First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 4. NUMERICAL QUADRATURE AND DIFFERENTIATION 152
Exercise 4.2-3:
Determine the value of n required to approximate
∫ 2
0
1
x +1 dx
to within 10 −4 , and compute the approximation, using the composite trapezoidal and composite
Simpson’s rule.
Composite rules and roundoff error
As we increase n in the composite rules to lower error, the number of function evaluations
increases, and a natural question to ask would be whether roundoff error could accumulate
and cause problems. Somewhat remarkably, the answer is no. Let’s assume the roundoff error
associated with computing f(x) is bounded for all x, by some positive constant ɛ. And let’s
try to compute the roundoff error in composite Simpson rule. Since each function evaluation
in the composite rule incorporates an error of (at most) ɛ, the total error is bounded by
⎡
⎤
n
2
h
−1 n/2
∑ ∑
⎣ɛ +2 ɛ +4 ɛ + ɛ⎦ ≤ h [ ( n
) ( n
) ]
ɛ +2
3
3 2 − 1 ɛ +4 ɛ + ɛ = h (3nɛ) =hnɛ.
2 3
j=1
j=1
However, since h =(b − a)/n, the bound simplifies as hnɛ =(b − a)ɛ. Therefore no matter
how large n is, that is, how large the number of function evaluations is, the roundoff error is
bounded by the same constant (b − a)ɛ which only depends on the size of the interval.
Exercise 4.2-4: (This problem shows that numerical quadrature is stable with respect
to error in function values.) Assume the function values f(x i ) are approximated by ˜f(x i ),so
that |f(x i ) − ˜f(x i )|