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Trapezoidal Rule" }}{PARA 215 "" 0 "" {TEXT 214 12 "Wei-Chi \+ Yang" }}{PARA 214 "" 0 "" {TEXT 215 25 "e-mail: wyang@radford.edu" }} {PARA 214 "" 0 "" {TEXT 215 34 "URL: http://www.radford.edu/~wyang" }} {PARA 213 "" 0 "" {TEXT 209 0 "" }}{PARA 213 "" 0 "" {TEXT 209 179 "We would like to use an adaptive trapezoidal rule with a regular matrix \+ to approximate the integral of a function f, which is monotone and has a singularity at the left end point." }}{PARA 212 "> " 0 "" {MPLTEXT 1 210 26 "f:=proc(x) 1/sqrt(x) end;" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 211 18 "plot(f(x),x=0..1);" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 210 24 "evalf(int(f(x),x=0..1));" }}}{EXCHG {PARA 213 "" 0 "" {TEXT 209 305 "We shall define the regular matrices amk and bmk. Th e symbols \"right\" and \"left\" corresopnd to right end evaluation po int and left end evaluation point with repect to amk. Similarly, the s ymbols \"Right\" and \"Left\" corresopnd to right end evaluation point and left end evaluation point with repect to bmk. " }}{PARA 212 "> " 0 "" {MPLTEXT 1 210 49 "amk:=proc(a,b,m,k1) (2*(b-a)*(k1))/(m*(m+1)) e nd;" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 211 13 "amk(0,1,4,1);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "A:=matrix(9,9, proc(i,j) i f j>i then 0 else amk(0,1,i,j) fi end);" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 210 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 65 "B:=matrix(9,9, proc(i,j) if j>i then 0 else bmk(0,1,i,j) fi end);" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 210 56 "right:=proc(a,b,j,k 1,m) a+sum(amk(a,b,m,j),j=1..k1) end;" }}}{EXCHG {PARA 213 "" 0 "" {TEXT 209 0 "" }}{PARA 212 "> " 0 "" {MPLTEXT 1 210 56 "Right:=proc(a, b,j,k1,m) a+sum(bmk(a,b,m,j),j=1..k1) end;" }}}{EXCHG {PARA 212 "> " 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 212 "> " 0 "" {MPLTEXT 1 210 58 "Left:=pr oc(a,b,j,k1,m) a+sum(bmk(a,b,m,j),j=0..k1-1) end;" }}}{EXCHG {PARA 213 "" 0 "" {TEXT 209 18 "Here is our first " }{TEXT 222 6 "closed" } {TEXT 209 58 " adaptive quadrature with the unformly regular matrix am k." }}{PARA 213 "" 0 "" {TEXT 209 0 "" }}{PARA 212 "> " 0 "" {MPLTEXT 1 210 133 "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 213 "" 0 "" {TEXT 209 93 "#Next we define the ri chardson extrapolation quadrature, \"richard\", by starting with \"tr y0\"." }}{PARA 213 "" 0 "" {TEXT 209 0 "" }}{PARA 212 "> " 0 "" {MPLTEXT 1 210 19 "#try0:=proc(a,b,m) " }}{PARA 212 "> " 0 "" {MPLTEXT 1 210 79 "#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 212 "> " 0 "" {MPLTEXT 1 210 101 "#richard:=proc(a,b,m) (1/2)*amk(a,b,m,1)*f(right(a,b,j,1,m))+(1/3 )*(4*try0(a,b,m)-try0(a,b,m/2)) end;" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 210 0 "" }}}{EXCHG {PARA 213 "" 0 "" {TEXT 209 136 "Here is another closed quadrature, \"Try\", using bmk as the uniformly regula r matrix. This is to compare two quadratures \"try\" and \"Try\"." }} {PARA 213 "" 0 "" {TEXT 209 0 "" }}{PARA 212 "> " 0 "" {MPLTEXT 1 210 128 "Try:=proc(a,b,m) bmk(a,b,m,1)*f(Right(a,b,j,1,m))+sum(bmk(a,b,m,k 1)*((f(Right(a,b,j,k1,m))+f(Left(a,b,j,k1,m)))/2),k1=2..m) end;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 203 120 "We compare two closed quadtratur e try1 and Try as follows, you need to determine which one gives a bet ter approximation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digit s:=20;" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 210 41 "evalf(try1(0, 1,300));evalf(Try(0,1,300));" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 210 41 "evalf(try1(0,1,400));evalf(Try(0,1,400));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 203 0 "" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 210 41 "evalf(try1(0,1,430));evalf(Try(0,1,430));" }}}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 210 41 "evalf(try1(0,1,530));evalf(Try(0,1,530));" } }}{EXCHG {PARA 212 "> " 0 "" {MPLTEXT 1 210 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 203 79 "Do you know which one gives the best estimate for t he integration in this case?" }}}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "Typesetting:-mi(\"`?`\", italic = \"true\", mathvariant = \"italic \");" }}}{PARA 201 "" 0 "" {TEXT 212 0 "" }}{PARA 201 "" 0 "" {TEXT 212 0 "" }}{PARA 200 "" 0 "" {TEXT 219 0 "" }}{PARA 202 "" 0 "" {TEXT 223 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }