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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 40 "Two Dimensional Adaptive \+
Open Quadrature" }}{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 88 "We would like to use a quadrature to find the double in
tegral of the following function." }}{PARA 0 "> " 0 "" {MPLTEXT 1 203 
31 "f:=proc(x,y) 1/(sqrt(x*y)) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"fGf*6$%\"xG%\"yG6\"F)F)*$-%%sqrtG6#*&9$\"\"\"9%F0!\"\"F)F)F)" }}}
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F`yFczF\\\\lF_]lFb^lFa_lFj`lF]blF`clFadlFbelF_flF\\glFgglF`hlFghl$\"+g
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6#F3" 1 2 5 1 10 1 2 1 1 2 2 1.000000 59.000000 -98.000000 1 0 "Curve \+
1" }}{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Now we defin
e two uniformly regular matrices." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 205 57 "ank:=proc(a,b,n,k) (6*(b-a)*(k^2))/(n*(n+1)*(2*n+1)
) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ankGf*6&%\"aG%\"bG%\"nG%
\"kG6\"F+F+,$*,,&9%\"\"\"9$!\"\"F09'\"\"#9&F2,&F5F0F0F0F2,&F5F4F0F0F2
\"\"'F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 206 57 "bnk:=proc(
c,d,n,k) (6*(d-c)*(k^2))/(n*(n+1)*(2*n+1)) end;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$bnkGf*6&%\"cG%\"dG%\"nG%\"kG6\"F+F+,$*,,&9%\"\"\"9$!
\"\"F09'\"\"#9&F2,&F5F0F0F0F2,&F5F4F0F0F2\"\"'F+F+F+" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 55 "We define the right end and left end evaluation
 points." }}{PARA 0 "> " 0 "" {MPLTEXT 1 207 54 "rx:=proc(a,b,j,k,n) a
 + sum(ank(a,b,n,j), j=1..k) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%#rxGf*6'%\"aG%\"bG%\"jG%\"kG%\"nG6\"F,F,,&9$\"\"\"-%$sumG6$-%$ankG6&F
.9%9(9&/F8;F/9'F/F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 208 
54 "lx:=proc(a,b,j,k,n) a+sum(ank(a,b,n,j), j=0..k-1) end;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#lxGf*6'%\"aG%\"bG%\"jG%\"kG%\"nG6\"F,F,,&
9$\"\"\"-%$sumG6$-%$ankG6&F.9%9(9&/F8;\"\"!,&9'F/!\"\"F/F/F,F,F," }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 209 51 "ry:=proc(c,d,j,l,n) c+sum(
bnk(c,d,n,j),j=1..l) end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ryGf*6
'%\"cG%\"dG%\"jG%\"lG%\"nG6\"F,F,,&9$\"\"\"-%$sumG6$-%$bnkG6&F.9%9(9&/
F8;F/9'F/F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 210 53 "ly:=pr
oc(c,d,j,l,n) c+sum(bnk(c,d,n,j),j=0..l-1) end;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#lyGf*6'%\"cG%\"dG%\"jG%\"lG%\"nG6\"F,F,,&9$\"\"\"-%$
sumG6$-%$bnkG6&F.9%9(9&/F8;\"\"!,&9'F/!\"\"F/F/F,F,F," }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 37 "Here is our open adaptive quadrature." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 211 218 "trap2:=proc(a,b,c,d,n) sum(ank(a
,b,n,k)*sum(bnk(c,d,n,l)*((f(rx(a,b,j,k,n),ry(c,d,j,l,n))+ f(lx(a,b,j,
k,n), ly(c,d,j,l,n))+f(lx(a,b,j,k,n), ry(c,d,j,l,n)) + f(rx(a,b,j,k,n)
, ly(c,d,j,l,n)))/4), l = 2..n), k =2..n) end;" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%&trap2Gf*6'%\"aG%\"bG%\"cG%\"dG%\"nG6\"F,F,-%$sumG6$*
&-%$ankG6&9$9%9(%\"kG\"\"\"-F.6$*&-%$bnkG6&9&9'F6%\"lGF8,*-%\"fG6$-%#r
xG6'F4F5%\"jGF7F6-%#ryG6'F?F@FIFAF6#F8\"\"%-FD6$-%#lxGFH-%#lyGFLFM-FD6
$FQFJFM-FD6$FFFSFMF8/FA;\"\"#F6F8/F7FZF,F,F," }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 212 25 "evalf(trap2(0,1,0,1,20));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+<j(R%R!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
213 25 "evalf(trap2(0,1,0,1,30));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$
\"+aeEsR!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 214 25 "evalf(tra
p2(0,1,0,1,80));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+(yC[*R!\"*" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 215 26 "evalf(trap2(0,1,0,1,100))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+IkW'*R!\"*" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 79 "**Don't try the following evalf(trap2(0,1,0,1,2
50)); it drained out the memory." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 216 26 "evalf(trap2(0,1,0,1,250));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+%H@#**R!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 61 "Now we define the Richardson Extrapol
ation in two dimensions." }}{PARA 0 "> " 0 "" {MPLTEXT 1 217 74 "richa
rd:=proc(a,b,c,d,n)(1/3)*(4*trap2(a,b,c,d,n)-trap2(a,b,c,d,n/2)) end;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(richardGf*6'%\"aG%\"bG%\"cG%\"d
