2.858849, -2.827433, -2.796017, -2.764602, -2.733186, -2.701770, -2.670354, -2.638938, -2.607522, -2.576106, -2.544690, -2.513274, -2.481858, -2.450442, -2.419026, -2.387610, -2.356194, -2.324779, -2.293363, -2.261947, -2.230531, -2.199115, -2.167699, -2.136283, -2.104867, -2.073451, -2.042035, -2.010619, -1.979203, -1.947787, -1.916372, -1.884956, -1.853540, -1.822124, -1.790708, -1.759292, -1.727876, -1.696460, -1.665044, -1.633628, -1.602212, -1.570796, -1.539380, -1.507964, -1.476549, -1.445133, -1.413717, -1.382301, -1.350885, -1.319469, -1.288053, -1.256637, -1.225221, -1.193805, -1.162389, -1.130973, -1.099557, -1.068142, -1.036726, -1.005310, -0.973894, -0.942478, -0.911062, -0.879646, -0.848230, -0.816814, -0.785398, -0.753982, -0.722566, -0.691150, -0.659734, -0.628319, -0.596903, -0.565487, -0.534071, -0.502655, -0.471239, -0.439823, -0.408407, -0.376991, -0.345575, -0.314159, -0.282743, -0.251327, -0.219911, -0.188496, -0.157080, -0.125664, -0.094248, -0.062832, -0.031416, -0.000000, 0.031416, 0.062832, 0.094248, 0.125664, 0.157080, 0.188496, 0.219911, 0.251327, 0.282743, 0.314159, 0.345575, 0.376991, 0.408407, 0.439823, 0.471239, 0.502655, 0.534071, 0.565487, 0.596903, 0.628319, 0.659734, 0.691150, 0.722566, 0.753982, 0.785398, 0.816814, 0.848230, 0.879646, 0.911062, 0.942478, 0.973894, 1.005310, 1.036726, 1.068142, 1.099557, 1.130973, 1.162389, 1.193805, 1.225221, 1.256637, 1.288053, 1.319469, 1.350885, 1.382301, 1.413717, 1.445133, 1.476549, 1.507964, 1.539380, 1.570796, 1.602212, 1.633628, 1.665044, 1.696460, 1.727876, 1.759292, 1.790708, 1.822124, 1.853540, 1.884956, 1.916372, 1.947787, 1.979203, 2.010619, 2.042035, 2.073451, 2.104867, 2.136283, 2.167699, 2.199115, 2.230531, 2.261947, 2.293363, 2.324779, 2.356194, 2.387610, 2.419026, 2.450442, 2.481858, 2.513274, 2.544690, 2.576106, 2.607522, 2.638938, 2.670354, 2.701770, 2.733186, 2.764602, 2.796017, 2.827433, 2.858849, 2.890265, 2.921681, 2.953097, 2.984513, 3.015929, 3.047345, 3.078761, 3.110177, 3.141593]

sin3data=[2.026120, 1.904050, 1.785049, 1.669087, 1.556133, 1.446155, 1.339122, 1.235005, 1.133771, 1.035389, 0.939829, 0.847060, 0.757050, 0.669769, 0.585185, 0.503268, 0.423986, 0.347309, 0.273205, 0.201644, 0.132595, 0.066026, 0.001906, -0.059795, -0.119109, -0.176066, -0.230697, -0.283035, -0.333108, -0.380950, -0.426589, -0.470059, -0.511389, -0.550610, -0.587754, -0.622852, -0.655934, -0.687032, -0.716177, -0.743399, -0.768730, -0.792200, -0.813841, -0.833684, -0.851759, -0.868098, -0.882731, -0.895691, -0.907006, -0.916710, -0.924832, -0.931404, -0.936457, -0.940021, -0.942128, -0.942809, -0.942094, -0.940016, -0.936603, -0.931889, -0.925903, -0.918678, -0.910242, -0.900629, -0.889869, -0.877992, -0.865030, -0.851013, -0.835974, -0.819942, -0.802950, -0.785027, -0.766204, -0.746514, -0.725986, -0.704653, -0.682544, -0.659691, -0.636125, -0.611876, -0.586977, -0.561457, -0.535349, -0.508682, -0.481488, -0.453798, -0.425643, -0.397054, -0.368061, -0.338697, -0.308992, -0.278976, -0.248682, -0.218139, -0.187379, -0.156434, -0.125333, -0.094108, -0.062791, -0.031411, -0.000000, 0.031411, 0.062791, 0.094108, 0.125333, 0.156434, 0.187379, 0.218139, 0.248682, 0.278976, 0.308992, 0.338697, 0.368061, 0.397054, 0.425643, 0.453798, 0.481488, 0.508682, 0.535349, 0.561457, 0.586977, 0.611876, 0.636125, 0.659691, 0.682544, 0.704653, 0.725986, 0.746514, 0.766204, 0.785027, 0.802950, 0.819942, 0.835974, 0.851013, 0.865030, 0.877992, 0.889869, 0.900629, 0.910242, 0.918678, 0.925903, 0.931889, 0.936603, 0.940016, 0.942094, 0.942809, 0.942128, 0.940021, 0.936457, 0.931404, 0.924832, 0.916710, 0.907006, 0.895691, 0.882731, 0.868098, 0.851759, 0.833684, 0.813841, 0.792200, 0.768730, 0.743399, 0.716177, 0.687032, 0.655934, 0.622852, 0.587754, 0.550610, 0.511389, 0.470059, 0.426589, 0.380950, 0.333108, 0.283035, 0.230697, 0.176066, 0.119109, 0.059795, -0.001906, -0.066026, -0.132595, -0.201644, -0.273205, -0.347309, -0.423986, -0.503268, -0.585185, -0.669769, -0.757050, -0.847060, -0.939829, -1.035389, -1.133771, -1.235005, -1.339122, -1.446155, -1.556133, -1.669087, -1.785049, -1.904050, -2.026120]

