Numerical Hestock Integration-Lecture Note 1

Transcription

1 Numerical Hestock Integration-Lecture Note Wei-Chi YANG Department of Mathematics and Statistics Radford University Radford, VA USA Abstract This is a brief version of the ejmt paper, for complete content, please refer to the full version of the ejmt paper. (The Electronic Journal of Mathematics and Technology, Volume 3, Number 3, ISSN ) In this note, we describe how we can compute the integrals of a class of so called non-absolute integrals, whcih are highly oscillatory and are not Lebesgue integrable. Facts about Riemann Integral. A function is Riemann integrable if and only if it is bounded an the set of discountinuity is countable.. Every Riemann integrable function is bounded. Henstock Integral-Motivation Consider f(x) = x sin x if x and f() = over the interval [, ]. y x

2 . f(x)dx does not exist in Lebesgue sense, hence does not exist in Riemann sense.. f(x)dx does exist in Hesnstock sense. 3. Henstock integral is an extension of Lebesgue integral. In fact, we have R(f) L(f) H(f), where R(f) is the set of Riemann integrable functions, L(f) and H(f) are defined analogously.. *In order for a function f to be absolutely Henstock integrable over [a, b] it is necessary and sufficient that its antiderivative (primitive) F be absolutely continuous on [a, b] and F (x) = f(x) almost everywhere. This is the descriptive definition of the Lebesgue integral. In other words, the Lebesgue and absolute Henstock integrals are equivalent in this case. We call P a partition of an interval [a, b], if we have P = {([t i, t i ], x i ) : x i [t i, t i ], t = a t... t n t n } Definition Let δ(x) : [a, b] R +. A δ fine partition {([t i, t i ], x i ) : i n} of [a, b] is a partition of [a, b] such that [t i, t i ] (x i δ(x i ), x i + δ(x i )) for each i n. Definition A real-valued function f defined on a fixed interval [a, b] is said to be Henstock integrable with the integral value I if for every given ɛ > there is a positive function δ(x) such that for every δ fine partition of [a, b] we have n f(x i ) (t i t i ) I < ɛ. i= Fundamental Theorem of Calculus Theorem 3 If F is differentiable function on [a, b], then F is Henstock integrable and b a F (x)dx = F (b) F (a). () 3 Numerical Integrations in Maple and Mathematica. The integral they integrate has to be Riemann integrable. Consider F (x) = x cos ( ) π ( x if x π ) ( π ) π sin and F () =, then F (x) = x cos + x if x and F () =. Most x x software packages can not integrate F (x)dx since F (x) is not even Lebesgue integrable (but is Henstock integrable) and yet it follows from the Fundamental Theorem of Calculus that F (x)dx = F () F () =.. Not reliable for treating functions with singularities in higher dimensions. 3. They both rely on iterated integrals which are not always the value of the double integral (by Fubini s theorem).

3 xy Remark For function f(x, y) = (x + y ) if x + y > and f(x, y) = if x + y = in the region [, ] [, ], f(x, y)da () [,] [,] does not exist and yet the value of its repeated integrals is. Both Maple and Mathematica give the wrong answer. f(x, y)dxdy. Both Mathematica and Maple can t handle singularities which lie on the diagonal of a region. 3. Numerical integration and theoretical integration. Numerical integration experts can handle functions which are so called absolute integrals.. The non-absolute integrals, such as the following highly oscillatory function ( ) sin xy f(x, y) =, xy is not Lebesgue integrable but is Henstock integrable. Most experts in numerical integration do not talk about how to integrate this type of function directly. Most of the time, they recommend to use the transformation technique. 3. I and others (Lee, Peng Yee and his students in Singapore and researchers in China) try to bridge the theoretical and numerical integrations.. Error Bound and speed up of convergence (Romberg and Richardson schemes) 3. Closed type in one dimension (Ignore the singularities) First we experiment a closed type quadrature in one dimension, which can be used in estimating the integral of a monotone function with one singularity at one end point. A closed type quadrature is to ignore the singularity, see [DR]. We shall see that an adaptive quadrature in treating this type of function is more efficient than quadratures which use uniform spaced intervals. We therefore consider the following definition which enables us to divide an interval unevenly. 3.3 Uneven partition Definition 5 A matrix A with positive a nk is called uniformly regular if the following conditions are satisfied: (i) lim n a nk = uniformly over k. (ii) n k= a nk =.

4 For example, we may use the finite sum formula, n k= km, m =,,..., to form uniform regular matrices. For m =, we define the matrix a nk = k. n(n+) We remark that in Scientific Workplace, we may use In Maple, we may use In Mathematica, we use a(n, k) = k n(n + ) ank = (n, k) > k/(n (n + ));. ank[n_, k_] := k/{n (n + )}. (Which one is more natural to the users?) To illustrate what a uniform regular matrix would look like. We use Scientific Workplace to show the matrix determined by a nk when n =, but first we modify a(n, k) as follows: { k if k n n(n+) a(n, k) = if k > n. We obtain Consider the following closed type quadrature: Q n(f) = a nf(u n ) + n k= We define the right and left endpoints as follows: [ a(n, k) = a nk (f(u n,k ) + f(u nk )). ] k n(n + ) r(n, k) = k a(n, j) j=

5 and k l(n, k) = a(n, j) j= which correspond to u n,k and u n,k respectively. We define our first closed type quadrature as follows: Q (n) = (/)a(n, )f(r(n, )) + n k= a(n, k) (f(l(n, k)) + f(r(n, k))) We note that the first term of Q (n), (/)a(n, )f(r(n, )), is a tail term to take care of functions with a singularity, and the second term of Q (n) is a trapezoidal sum. Thus, we may call the quadrature, Q (n), to be the adaptive trapezoidal sum. Example 6 Consider the function f(x) = ln( cos x), if x, and f() =. (We notice that f has a singularity at x =.) Use Q (n) to approximate ln( cos x)dx. If we use Evaluate numerically with Scientific Workplace under Maple, we get the following numeric results: Q (3) = Q () = Q (3) = We note that when we increase n, we will be warned of the existence of the singularity at x =. To further investigate the convergence or divergence of this integral, we could write a separate program to run our quadrature. Remark 7 Consider the function f(x) = x, if x ±, and f() =. We notice that f has a singularity at x = ±, and lim x + f(x) = +, we use Q (n) to approximate f(x)dx ( from Maple and Mathematica). Since our interval is not [, ], we need to modify the followings first [ c(a, b, n, k) = and r(a, b, n, k) = + ] (b a)k n(n + ) k c(a, b, n, j) j= k l(a, b, n, k) = + c(a, b, n, j) j=

6 Q (a, b, n) = (/)c(a, b, n, )f(r(a, b, n, )) + n c(a,b,n,k) k= (f(l(a, b, n, k)) + f(r(a, b, n, k))) Q (,, 3) = Q (,, ) = Q (,, 5) = Q (,, 7) = Q (,, ) = Q (,, ) = Q (,, ) = Q (,, 8) =? Let s investigate the results by using different uniform regular matrix: and d(a, b, n, k) = r(a, b, n, k) = + 6(b a)k n(n + )(n + ) k d(a, b, n, j) j= k l(a, b, n, k) = + d(a, b, n, j) Q (a, b, n) = (/)d(a, b, n, )f(r(a, b, n, )) + n d(a,b,n,k) k= (f(l(a, b, n, k)) + f(r(a, b, n, k))) We obtain the following data: j= Q (,, 3) = Q (,, ) = Q (,, 5) = Q (,, 7) = Q (,, ) = Q (,, ) = Q (,, ) = Q (,, 8) =? We note that for this function f, the convergence is much faster (compared with the answer obtained from Maple or Mathematica) if we use the second order uniform regular matrix, d(n, k). 3. D Numerical Method for Henstock integral Example 8 Let f(x, y) = sin( ) xy xy if xy A nonabsolute integral f(x, y)da. [,] if xy =

7 . Partition [, ] [, ] as follows. Apply the closed type quadrature on each D ij and sum up the integrals. (See below) 3. Write D ij = [x i+, x i ] [y j+, y j ], where x i =, y i j = for i, j =,,... j Closed type in two dimensions { For f(x, y) = xy if x, and y., to speed up the rate of convergence for this type of if x = y = function, naturally, we consider a closed type quadrature, which is an extension of Q n(f), as follows: Q 5 n(f) = m l= n k= a nk b ml (f(u n,k, v m,l ) + f(u nk, v m,l ) + f(u n,k, v ml ) + f(u nk, v ml )) + a nb m f(u n, v m ) n a nk b m (f(u n,k, v m ) + f(u nk, v m )) + k= m a n b ml (f(u n,, v m,l ) + f(u n, v ml )) l= 6k If we use a nk = and b 6l n(n+)(n+) ml = Maple: m(m+)(m+) Q 5 7(f) = Q 5 8(f) = Q 5 9(f) = , we obtain the following information from By comparing the open type and closed type quadratures, we see that closed type quadrature is more efficient in this case.

8 5 Conclusion. Computer algebra systems are great tools for teaching and research. Users can use them to explore mathematics, making conjectures, verifying conjectures, and consequently formulating exciting new theorems.. CAS enables us to narrow the gap between the pure and applied mathematics. References [DR] Davis and Rabinowitz, method of Numerical Integration, nd ed., Academic Press 983. [LY] P.Y. Lee and W.-C. Yang, Henstock Integral and Numerical Integration, preprint. [Y] W.-C. Yang, The Errors for the Closed and Open Type Adapted Quadratures, preprint.