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1 0 0 0 0 2 0 2 0 2 2 -1 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 73 "One Dimensional Adaptive \+ Trapezoidal Rule for highly osscilating function" }}{PARA 19 "" 0 "" {TEXT -1 12 "Wei-Chi Yang" }}{PARA 203 "" 0 "" {TEXT -1 23 "e-mail: wy ang@runet.edu" }}{PARA 203 "" 0 "" {TEXT -1 32 "URL: http://www.runet. edu/~wyang" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "We would like to use an adaptive trapezoidal rule with a regul ar matrix to approximate the integral of f over [0,1]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 202 27 "f:=proc(x) \+ sin(1/x)/x end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 203 5 "f(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 204 33 "trueint:=evalf(int(f(x) ,x=0..1));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 305 "We shall define th e regular matrices amk and bmk. The symbols \"right\" and \"left\" cor resopnd to right end evaluation point and left end evaluation point wi th repect to amk. Similarly, the symbols \"Right\" and \"Left\" corres opnd to right end evaluation point and left end evaluation point with \+ repect to bmk. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 205 49 "amk:=proc(a,b, m,k1) (2*(b-a)*(k1))/(m*(m+1)) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "simplify(sum(amk(a,b,m,k),k=1..m));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 82 " The command above is to demonstrate that the sum of all the subinterva ls is (b-a)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 206 59 "bmk:=proc(a,b,m,k1) (6*(b-a)*(k1^2))/(m*(m+1)*(2*m+1)) end;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "amk(0,1,4,1);amk(0,1,4,2); amk(0,1,4,3);amk(0,1,4,4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 207 56 "right:=proc(a,b,j,k1,m) a+sum(amk(a,b,m,j),j=1..k1) end;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "simplify(right(0,1,j,1,4)), simplify(right(0,1,j,2,4)),simplify(right(0,1,j,3,4)),simplify(right(0 ,1,j,4,4));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 208 63 "right2:=pr oc(a,b,j,k1,m) a+sum((1/2)*amk(a,b,m,j),j=1..k1) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "simplify(right2(0,1,j,1,4)),simpli fy(right2(0,1,j,2,4)),simplify(right2(0,1,j,3,4)),simplify(right2(0,1, j,4,4));" }}{PARA 206 "> " 0 "" {MPLTEXT 1 0 224 "simplify(right2(0,1, j,1,8)),simplify(right2(0,1,j,2,8)),simplify(right2(0,1,j,3,8)),simpli fy(right2(0,1,j,4,8)),simplify(right2(0,1,j,5,8)),simplify(right2(0,1, j,6,8)),simplify(right2(0,1,j,7,8)),simplify(right2(0,1,j,8,8));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 209 56 "Right:=proc(a,b,j,k1,m) a+sum(bmk(a,b,m,j),j=1..k1) end;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 210 58 "left:=proc(a,b,j,k1,m) a+ sum(amk(a,b,m,j),j=0..k1-1) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 211 65 "left2:=proc(a,b,j,k1,m) a+sum((1/2)*amk(a,b,m,j),j =0..k1-1) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 212 58 "Left:=p roc(a,b,j,k1,m) a+sum(bmk(a,b,m,j),j=0..k1-1) end;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "'try1' below is the closed adaptive quadrature \+ with the unformly regular matrix amk." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 213 133 "try1:=proc(a,b,m) (amk(a,b,m ,1)*f(right(a,b,j,1,m)))/2+sum(amk(a,b,m,k1)*((f(right(a,b,j,k1,m))+f( left(a,b,j,k1,m)))/2),k1=2..m) end;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "'try0' is the open quadrature. We shall define the richardson \+ extrapolation quadrature, \"richard\", by starting with \"try0\"." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 214 18 "t ry0:=proc(a,b,m) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 215 78 "sum(amk(a,b, m,k1)*((f(right(a,b,j,k1,m))+f(left(a,b,j,k1,m)))/2),k1=2..m) end;" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 216 100 "richard:=proc(a,b,m) (1/ 2)*amk(a,b,m,1)*f(right(a,b,j,1,m))+(1/3)*(4*try1(a,b,m)-try1(a,b,m/2) ) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 217 104 "try2:=proc(a,b ,m) sum((1/2)*amk(a,b,m,k1)*((f(right2(a,b,j,k1,m))+f(left2(a,b,j,k1,m )))/2),k1=2..m) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{PARA 11 "" 0 "" {TEXT 20 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 218 118 "Romberg1:=proc(a,b,m,n) if n<2 then try1(a,b,m) else try1(a,b ,m-1)+(try1(a,b,m-1)-try2(a,b,m-1))/(4^(n-1)-1) end; end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 