8.3 Numerical Quadrature, Continued

Transcription

1 8.3 Numerical Quadrature, Continued Ulrich Hoensch Friday, October 31, 008

2 Newton-Cotes Quadrature General Idea: to approximate the integral I (f ) of a function f : [a, b] R, use equally spaced nodes in the interval [a, b] and use the integral of the interpolating polynomial of degree n 1 as an approximation of I (f ) (interpolary quadrature). There are two types of Newton-Cotes Quadrature: the open n-point rule has nodes x i = a + i(b a)/(n + 1), i = 1,,..., n; the closed n-point rule has nodes x i = a + (i 1)(b a)/(n 1), i = 1,,..., n.

3 Examples of Newton-Cotes Formulas The midpoint rule (open 1-point rule): ( ) a + b M(f ) = (b a)f. The trapezoid rule (closed -point rule): T (f ) = b a (f (a) + f (b)). Simpson s rule (closed 3-point rule): S(f ) = b a ( ( ) ) a + b f (a) + 4f + f (b). 6

4 Degree of a Quadrature Rule Definition The degree of a quadrature rule Q(f ) is the largest positive integer d so that any polynomial of degree d or less is integrated exactly by the quadrature rule. That is, Q(p) = I (p) for all p P d. By design, any n-point interpolary quadrature rule Q n (f ) has degree at least n 1. So, degree(m(f )) 0 degree(t (f )) 1 degree(s(f )). We will see later that for some of these rules the degree is actually higher than estimated here.

5 Example 1 We want to approximate the integral I (f ) = 1 0 e x dx using each of the three Newton-Cotes quadrature rules given above. M(f ) = (1 0)e T (f ) = 1 ( e 0 + e 1) S(f ) = 1 ( e 0 + 4e e 1) The value of the integral, rounded to 6 decimal places, is Thus, the error when using the trapezoid rule is larger than when using the midpoint rule. We will see the reason for this presently.

6 Error Estimates for Newton-Cotes Quadrature To investigate error estimates for the midpoint rule, we use the Taylor series expansion about m = (a + b)/: f (x) = f (m) + f (m)(x m) + f (m) (x m) + (1)! f (3) (m) (x m) 3 + f (4) (m) (x m) 4 + O ( (x m) 5). 3! 4! Note that for m = (a + b)/, b a (x m) n dx = { (b a) n+1 /( n (n + 1)) if n is even 0 if n is odd If we integrate both sides of the Taylor series expansion from a to b, the odd-order terms will consequently cancel out, and we obtain the following.

7 Error Estimates for Newton-Cotes Quadrature I (f ) = f (m)(b a) + f (m) 4 (b a)3 + f (4) (m) 190 (b a)5 + O ( (b a) 7) = M(f ) + E 1 (f ) + E (f ) + O ( (b a) 7). () Substituting x = a and x = b into (1) gives f (a) = f (m) + f (m)(a m) + f (m) (a m) +! f (3) (m) (a m) 3 + f (4) (m) (a m) 4 + O ( (a m) 5). 3! 4! and f (b) = f (m) + f (m)(b m) + f (m) (b m) +! f (3) (m) (b m) 3 + f (4) (m) (b m) 4 + O ( (b m) 5). 3! 4!

8 Error Estimates for Newton-Cotes Quadrature Since a m = a a b = a b = b a = b a b adding the two last equations on the previous slide gives = (b m), ( ) b a f (a) + f (b) = f (m) + f (m) + f (4) ( ) (m) b a 4 ( + O (b a) 6). 1 Consequently, f (a) + f (b) f (m) = f (m) (b a) + 8 f (4) (m) 384 (b a)4 + O ( (b a) 6).

9 Error Estimates for Newton-Cotes Quadrature Multiplying the last equation by (b a) and comparing the result to () gives T (f ) M(f ) = 3E 1 (f ) + 5E (f ) + O((b a) 7 ), or, assuming that E (f ) E 1 (f ), Also, it follows from E 1 (f ) T (f ) M(f ). 3 I (f ) = M(f ) + E 1 (f ) + E (f ) + O ( (b a) 7) that I (f ) = T (f ) E 1 (f ) 4E (f ) + O ( (b a) 7).

10 Error Estimates for Newton-Cotes Quadrature This leads to the following observations. If E (f ) E 1 (f ), the midpoint rule is about twice as accurate as the trapezoid rule (as we saw in example 1). The difference between the midpoint rule and the trapezoid rule can be used to estimate the error E 1 (f ) in either of them (a-priori error estimate). Halving the length of the interval in either rule decreases the error by a factor of about /8. Note: We can obtain Simpson s rule S(f ) as a weighted mean of M(f ) and T (f ) that causes the error term E 1 (f ) to drop out: I (f ) = 3 M(f ) T (f ) 3 E (f ) + O ( (b a) 7) = S(f ) 3 E (f ) + O ( (b a) 7).

11 Example Use these error estimate for the integral I (f ) = Then, M(f ) = (1 0) T (f ) = 1 0 We have the a priori error estimate E 1 (f ) T (f ) M(f ) 3 ( ) 1 = 1 4 ( ) = = 1/4 3 = 1 1. x dx ( = 1 ). 3 So, the error estimates are I (f ) M(f ) 1/1 and I (f ) T (f ) 1/6. Also, S(f ) = 1/3, which in this case as for all other estimates is exact (since the function is quadratic).

12 Properties of Newton-Cotes Interpolation We see from equation () that the midpoint rule actually integrates linear polynomials exactly (because the second derivative vanishes), and also that Simpson s rule integrates cubic functions exactly (because their fourth derivatives vanish). In general, we have that if n is odd, an n-point Newton-Cotes rule has degree n; if n is even, an n-point Newton-Cotes rule has degree n 1. This can be seen formally by considering Taylor expansions as in (1), which gives cancelation of error terms; or geometrically as in Figure 8.3 on p. 349 of the book, which shows cancelation of areas.

13 Properties of Newton-Cotes Interpolation Newton-Cotes rules should not be used for large n, because of the convergence of the underlying interpolation polynomial to the function f is not guaranteed. In practice, Newton-Cotes formulas applied to small subintervals of the given interval [a, b].