The error bound formula for Simpson’s rule is

EBSN = K(b − a)^5/180N^4

where N is the number of subintervals used and K = 12 specifically for our h(x) on the interval [0, 1]. However, it can be shown that the error bound using an nth degree polynomial approximation is given by

EBTn = 1/((2n + 3)(n + 1)!

(The “T” is for “Taylor” after whom these polynomials are named.) Our example above used n = 5. Discuss and compare the errors predicted by these formula for n = 5. If accurate estimates are needed, say to 12 decimal places (i.e. an error less than 10−12 ), which method would you use? How many terms (or subintervals) are necessary for 20 decimal places of accuracy with each method? (Standard encoding of floating point numbers are capable of an accuracy of approximately 16 decimal places.) Is one method always more accurate than the other? Other than accuracy, are there other advantages or disadvantages to using one form of approximation over the other?

Thanks in advance