Numerical analysis homework help>Least Squares and Fourier Series
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.