Least Squares and Fourier Series
jon371449
Project-3-Least-Squares-and-FourierSeries.docx|
x-values |
y-values |
|
-3.14159265358979 |
0.00000000000000 |
|
-2.81089869005402 |
-0.43702230525987 |
|
-2.48020472651826 |
-0.78170987707147 |
|
-2.14951076298249 |
-0.85207605602719 |
|
-1.81881679944672 |
-0.52335935447908 |
|
-1.48812283591095 |
0.22121102681686 |
|
-1.15742887237519 |
1.27130734031775 |
|
-0.82673490883942 |
2.41089249371528 |
|
-0.49604094530365 |
3.37539264122665 |
|
-0.16534698176788 |
3.92749194128624 |
|
0.16534698176788 |
3.92749194128624 |
|
0.49604094530365 |
3.37539264122665 |
|
0.82673490883942 |
2.41089249371528 |
|
1.15742887237519 |
1.27130734031775 |
|
1.48812283591095 |
0.22121102681686 |
|
1.81881679944672 |
-0.52335935447908 |
|
2.14951076298249 |
-0.85207605602719 |
|
2.48020472651826 |
-0.78170987707147 |
|
2.81089869005402 |
-0.43702230525987 |
|
3.14159265358979 |
0.00000000000000 |
|
We want to fit two Least Squares polynomials to the 20-point data on the left. Notice that the interval of interest is . You will have to cut and paste the data into your computer program.
Part-I Goal: Approximate the data by a 4th degree polynomial, P4(x) = ao+ a1x+ a2x2+ a3x3+ a4x4 and write your coefficients in the space provided. Round your final answers to 4 decimal digits. Also plot the data points and the curve y= P4(x) together, in the spaces provided below. Do not connect the data points.
Part-II Goal: Approximate the data by a Trig Polynomial, S3(x)=a0/2+(a3/2)Cos3x+a1Cosx+a2Cos2x+b1Sinx+b2Sin2x and write your coefficients in the space provided. The final answers must be rounded to 4 decimal digits. Notice that the forms of ao and a3 are compatible with our book and well suited for Cooley-Tukey Fast Fourier Transform. Also plot the data points and the curve y= S3(x) together, in the spaces provided below. Do not connect the data points. |
Your output/notes should be printed below:
(a) Print the Coefficients for Part-I here, rounded to 4 decimal digits:
|
ao =
|
a1= |
a2= |
a3= |
a4= |
(b) Print the Coefficients for Part-II here, rounded to 4 decimal digits:
|
=
|
a1= |
a2= |
=
|
b1= |
b2= |
(c) Plot the graph for Part-I here, For full credit, your graph should be labelled properly.
(d) Plot the graph for Part-II here, For full credit, your graph should be labelled properly.
e) Your notes for Part-I must be typed up here. Any assumptions, equations, techniques that you used.
f) Your notes for Part-II must be typed up here. Any assumptions, equations, techniques that you used.
