Nicole Typaldos

Basic Simpson’s Rule:
• NOTE: P
(x)
≈f
(x) then ∫P
(x)dx
≈ ∫f
(x)dx
• Using two equal subintervals say,
[a, a+h] and [a+h, a+2h] with the interpolating polynomial P(x) of degree two, on the integrated interval.
• We have:
• With error term:

Questions about approximating:
• What if the graph oscillates greatly in some subintervals?
• What if the oscillation of the interpolating polynomial P(x) is greater than the function we are trying to approximate?

Adaptive Simpson’s Rule:
• With the basic Simpson’s Rule those subintervals with large magnitude first derivatives are not very accurate and grossly underestimate the amplitude and number of oscillations of the function.
• Let
be equal to the total error over the entire interval.

• Let
be equal to the total error over the entire interval.
• Then with the 1 st Simpson’s we have 2
A subintervals:
B
• With ≈ errors: ε /2 ε /2
• If the error in say “A”≤ ε /2 then that subinterval is left alone and the next is evaluated.
• If the error in “B”> ε /2, then “B” is broken up into further subintervals until the total error of “B” ≤ ε /2 .

Adaptive Simpson’s Rule:
• For example:
• We want to approximate a function using
Simpson’s Adaptive Rule y=exp(t).*sin(t.*cos(exp(t))) using the error of 1*e^(-5)
• To the example using MATLAB.