Suppose that is a real function defined on a closed interval from to . Suppose that is an integer that is no less than 2, and define . Then, for each integer , define and , respectively. Namely, are the points that divide the interval from to into equal parts, and is the value of the function at .
Next, define , and define as the approximate value calculated by the composite trapezoid rule applied on using the points which divide the interval from to into equal parts.
Answer the following questions.
(1) Assume that is a four times continuously differentiable function. Let be an integer such that and define as the second order differential of at . Express an approximate value of whose error is , as a linear combination of , and .
(2) The approximation obtained by question (1) seems to become accurate when approaches zero. Answer, with a reason, whether this is correct or not in the calculation with the IEEE 754 double precision floating point operations.
(3) Express using , and .
(4) Assume that can be expressed by a quadratic function in each interval formed by the division into equal parts. Then, define similarly using the division into equal parts composed of the division of each original part into two halves. Express using and .
The approximation for becomes more accurate as approaches zero theoretically. However, in the context of IEEE 754 double precision floating point operations, as decreases, the difference between and also becomes very small. This can lead to a significant loss of precision due to rounding errors, which is known as catastrophic cancellation. Therefore, in practice, there is a limit to the accuracy of this approximation when becomes too small.
When is approximated by a quadratic function on each subinterval, the error can be related to the integral approximations and as follows:
This formula arises from the fact that the error in the trapezoidal rule is proportional to , and doubling the number of intervals reduces the error by a factor of 4. By subtracting the two approximations, we can eliminate the leading order error term, leaving a smaller error proportional to .