##
Numerical Methods, MAT 350

Autumn 2001

## Homework: Polynomial Interpolation

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

- 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}.
- By hand, find a cubic polynomial that goes through these points.
Do not try to simplify this polynomial.
- Use your polynomial to estimate the function at
`x`=0.5.
Compare your answer to the exact function
`y`=2^{x} at this point.
- Use the polynomial to estimate the function at
`x`=3.
Compare your answer to the exact function at this point.
- 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).
- 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).
P_{1} |

| P_{12} |

P_{2} | | P_{123} |

| P_{23} | | P_{1234} |

P_{3} | | P_{234} |

| P_{34} |

P_{4} |

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