219 122 "Romberg2:=proc(a,b,m,n) if n<2 t hen try1(a,b,m) else try1(a,b,m-1)+(try1(a,b,m-1)-try1(a,b,(m-1)/2))/( 4^(n-1)-1) end; end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 220 13 "6 25*5;625*25;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Now we divide [0 ,1] into a union of subintervals [x(i+1), x(i)], where x(1)=0, x(i)=5^ (-(i-1)). Here we let " }}{PARA 0 "" 0 "" {TEXT -1 20 "I1=[x2,x1]=[1/5 ,1], " }}{PARA 0 "" 0 "" {TEXT -1 21 "I2=[x3,x2]=[1/25,1/5]" }}{PARA 0 "" 0 "" {TEXT -1 23 "I3=[x4,x3]=[1/125,1/25]" }}{PARA 0 "" 0 "" {TEXT -1 24 "I4=[x5,x4]=[1/625,1/125]" }}{PARA 0 "" 0 "" {TEXT -1 25 " I5=[x6,x5]=[1/3125,1/625]" }}{PARA 0 "" 0 "" {TEXT -1 27 "I6=[x7,x6]=[ 1/15625,1/3125]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 221 35 "A1:=ev alf(Romberg1(1/5,1,2000,16));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 222 38 "A2:=evalf(Romberg1(1/25, 1/5,2000,16));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 223 40 "A3:=eva lf(Romberg1(1/125,1/25,2000,16));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 224 41 "A4:=evalf(Romberg1(1/625,1/125,2500,16));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 225 42 "A5:=evalf(Romberg1(1/3125, 1/625,2500,16));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 226 44 "A6:=e valf(Romberg1(1/15625,1/3125,4500,16));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 227 30 "rombergans:=A1+A2+A3+A4+A5+A6;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 228 33 "errorromberg:=trueint-rombergans;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 229 35 "C1:=evalf(Romberg2(1/5,1,2 000,16));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 230 38 "C2:=evalf(Romberg2(1/25,1/5,2000,16));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 231 40 "C3:=evalf(Romberg2(1/125,1 /25,2000,16));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 232 41 "C4:=eva lf(Romberg2(1/625,1/125,2500,16));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 233 42 "C5:=evalf(Romberg2(1/3125,1/625,2500,16));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 234 44 "C6:=evalf(Romberg2(1/15625 ,1/3125,4500,16));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 235 31 "rom bergans2:=C1+C2+C3+C4+C5+C6;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 236 35 "errorromberg2:=trueint-rombergans2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 237 38 "C11:=evalf(Romberg2(1/5,1,2000,2000));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 238 41 "C22:=evalf(Romberg2(1/25,1/5,2000,2000));" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 239 43 "C33:=evalf(Romberg2(1/125,1/25,2000,2000)) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 240 44 "C44:=evalf(Romberg2( 1/625,1/125,2500,2500));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 241 45 "C55:=evalf(Romberg2(1/3125,1/625,2500,2500));" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 242 47 "C66:=evalf(Romberg2(1/15625,1/3125,4500,45 00));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 243 37 "rombergans3:=C11 +C22+C33+C44+C55+C66;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 244 35 " errorromberg3:=trueint-rombergans3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "Here we try the Richardson's." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 245 31 "B1:=evalf(richard(1/5,1,2000));" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 246 34 "B2: =evalf(richard(1/25,1/5,2000));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 247 36 "B3:=evalf(richard(1/125,1/25,2000));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 248 37 "B4:=evalf(richard(1/625,1/125,2500));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 249 38 "B5:=evalf(richard(1/3125,1 /625,2500));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 250 40 "B6:=evalf (richard(1/15625,1/3125,4500));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 251 30 "richardans:=B1+B2+B3+B4+B5+B6;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 252 33 "errorrichard:=trueint-richardans;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "C onclusion: For this problem, Romberg's is better than that of Richards on's." }}}{PARA 207 "" 0 "" {TEXT -1 0 "" }}{PARA 208 "" 0 "" {TEXT -1 0 "" }}{PARA 209 "" 0 "" {TEXT -1 0 "" }}{PARA 210 "" 0 "" {TEXT -1 0 "" }}{PARA 211 "" 0 "" {TEXT -1 0 "" }}{PARA 212 "" 0 "" {TEXT -1 0 "" }}{PARA 213 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }