Tham khảo tài liệu 'burden - numerical analysis 5e (pws, 1993) epside 1 part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 218 CHAPTER 4 Numerical Differentiation and Integration where as before the roots rnj and coefficients cnj come from Table on page 209 Now the interval a is translated to -1 1 and Gaussian quadrature is applied to approximate the integral on the right side of this equation. The details are given in Algorithm . Gaussian Double Integral rb fd x To approximate the integral f x ý dy dx Ja Jdx INPUT endpoints a b positive integers m n assume that the roots ritj and coefficients Cịj are available for Ì equals m and n and for 1 j ị OUTPUT approximation J to L Step 1 Set q b ứ 2 h2 b a Ịl J 0. Step 2 For i 1 2 . . m do Steps 3-5. steps SetJX 0 X hỵrmẠ . hf dỵ d x C1 c x kỉ dỵ - Cj 2 2 41 cl 2. Step 4 For j 1 2 . . n do sety kfaj kz Q f x ỳ JX JX cnJ Q. Step 5 Set J J cm i kỵ JX. Step 6 Set J hỵj Step 7 OUTPUT J STOP. EXAMPLE 3 The volume of the solid in Figure is approximated by applying Simpson s Double jr Integral Algorithm with n m 5 to u . N2 I I ey x dy dx. .A Jx3 . This requires 121 evaluations of the function f x ỳ eylx and produces the result which is accurate to nearly seven decimal places. Applying the Gaussian 7 U-Quadrature Algorithm with n m 5 requires only 25 function evaluations and in addition gives the approximation which is accurate to 11 decimal places. Multiple Integrals 219 09. to go- ỉịSa 1 . T ALGORITHM mấl. suit ian Triple integrals of the form p -d x p x ỳ f x 1 z dz dy dx a Jc x Ja x y see Figure are approximated in a similar manner. Because of the number of calculations involved Gaussian quadrature is the method of choice. Algorithm implements this procedure. Gaussian Triple Integral Jrb pd x r@ x y f x y z dz dy dx a Jc x Ja x ỳ INPUT endpoints a b positive integers m ỉt p assume that the roots r j and coefficients Cij are available for i equals m n and p and for 1 .7 0- OUTPUT approximation J to L Step 1 Set hỵ b a 2 h2 b d 2 7 0. Step 2 For Ỉ 1 2 . . . m do Steps 3-8. 220 A . 0 Í
