Do in Maple

$8$ Non$-$Linear Regression

Study carefully the Maple code provided in the next $8$ pages, that applies non$-$linear

regression to approximate the trigonometric function

$cos(\frac{x}{50})sin(\frac{x}{50})$

in the interval $0,4.$

Modify the code to apply non$-$linear regression to approximate an arbitrary trigonometric

function made of cos and sin terms. Your code should be able to work with an arbitrary

number of higher frequency terms.

Test your code with the following trigonometric function:

$cos(x^2)+cos(2x^2)+sin(x^2)+sin(2x^2)$

in the interval $0,4.$

How many higher frequency terms do you need to add into your model, in order to get a

reasonable approximation of this function? Illustrate the quality of your approximation

by a plot similar to the ones contained in the Maple code, i$.$e$.$ overlapping the sampling

data and the data produced by the model.

Code:

n :$=50$;

x :$=[$seq$($Pi$*$i$/$n$,$ i $=1..$ n$)]$;

y :$=[$seq$($evalf$($cos$($Pi$*$i$/$n$)*$sin$($Pi$*$i$/$n$)),$ i $=1..$ n$)]$;

S :$=$ evalf$($sum$(($y$[$i$]-$ a$1*$sin$($omega$*$x$[$i$])-$ a$2*$cos$($omega$*$x$[$i$]))^2,$ i $=1..$ n$))$;

Sa$1$ :$=$ diff$($S$,$ a$1)$;

Sa$2$ :$=$ diff$($S$,$ a$2)$;

sols :$=$ solve$(\{$Sa$1,$ Sa$2\},\{$a$1,$ a$2\})$;

assign$($sols$)$;

a$1$;

a$2$;

dataPoints :$=$ plot$([$seq$([$x$[$i$],$ y$[$i$]],$ i $=1..$ n$)],$ style $=$ point$)$;

lb :$=$ min$($op$($x$))$;

ub :$=$ max$($op$($x$))$;

quadraticRegression :$=$ plot$($a$1*$sin$($omega$*$z$)+$ a$2*$cos$($omega$*$z$),$ z $=$ lb $..$ ub$,$ color $=$ blue$)$;

plots$[$display$](\{$dataPoints$,$ quadraticRegression$\})$;