Question

Choose one of the following projects to do. Follow the directions, write up clear project reports (typed is preferred, with inclusion of any necessary pictures or attached pages of written work). Have fun! Note I found these from a paper written by Olivia M. Carducci of Lafayette College which was published in the Volume III, Number 4 edition of Primus in December 1993 1) Polynomial Curve Fitting Given a collection of n dat a points, {(zı,y1), (T2W2), . . . , (zn,Yn), it can be shown that if all the x- coordinates are distinct, then there is a unique polynomial of degree n 1 or less that passes through the n points. Let p()aa2+ an-1a" be the general polynomial. We can find this polynomial by solving the following system of n equations in n unknowns. ao ai2a22 +an-12 a) To experience how this works, find the polynomial that fits the following 3 data points: Write out your work, showing the system of equations and how you found your solution. Then state the polynomial and show a graph of the three data points and the polynomial on the same plane b) To see how this can be extended to a much larger set of data points, find the polynonial that fits the following data points: monthly closing prices of Apple stock for 20 straight months: x = month 10 y price 106.99 112.69 118.07 133.29 140.39 140.38 149.28 141.32 145.94 160.93 17 x month 11 12 13 14 18 20 y price 151.82 166.52 169.29 167.31 165.53 176.1 166.55 164.05 185.5 184.46 You DO NOT want to do this by hand. I recommend using a Computer Algebra System (like Math- ematica) or a Graphing Calculator to handle the large system of equations. For an understanding of how to use Mathematica for this project, see https://mathematica.stackexchange.com/questions/137478/solving-a-system-of-equations- for-an-array-of-unknowns. We have Mathematica in the computer lab on campus. Feel free to contact me about how to use Mathematica on campus Take screen shots or pictures of the Matematica, Graphing Calculator, etc work and attach it to your work. You should show what the system of equations is, what the polynomial is, and also show a graph with al the data points and the polynomial on the same plane

Answer 1

MATLAB Code:

close all
clear
clc

% Problem (a)
fprintf('Problem (a)\n----------------------------------------\n')
% Data Points
x = [-2 1 3]';
y = [3 5 8]';

A = [ones(size(x)) x x.^2]; % system of equations
p = A\y;
fprintf('Model: y = (%.4f) + (%.4f)*x + (%.4f)*x^2\n', p)

xx = -4:0.01:4;
yy = p(1) + p(2)*xx + p(3)*xx.^2;
plot(x,y,'o',xx,yy), xlabel('x'), ylabel('y')
title('Degree 2 Polynomial Fit (Problem (a))')
legend('Data Points', 'Polynomial Fit', 'Location', 'northwest')

% Problem (b)
fprintf('\nProblem (b)\n----------------------------------------\n')
x = 1:20; x = x';
y = [106.99 112.69 118.07 133.29 140.39 140.38 149.28 141.32 145.94 160.93 151.82 166.52 169.29 167.31 165.53 176.1 166.55 164.05 185.5 184.46]';
A = [ones(size(x)) x x.^2 x.^3 x.^4 x.^5 x.^6 x.^7 x.^8 x.^9 x.^10 x.^11 x.^12 x.^13 x.^14 x.^15 x.^16 x.^17 x.^18 x.^19]; % system of equations
p = A\y;
fprintf('Model: y = (%.4f) + (%.4f)*x + (%.4f)*x^2 + (%.4f)*x^3 + (%.4f)*x^4 + (%.4f)*x^5 + (%.4f)*x^6 + (%.4f)*x^7 + (%.4f)*x^8 + (%.4f)*x^9 + (%.4f)*x^10 + (%.4f)*x^11 + (%.4f)*x^12 + (%.4f)*x^13 + (%.4f)*x^14 + (%.4f)*x^15 + (%.4f)*x^16 + (%.4f)*x^17 + (%.4f)*x^18 + (%.4f)*x^19\n', p)

xx = 0:0.01:25;
yy = p(1) + p(2)*xx + p(3)*xx.^2 + p(4)*xx.^3 + p(5)*xx.^4 + p(6)*xx.^5 + p(7)*xx.^6 + p(8)*xx.^7 + p(9)*xx.^8 + p(10)*xx.^9 + p(11)*xx.^10 + p(12)*xx.^11 + p(13)*xx.^12 + p(14)*xx.^13 + p(15)*xx.^14 + p(16)*xx.^15 + p(17)*xx.^16 + p(18)*xx.^17 + p(19)*xx.^18 + p(20)*xx.^19;
figure, plot(x,y,'o',xx,yy), xlabel('x'), ylabel('y')
title('Degree 19 Polynomial Fit (Problem (b))')
legend('Data Points', 'Polynomial Fit', 'Location', 'northwest')
axis([0 25 -20000 20000])

Output:

Problem (a)
----------------------------------------
Model: y = (4.0000) + (0.8333)*x + (0.1667)*x^2

Problem (b)
----------------------------------------
Model: y = (-1516876.2747) + (5341329.6562)*x + (-8194918.8757)*x^2 + (7388782.0758)*x^3 + (-4433433.6649)*x^4 + (1894828.5409)*x^5 + (-601596.9639)*x^6 + (145850.8328)*x^7 + (-27501.3898)*x^8 + (4081.2961)*x^9 + (-479.9333)*x^10 + (44.8234)*x^11 + (-3.3179)*x^12 + (0.1933)*x^13 + (-0.0087)*x^14 + (0.0003)*x^15 + (-0.0000)*x^16 + (0.0000)*x^17 + (-0.0000)*x^18 + (0.0000)*x^19

Plots:

Degree 19 Polynomial Fit (Problem (b)) x 10 2 O Data Points Polynomial Fit 1.5 0.5 0 -0.5 -1 -2 10 15 20 25