# Data obtained from poly_sin5
sin5data=[-0.524044, -0.521131, -0.520097, -0.520826, -0.523202, -0.527114, -0.532453, -0.539115, -0.546995, -0.555995, -0.566017, -0.576967, -0.588753, -0.601286, -0.614481, -0.628253, -0.642523, -0.657211, -0.672242, -0.687543, -0.703043, -0.718674, -0.734371, -0.750069, -0.765709, -0.781232, -0.796580, -0.811702, -0.826544, -0.841058, -0.855195, -0.868912, -0.882166, -0.894914, -0.907120, -0.918745, -0.929756, -0.940120, -0.949806, -0.958785, -0.967030, -0.974517, -0.981223, -0.987125, -0.992204, -0.996444, -0.999826, -1.002337, -1.003965, -1.004697, -1.004525, -1.003440, -1.001436, -0.998508, -0.994652, -0.989867, -0.984151, -0.977505, -0.969932, -0.961434, -0.952017, -0.941686, -0.930449, -0.918313, -0.905288, -0.891386, -0.876616, -0.860994, -0.844531, -0.827243, -0.809146, -0.790257, -0.770593, -0.750173, -0.729016, -0.707143, -0.684574, -0.661332, -0.637439, -0.612918, -0.587793, -0.562089, -0.535830, -0.509044, -0.481755, -0.453992, -0.425780, -0.397148, -0.368125, -0.338738, -0.309017, -0.278991, -0.248690, -0.218143, -0.187381, -0.156434, -0.125333, -0.094108, -0.062791, -0.031411, -0.000000, 0.031411, 0.062791, 0.094108, 0.125333, 0.156434, 0.187381, 0.218143, 0.248690, 0.278991, 0.309017, 0.338738, 0.368125, 0.397148, 0.425780, 0.453992, 0.481755, 0.509044, 0.535830, 0.562089, 0.587793, 0.612918, 0.637439, 0.661332, 0.684574, 0.707143, 0.729016, 0.750173, 0.770593, 0.790257, 0.809146, 0.827243, 0.844531, 0.860994, 0.876616, 0.891386, 0.905288, 0.918313, 0.930449, 0.941686, 0.952017, 0.961434, 0.969932, 0.977505, 0.984151, 0.989867, 0.994652, 0.998508, 1.001436, 1.003440, 1.004525, 1.004697, 1.003965, 1.002337, 0.999826, 0.996444, 0.992204, 0.987125, 0.981223, 0.974517, 0.967030, 0.958785, 0.949806, 0.940120, 0.929756, 0.918745, 0.907120, 0.894914, 0.882166, 0.868912, 0.855195, 0.841058, 0.826544, 0.811702, 0.796580, 0.781232, 0.765709, 0.750069, 0.734371, 0.718674, 0.703043, 0.687543, 0.672242, 0.657211, 0.642523, 0.628253, 0.614481, 0.601286, 0.588753, 0.576967, 0.566017, 0.555995, 0.546995, 0.539115, 0.532453, 0.527114, 0.523202, 0.520826, 0.520097, 0.521131, 0.524044]


fontprop=font_manager.FontProperties(family="CMU Serif", size=12)

plt.plot(x_values, y_values, label="sin(x)")
plt.xlabel("x", fontname="CMU Serif")
plt.ylabel("sin(x)", fontname="CMU Serif")
plt.plot(c_program_x_range, sin3data, label="Maclaurin degree 3")
plt.plot(c_program_x_range, sin5data, label="Maclaurin degree 5")
plt.xticks(fontname="CMU Serif")
plt.yticks(fontname="CMU Serif")
plt.title("Maclaurin Approximations of sin(x)", fontname="CMU Serif")
plt.legend(prop=fontprop)
plt.show()