Homework: Polynomial Interpolation

Due Friday, October 5, 2000 at 2:00 PM

1. Consider the set of points in the xy-plane: (-1,0.5), (0,1), (1,2), (2,4). Notice that these points lie on the curve y=2^x.
   1. By hand, find a cubic polynomial that goes through these points. Do not try to simplify this polynomial.
   2. Use your polynomial to estimate the function at x=0.5. Compare your answer to the exact function y=2^x at this point.
   3. Use the polynomial to estimate the function at x=3. Compare your answer to the exact function at this point.
   4. Which of the above is interpolation and which is extrapolation? Which has the larger percentage error?
   In this exercise, it is easier to evaluate the function itself rather than the interpolating polynomial. In a real-life situation, the opposite would be true (else you wouldn't have bothered to interpolate in the first place).
2. Consider the set of points in the xy-plane: (0,1), (0.6,0.8), (0.8,0.6), (1,0). Note that these points all lie on a circle of radius one. Use the "pyramid" method to evaluate the interpolating polynomial at x=0.5 and at x=sqrt(3)/2. Compare your answer with the exact result (points on a circle).

   P1
   	P12
   P2		P123
   	P23		P1234
   P3		P234
   	P34
   P4

   Note that you never explicitly construct the polynomial itself. You might want to write a short computer program to perform this calculation.