{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "" -1 202 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "" -1 203 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "" -1 204 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 } {CSTYLE "" -1 205 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 } {CSTYLE "" -1 206 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 } {CSTYLE "" -1 207 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 } {CSTYLE "" -1 208 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 } {CSTYLE "" -1 209 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 } {CSTYLE "" -1 210 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 } {CSTYLE "" -1 211 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 } {CSTYLE "" -1 212 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 } {CSTYLE "" -1 213 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 } {CSTYLE "" -1 214 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 } {CSTYLE "" -1 215 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 } {CSTYLE "" -1 218 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 } {CSTYLE "" -1 219 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 } {CSTYLE "" -1 220 "Courier" 1 14 255 0 0 1 2 1 2 2 1 2 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 2 0 2 0 2 2 0 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 2 0 2 0 2 2 0 1 } {PSTYLE "Normal258" -1 203 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 2 0 2 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 48 "One Dimensional Closed Ad aptive Trapezoidal Rule" }}{PARA 19 "" 0 "" {TEXT -1 12 "Wei-Chi Yang " }}{PARA 203 "" 0 "" {TEXT -1 23 "e-mail: wyang@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 regular matrix to approximate \+ the integral of f over [0,1]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 202 30 "f:=proc(x) 1/sqrt(1-x^2) end;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plot(f(x),x=-1..0);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 203 25 "evalf(int(f(x),x=-1..0)); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 341 "(I have no idea what this va lue is) We shall define the regular matrices amk and bmk. The symbols \+ \"right\" and \"left\" corresopnd to right end evaluation point and le ft end evaluation point with repect to amk. Similarly, the symbols \"R ight\" and \"Left\" corresopnd to right end evaluation point and left \+ end evaluation point with repect to bmk. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 204 49 "amk:=proc(a,b,m,k1) (2*(b-a)*(k1))/(m*(m+1)) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "amk(0,1,4,1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 205 59 "bmk:=proc(a,b,m,k1) (6*(b-a)*(k1^ 2))/(m*(m+1)*(2*m+1)) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 206 56 "right:=proc(a,b,j,k1,m) a+sum(amk(a,b,m,j),j=1..k1) end;" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 207 56 "Right:=proc(a,b,j,k1,m) a+sum(bmk(a,b,m,j),j=1..k1) end;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 208 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 1 209 58 "Left:=proc(a,b,j,k1,m) a+sum(bmk(a,b,m,j),j=0..k1- 1) end;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Here is our first adap tive quadrature with the unformly regular matrix amk." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 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 0 "" 0 "" {TEXT -1 92 "Next we define the richardson extrapolation quadr ature, \"richard\", by starting with \"try0\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 211 18 "try0:=proc(a,b,m) " }}{PARA 0 "> " 0 "" {MPLTEXT 1 212 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 1 213 100 "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 0 "> " 0 "" {MPLTEXT 1 214 42 "fac:=n->if n<2 then 1 else n*fac(n-1) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 215 67 "FAC:= proc(n::posint);\nif n<2 then 1 else n*fac(n-1) end;\n end;" } {MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 218 7 "FAC( 5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 219 121 "Romberg:=proc(a,b ,m,n) if n<2 then 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 0 28 "evalf(Romberg(-1,0,400,20));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "evalf(Romberg(-1,0,430,20));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "evalf(Romberg(-1,0,440,20));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "evalf(Romberg(-1,0,450,20));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "evalf(Romberg(-1,0,10000,4)) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "Here \+ is another quadrature, \"Try\", using bmk as the uniformly regular mat rix. This is to compare two quadratures \"try\" and \"Try\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 220 130 "Try:=p roc(a,b,m) bmk(a,b,m,1)*f(Right(a,b,j,1,m))/2+sum(bmk(a,b,m,k1)*((f(Ri ght(a,b,j,k1,m))+f(Left(a,b,j,k1,m)))/2),k1=2..m) end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "Romberg2:=proc(a,b,m,n) if n<2 the n Try(a,b,m) else Try(a,b,m-1)+(Try(a,b,m-1)-Try(a,b,(m-1)/2))/(4^(n-1 )-1) end; end;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "evalf(Rom berg2(-1,0,400,20));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eva lf(Romberg2(-1,0,430,20));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "evalf(Romberg2(-1,0,440,20));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "evalf(Romberg2(-1,0,450,20));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 30 "evalf(Romberg2(-1,0,10000,4));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "Remark: By comparing with Romberg above, \+ Romberg2 here is better when we use bnk." }}}}{MARK "32" 0 }{VIEWOPTS 1 1 0 3 2 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }