04 Numerical Integration.ppt

Introduction to puting-- A Matrix Vector Approach Using Matlab Written by Charles Loan陈文斌复旦大学Chapter 4 Numerical Integration?The Newton-Cotes posite Rules?Adaptive Quadrature?Special Topics?Shared Memory Adaptive Quadrature??badxxfI)(An m-point quadrature rule Q for the definite integral is an approximation of the form????mkkkxfwabQ1)()(abscissasweightsEfficiency essentially depends upon the number of function Newton-Cotes Rules)()(xfxp?dxxfdxxpbaba???)()(M-point Newton-Cotes ruledxxpQbammNC???)(1)( where pm-1(x) interpolates f(x) atmiabmiaxi:1),(11?????? m=2 trapezoidal rule?????????????????????)(21)(21)()()()()()2(bfafabdxaxabafbfafQbaNC m=3 and c=(a+b)/2Simpson rule??????????????????????????????????????????????)(24)(6))(()()()()()()()()()3(bfbafafabdxcxaxabacafcfcbcfbfaxacafcfafQbaNC????????????????mkkiikmxxcxp1111)(???????????????????babakiimmkmmNCdxxxcdxxpQ1111)()()(????????1011)()(mmkmkkkmmNCShcdsshaphQ??????????????????10111mkimkdsisSEqual spacing, x=a+sh)!3/()23()2/()2(,/)(,3123442123312211hffffchfffchffcfc???????????????????????????????????10224102310102114/)3()1()2)(1(,3/)2/5()1()1(2/)1(),1(1mmmmmmmmmmdssssSmmdsssSmsdsSmdsS8/)33)(()33(8343214321)4(ffffabffffhQNC?????????Weight [1 3 3 1]/8 for QNC(4)function w = NCweights(m) if m==2, w=[1 1]'/2;elseif m==3, w=[1 4 1]'/6;elseif m==4, w=[1 3 3 1]'/8;elseif m==5, w=[7 32 12 32 7]'/90;elseif m==6, w=[19 75 50 50 75 19]'/288;elseif m==7, w=[41 216 27 272 27 216 41]'/840;elseif m==8, w=[751 3577 1323 2989 2989 1323 3577 751]'/17280;elseif m==9, w=[989 5888 -928 10496 -4540 10496 -928 5888 … 989]'/28350;elseif m==10, w=[2857 15741 1080 19344 5778 5778 19344 1080… 15741 2857]'/89600; else w=[16067 106300 -48525 272400 -260550 427368 … -260550 272400 -48525 106300 16067]'/598752; end w(1:m)=w(m:-1:1)????????????????)()(],,)[()(111)(mmmiiimNCxfxfwwabfwabQ??function numI = QNC(fname,a,b,m)w = NCweights(m);x = linspace(a,b,m)';