(a) Suppose you are using the trapezoid rule to approximate an integral over an interval [a, b]….
(a) Suppose you are using the trapezoid rule to approximate an integral over an interval [a, b]. If you wish to obtain a more accurate approximation of the integral, which will gain more accuracy: (1) dividing the interval in half and using the trapezoid rule on each subinterval, or (2) using Simpson’s rule on the original interval? Note that either approach will use the same three function values, at the endpoints and the midpoint of the original interval. Support your answer with an error analysis. Test your conclusions experimentally with a few sample integrals.
(b) Suppose you are using Simpson’s rule to approximate an integral over an interval [a, b]. If you wish to obtain a more accurate approximation of the integral, which will gain more accuracy: (1) dividing the interval in half and using Simpson’s rule on each subinterval, or (2) using a closed NewtonCotes rule with the same five points as nodes? Support your answer with an error analysis. Test your conclusions experimentally with a few sample integrals.
(c) In general, for a closed n-point quadrature rule Qn, is more accuracy gained by halving the step size and using Qn on each subinterval, or using the rule Q2n−1 on the original interval? Use the general error bound from Section 8.3 to support your conclusion.
