8.4: Practice Curve Fitting Example

Consider the following polynomial with constant scalars \(a\), \(b\), and \(c\), that falls on the \(xy\)-plane:

\[f(x) = ax^2 + bx + c \nonumber \]

Question

Is this function linear? Why or why not?

Assume that we do not know the values of \(a\), \(b\) and \(c\), but we do know that the points (1,2), (-1,12), and (2,3) are on the polynomial. We can substitute the known points into the equation above. For eample, using point (1,2) we get the following equation:

\[2 = a1^2 + b1 + c \nonumber \]

\[\text{or} \nonumber \]

\[2 = a + b + c \nonumber \]

Question

Generate two more equations by substituting points (-1,12) and (2,3) into the above equation:

Question

If we did this right, we should have three equations and three unknowns (\(a\),\(b\),\(c\)). Note also that these equations are linear (how did that happen?). Transform this system of equations into two matrices \(A\) and \(b\) like we did above.

#Put your answer to the above question here.

from urllib.request import urlretrieve

urlretrieve('https://raw.githubusercontent.com/colbrydi/jupytercheck/master/answercheck.py', 
            'answercheck.py');

from answercheck import checkanswer

checkanswer.matrix(A,'1896041ded9eebf1cba7e04f32dd1069');

from answercheck import checkanswer

checkanswer.matrix(b,'01e594bb535b4e2f5a04758ff567f918');

Question

Write the code to solve for \(x\) (i.e., (\(a\),\(b\),\(c\))) using numpy.

#Put your answer to the above question here.

from answercheck import checkanswer

checkanswer.vector(x,'1dab22f81c2c156e6adca8ea7ee35dd7');

Question

Given the value of your x matrix derived in the previous question, what are the values for \(a\), \(b\), and \(c\)?

#Put your answer to the above question here.
a = 0
b = 0
c = 0

Assuming the above is correct, the following code will print your 2nd order polynomial and plot the original points:

import numpy as np
import matplotlib.pylab as plt
x = np.linspace(-3,3)
y = a*x**2 + b*x + c

#plot the function. (Transpose is needed to make the data line up).
plt.plot(x,y.transpose())

#Plot the original points
plt.scatter(1, 2)
plt.scatter(-1, 12)
plt.scatter(2, 3)
plt.xlabel('x-axis')
plt.ylabel('y-axis')

Question

The following program is intended to take four points as inputs (\(p1\), \(p2\), \(p3\), \(p4\) \( \in R^2 \)) and calculate the coefficients \(a\), \(b\), \(c\), and \(d\) so that the graph of \(f(x) = ax^3 + bx^2 + cx + d\) passes smoothly through the points. Test the function with the following points (1,2), (-1,6), (2,3), (3,2) as inputs and print the values for \(a\), \(b\), \(c\), and \(d\).

def fitPoly3(p1,p2,p3,p4):
    A = np.matrix([[p1[0]**3, p1[0]**2, p1[0], 1],
                   [p2[0]**3, p2[0]**2, p2[0], 1],
                   [p3[0]**3, p3[0]**2, p3[0], 1],
                   [p4[0]**3, p4[0]**2, p4[0], 1]])
    
    b = np.matrix([p1[1],p2[1],p3[1],p4[1]]).T

    X = np.linalg.solve(A, b)
    a = X[0]
    b = X[1]
    c = X[2]
    d = X[3]
    #Try to put your figure generation code here for the next question 
    #####Start your code here #####
    
    
    #####End of your code here#####       
    return (a,b,c,d)

#put your answer to the above question here

Question

Modify the above fitpoly3 function to also generate a figure of the input points and the resulting polynomial in range x=(-3,3).

# Put the answer to the above question above or copy and paste the above function and modify it in this cell. 

Question

Give any four \(R^2\) input points to fitPoly3, is there always a unique solution? Explain your answer.