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
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
N MATLAB: This is an easy problem to demonstrate how polyfit and polyval work! The values...
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