Question

N MATLAB:

This is an easy problem to demonstrate how polyfit and polyval work!

The values in y are basically a sine function on x. You can validate by checking in MATLAB

>> x = 0:9

x =

0 1 2 3 4 5 6 7 8 9

>> y = sin(x)

y =

0 0.8415 0.9093 0.1411 -0.7568 -0.9589 -0.2794 0.6570 0.9894 0.4121

Now, let us try to see what happens if we use polyfit on the values of y and see whether 1st order, 2nd order, 3rd or 7th order polynomials can approximate the sine function

If you implement this correctly, you will see that polyfit works very well for interpolation, but not for extrapolation

SCRIPT:

x = [0 1 2 3 4 5 6 7 8 9];

y = [0 0.8415 0.9093 0.1411 -0.7568 -0.9589 -0.2794 0.6570 0.9894 0.4121];

%use polyfit to find the coefficients of 1st order polynomial using x and y, and store them in p1

p1 =

%use polyfit to find the coefficients of 2nd order polynomial using x and y, and store them in p2

p2 =

%Similarly do it for order 3, 7 and store it in p3, p7

p3 =

p7 =

% Now visualize the polynomials estimated

x1 = linspace(0,10);

y1 = polyval(p1, x1);

y2 = polyval(p2, x1);

y3 = polyval(p3, x1);

y7 = polyval(p7, x1);

subplot(2,2,1)

plot(x,y,'o')

hold on

plot(x1,y1)

hold on

plot(x1,sin(x1),'g--')

title(' p = 1')

hold off

subplot(2,2,2)

plot(x,y,'o')

hold on

plot(x1,y2)

hold on

plot(x1,sin(x1),'g--')

title(' p = 2')

hold off

subplot(2,2,3)

plot(x,y,'o')

hold on

plot(x1,y3)

hold on

plot(x1,sin(x1),'g--')

title(' p = 3')

hold off

subplot(2,2,4)

plot(x,y,'o')

hold on

plot(x1,y7)

hold on

plot(x1,sin(x1),'g--')

title(' p = 7')

hold off

Answer 1

Code:

clc
clear all

x = [0 1 2 3 4 5 6 7 8 9];

y = [0 0.8415 0.9093 0.1411 -0.7568 -0.9589 -0.2794 0.6570 0.9894 0.4121];

%use polyfit to find the coefficients of 1st order polynomial using x and y, and store them in p1

p1 = polyfit(x, y, 1)

%use polyfit to find the coefficients of 2nd order polynomial using x and y, and store them in p2

p2 = polyfit(x, y, 2)

%Similarly do it for order 3, 7 and store it in p3, p7

p3 =polyfit(x, y, 3)

p7 = polyfit(x, y, 7)

% Now visualize the polynomials estimated

x1 = linspace(0,10);

y1 = polyval(p1, x1);

y2 = polyval(p2, x1);

y3 = polyval(p3, x1);

y7 = polyval(p7, x1);

subplot(2,2,1)

plot(x,y,'o')

hold on

plot(x1,y1)

hold on

plot(x1,sin(x1),'g--')

title(' p = 1')

hold off

subplot(2,2,2)

plot(x,y,'o')

hold on

plot(x1,y2)

hold on

plot(x1,sin(x1),'g--')

title(' p = 2')

hold off

subplot(2,2,3)

plot(x,y,'o')

hold on

plot(x1,y3)

hold on

plot(x1,sin(x1),'g--')

title(' p = 3')

hold off

subplot(2,2,4)

plot(x,y,'o')

hold on

plot(x1,y7)

hold on

plot(x1,sin(x1),'g--')

title(' p = 7')

hold off

Output:

p1 =

0.0122 0.1405

p2 =

0.0449 -0.3916 0.6789

p3 =

0.0077 -0.0588 -0.0376 0.4854

p7 =

0.0000 -0.0001 -0.0096 0.1295 -0.5553 0.5249 0.7487 0.0004