G%\"nG6\"F,F,,&*&#\"\"%\"\"$\"\"\"-%&trap2G6'9$9%9&9'9(F2F2*&#F2F1F2-F
46'F6F7F8F9,$*&#F2\"\"#F2F:F2F2F2!\"\"F,F,F," }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 218 55 "evalf(richard(0,1,0,1,25)); evalf(richard(0,1,
0,1,30));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+h.sPT!\"*" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+2:!Q*R!\"*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 219 27 "evalf(richard(0,1,0,1,50));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+M\"Gt*R!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 220 28 "evalf(richard(0,1,0,1,100));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#$\"+A$*4**R!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Clearly, these results are much \+
faster than those from trap2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 29 "** Romberg Integration in 2D." }}{PARA 0 
"> " 0 "" {MPLTEXT 1 221 29 "f:=(x,y)->sin(1/(x*y))/(x*y);" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 222 192 "Romberg:=proc(a,b,c,d,m,n,kx,ky) if m<2 \+
or n<2 then trap2(a,b,c,d,m,n) else trap2(a,b,c,d,m-1,n-1)+(trap2(a,b,
c,d,m-1,n-1)-trap2(a,b,c,d,(m-1)/2,(n-1)/2))/((4^(kx-1)-1)*(4^(ky-1)-1
)) end; end;" }}{PARA 0 "" 0 "" {TEXT 223 16 "Another Example:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG
f*6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF)*(-%$sinG6#*&9$!\"\"9%F3\"\"\"
F2F3F4F3F)F)F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(RombergGf*6*%\"aG
%\"bG%\"cG%\"dG%\"mG%\"nG%#kxG%#kyG6\"F/F/@%529(\"\"#29)F4-%&trap2G6(9
$9%9&9'F3F6,&-F86(F:F;F<F=,&F3\"\"\"!\"\"FB,&F6FBFCFBFB*(,&F?FB-F86(F:
F;F<F=,&F3#FBF4#FCF4FB,&F6FJFKFBFCFB,&)\"\"%,&9*FBFCFBFBFCFBFC,&)FO,&9
+FBFCFBFBFCFBFCFBF/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 224 48 "A1:=evalf(Romberg(1/5
,1,1/5,1,200,200,200,200));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A1G$
\"+D*)Rt#*!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 225 54 "A2:=eval
f(Romberg(1/25,1/5,1/25,1/5,200,200,200,200));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#A2G$\"+n\"HAA$!#8" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 226 58 "A3:=evalf(Romberg(1/125,1/25,1/125,1/25,200,200,200
,200));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A3G$\"+&4)\\=`!#7" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 227 60 "A4:=evalf(Romberg(1/625,1/
125,1/625,1/125,200,200,200,200));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#A4G$!+0IhsA!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 228 62 "A5:=
evalf(Romberg(1/3125,1/625,1/3125,1/625,200,200,200,200));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#A5G$\"+se3yv!#7" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 229 66 "A6:=evalf(Romberg(1/15625,1/3125,1/15625,1/312
5,200,200,200,200));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A6G$!+6`:9X
!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "A6;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#$!+6`:9X!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
230 27 "romberg:=A1+A2+A3+A4+A5+A6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%(rombergG$!+\\&z![Z!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 231 
38 "evalf(int(int(f(x,y),x=0..1),y=0..1));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+?`D7v!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
232 47 "A11:=evalf(Romberg(1/5,1,1/5,1,200,200,20,20));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$A11G$\"+0!*Rt#*!#7" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 233 53 "A22:=evalf(Romberg(1/25,1/5,1/25,1/5,300,300,20,
20));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$A22G$\"+RtxjJ!#8" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 234 57 "A33:=evalf(Romberg(1/125,1
/25,1/125,1/25,400,400,20,20));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$
A33G$!+6Mo`=!#6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 235 59 "A44:=e
valf(Romberg(1/625,1/125,1/625,1/125,500,500,20,20));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$A44G$\"+0xk39!#6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 236 62 "A5:=evalf(Romberg(1/3125,1/625,1/3125,1/625,200,200
,200,200));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#A5G$\"+se3yv!#7" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 237 66 "A6:=evalf(Romberg(1/15625,
1/3125,1/15625,1/3125,200,200,200,200));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#A6G$!+6`:9X!#7" }}}{PARA 206 "" 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 
}