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{SECT 0 {EXCHG {PARA 210 "" 0 "" {TEXT 215 48 "One Dimensional Closed \+
Adaptive Trapezoidal Rule" }}{PARA 211 "" 0 "" {TEXT 216 12 "Wei-Chi Y
ang" }}{PARA 212 "" 0 "" {TEXT 217 23 "e-mail: wyang@runet.edu" }}
{PARA 212 "" 0 "" {TEXT 217 32 "URL: http://www.runet.edu/~wyang" }}
{PARA 213 "" 0 "" {TEXT 218 0 "" }}{PARA 213 "" 0 "" {TEXT 218 117 "We
 would like to use an adaptive trapezoidal rule with a regular matrix \+
to approximate the integral of  f over [0,1]." }}{PARA 213 "" 0 "" 
{TEXT 218 0 "" }}{PARA 214 "> " 0 "" {MPLTEXT 1 219 26 "f:=proc(x)  1/
sqrt(x) end;" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 220 18 "plot(f(
x),x=0..1);" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 219 24 "evalf(in
t(f(x),x=0..1));" }}}{EXCHG {PARA 213 "" 0 "" {TEXT 218 305 "We shall \+
define the regular matrices amk and bmk. The symbols \"right\" and \"l
eft\" corresopnd to right end evaluation point and left end evaluation
 point with repect to amk. Similarly, the symbols \"Right\" and \"Left
\" corresopnd to right end evaluation point and left end evaluation po
int with repect to bmk. " }}{PARA 214 "> " 0 "" {MPLTEXT 1 219 49 "amk
:=proc(a,b,m,k1) (2*(b-a)*(k1))/(m*(m+1)) end;" }}}{EXCHG {PARA 216 ">
 " 0 "" {MPLTEXT 1 220 13 "amk(0,1,4,1);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 65 "A:=matrix(9,9, proc(i,j) if j>i then 0 else amk(0,1,i
,j) fi end);" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 219 59 "bmk:=pr
oc(a,b,m,k1) (6*(b-a)*(k1^2))/(m*(m+1)*(2*m+1)) end;" }}}{EXCHG {PARA 
214 "> " 0 "" {MPLTEXT 1 219 56 "right:=proc(a,b,j,k1,m) a+sum(amk(a,b
,m,j),j=1..k1) end;" }}}{EXCHG {PARA 213 "" 0 "" {TEXT 218 0 "" }}
{PARA 214 "> " 0 "" {MPLTEXT 1 219 56 "Right:=proc(a,b,j,k1,m) a+sum(b
mk(a,b,m,j),j=1..k1) end;" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 
219 58 "left:=proc(a,b,j,k1,m)  a+sum(amk(a,b,m,j),j=0..k1-1) end;" }}
}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 219 58 "Left:=proc(a,b,j,k1,m) \+
 a+sum(bmk(a,b,m,j),j=0..k1-1) end;" }}}{EXCHG {PARA 213 "" 0 "" 
{TEXT 218 75 "Here is our first adaptive quadrature with the unformly \+
regular matrix amk." }}{PARA 213 "" 0 "" {TEXT 218 0 "" }}{PARA 214 ">
 " 0 "" {MPLTEXT 1 219 133 "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 218 92 "Next \+
we define the richardson extrapolation quadrature, \"richard\",  by st
arting with \"try0\"." }}{PARA 213 "" 0 "" {TEXT 218 0 "" }}{PARA 214 
"> " 0 "" {MPLTEXT 1 219 18 "try0:=proc(a,b,m) " }}{PARA 214 "> " 0 "
" {MPLTEXT 1 219 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 214 "> " 0 "" {MPLTEXT 
1 219 100 "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 214 "> " 0 
"" {MPLTEXT 1 219 0 "" }}}{EXCHG {PARA 213 "" 0 "" {TEXT 218 129 "Here
 is another quadrature, \"Try\", using bmk as the uniformly regular ma
trix. This is to compare two quadratures \"try\" and \"Try\"." }}
{PARA 213 "" 0 "" {TEXT 218 0 "" }}{PARA 214 "> " 0 "" {MPLTEXT 1 219 
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 214 "> " 0 "" {MPLTEXT 1 219 21 "evalf(try1(0,1,300));" }
}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 219 21 "evalf(try1(0,1,400));
" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 219 21 "evalf(try1(0,1,430)
);" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 219 21 "evalf(try1(0,1,53
0));" }}}{EXCHG {PARA 214 "> " 0 "" {MPLTEXT 1 219 0 "" }}}{PARA 224 "
" 0 "" {TEXT 204 0 "" }}{PARA 224 "" 0 "" {TEXT 204 0 "" }}{PARA 225 "
" 0 "" {TEXT 229 0 "" }}{PARA 226 "" 0 "" {TEXT -1 0 "" }}}{MARK "25" 
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