Chapter 6. Quadrature - MathWorks

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20 Chapter 6. Quadrature The statement k = inpolygon(u,v,x,y) returns an array the same size as u and v whose elements are one for the points in the polygon and zero for the points outside. The total number of points in the region is the number of nonzeros in k, that is, nnz(k), so the area of the corresponding portion of the grid is h^2*nnz(k) (c) Two-dimensional adaptive quadrature. The characteristic function of the region χ(u, v) is equal to one for points (u, v) in the region and zero for points outside. The area of the region is χ(u, v)dudv. The Matlab function inpolygon(u,v,x,y) computes the characteristic function if u and v are scalars, or arrays of the same size. But the quadrature functions have one of them a scalar and the other an array. So we need an M-file, chi.m, containing function k = chi(u,v,x,y) if all(size(u) == 1), u = u(ones(size(v))); end if all(size(v) == 1), v = v(ones(size(u))); end k = inpolygon(u,v,x,y); Two-dimensional adaptive numerical quadrature is obtained with dblquad(@chi,xmin,xmax,ymin,ymax,tol,[],x,y) This is the least efficient of the three methods. Adaptive quadrature expects the integrand to be reasonably smooth, but χ(u, v) is certainly not smooth. Consequently, values of tol smaller than 10 −4 or 10 −5 require a lot of computer time. Figure 6.5 shows that the estimates of the area obtained by these three methods agree to about two digits, even with fairly large grid sizes and tolerances. Experiment with your own data, use a moderate amount of computer time, and see how close the three estimates can be to each other.
Bibliography [1] F. Bornemann, D. Laurie, S. Wagon, and J. Waldvogel, The SIAM 100-Digit Challenge: A Study in High-Accuracy Numerical Computing, SIAM, Philadelphia, 2004. [2] L. N. Trefethen, A hundred-dollar, hundred-digit challenge, SIAM News, 35(1)(2002). Society of Industrial and Applied Mathematics. http://www.siam.org/pdf/news/388.pdf http://www.siam.org/books/100digitchallenge [3] W. Gander and W. Gautschi, Adaptive Quadrature—Revisited, BIT Numerical Mathematics, 40 (2000), pp. 84–101. http://www.inf.ethz.ch/personal/gander 21
Page 1 and 2: Chapter 6 Quadrature The term numer
Page 3 and 4: 6.2. Basic Quadrature Rules 3 Midpo
Page 5 and 6: 6.3. quadtx, quadgui 5 6.3 quadtx,
Page 7 and 8: 6.4. Specifying Integrands 7 Our te
Page 9 and 10: 6.5. Performance 9 calls for linear
Page 11 and 12: 6.6. Integrating Discrete Data 11 1
Page 13 and 14: 6.7. Further Reading 13 The quantit
Page 15 and 16: Exercises 15 6.9. (a) What is the e
Page 17 and 18: Exercises 17 (a) Derive and solve t
Page 19: Exercises 19 x = round(100*[0 sort(

Chapter 6. Quadrature - MathWorks

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