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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "s:=proc(x,y);\n  a:=
x+y;\n  RETURN(a);\nend;" }}{PARA 7 "" 1 "" {TEXT -1 41 "Warning, `a` \+
is implicitly declared local" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG
R6$%\"xG%\"yG6#%\"aG6\"F+C$>8$,&9$\"\"\"9%F1-%'RETURNG6#F.F+F+F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "s(1,2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "a;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\"aG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "s:=proc(x,y)\n  local a;\n  a:=x+y;\n  RETURN(a);\nen
d;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sGR6$%\"xG%\"yG6#%\"aG6\"F+C
$>8$,&9$\"\"\"9%F1-%'RETURNG6#F.F+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 80 "s2:=proc(x,y,z,t)\n  local a,b,c;\n  a:=x+y;\n  b:=a+
z;\n  c:=b+t;\n  RETURN(c);\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>
%#s2GR6&%\"xG%\"yG%\"zG%\"tG6%%\"aG%\"bG%\"cG6\"F/C&>8$,&9$\"\"\"9%F5>
8%,&F2F59&F5>8&,&F8F59'F5-%'RETURNG6#F<F/F/F/" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 13 "s2(2,5,-1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "a;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%\"aG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
81 "s2:=proc(x,y,z,t)\n  global a,b,c;\n  a:=x+y;\n  b:=a+z;\n  c:=b+t
;\n  RETURN(c);\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#s2GR6&%\"x
G%\"yG%\"zG%\"tG6\"F+F+C&>%\"aG,&9$\"\"\"9%F1>%\"bG,&F.F19&F1>%\"cG,&F
4F19'F1-%'RETURNG6#F8F+6%F.F4F8F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "s2(2,5,-1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"
)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "a;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "b;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2 "c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "s3:=proc(x,y,z,t)\n  global \+
a,b,c;\n  a:=s(x,y);\n  b:=s(a,z);\n  c:=s(b,t);\n  RETURN(c);\nend;" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#s3GR6&%\"xG%\"yG%\"zG%\"tG6\"F+F+
C&>%\"aG-%\"sG6$9$9%>%\"bG-F06$F.9&>%\"cG-F06$F59'-%'RETURNG6#F:F+6%F.
F5F:F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "s3(2,5,-1,2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2 "a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "b;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "c;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\")" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 52 "myproc:=proc(a);\n  b:=3;\n  r:=a*b;\n  RETURN(r);\ne
nd;" }}{PARA 7 "" 1 "" {TEXT -1 41 "Warning, `b` is implicitly declare
d local" }}{PARA 7 "" 1 "" {TEXT -1 41 "Warning, `r` is implicitly dec
lared local" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'myprocGR6#%\"aG6$%\"
bG%\"rG6\"F+C%>8$\"\"$>8%*&9$\"\"\"F.F4-%'RETURNG6#F1F+F+F+" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "myproc(4);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#\"#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "m
yproc2:=proc(a);\n  r:=a*b;\n  b:=3;\n  RETURN(r);\nend;" }}{PARA 7 "
" 1 "" {TEXT -1 41 "Warning, `r` is implicitly declared local" }}
{PARA 7 "" 1 "" {TEXT -1 41 "Warning, `b` is implicitly declared local
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(myproc2GR6#%\"aG6$%\"rG%\"bG6\"
F+C%>8$*&9$\"\"\"8%F1>F2\"\"$-%'RETURNG6#F.F+F+F+" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 11 "myproc2(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#,$%\"bG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "b:='b';" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bGF$" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 7 "r:=4*b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG,$%
\"bG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "b:=3;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2 "r;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#7" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 167 "The commands above are examples of how t
o use local and global variables and how to deal .\n\nWe now continue \+
our quest for a procedure that finds the min of a function\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 245 "gradstep:=proc(f,pt,epsilon
)\n  local x1,y1,gradx,grady,x2,y2;\n  x1:=pt[1];\n  y1:=pt[2];\n  gra
dx:=eval(diff(f(x,y),x),\{x=x1,y=y1\});\n  grady:=eval(diff(f(x,y),y),
\{x=x1,y=y1\});\n  x2:=x1-epsilon*gradx;\n  y2:=y1-epsilon*grady;\n  R
ETURN([x2,y2]);\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)gradstepGR
6%%\"fG%#ptG%(epsilonG6(%#x1G%#y1G%&gradxG%&gradyG%#x2G%#y2G6\"F1C)>8$
&9%6#\"\"\">8%&F66#\"\"#>8&-%%evalG6$-%%diffG6$-9$6$%\"xG%\"yGFI<$/FIF
4/FJF:>8'-FA6$-FD6$FFFJFK>8(,&F4F8*&9&F8F?F8!\"\">8),&F:F8*&FX\"\"\"FO
F8FY-%'RETURNG6#7$FUFenF1F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 261 "findmin:=proc(f,guess,epsilon)\n  local npt,nv,opt,ov,i;\n  npt
:=guess;\n  nv:=f(npt[1],npt[2]);\n  ov:=nv+1;\n  for i from 1 to 200 \+
while (nv<ov) do\n    opt:=npt;\n    ov:=nv;\n    npt:=gradstep(f,opt,
epsilon);\n    nv:=f(npt[1],npt[2]);\n  od;\n  RETURN(opt,ov,i);\nend;
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(findminGR6%%\"fG%&guessG%(epsil
onG6'%$nptG%#nvG%$optG%#ovG%\"iG6\"F0C'>8$9%>8%-9$6$&F36#\"\"\"&F36#\"
\"#>8',&F6F<F<F<?(8(F<F<\"$+#2F6FAC&>8&F3>FAF6>F3-%)gradstepG6%F8FI9&>
F6F7-%'RETURNG6%FIFAFDF0F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 287 "gradstep2:=proc(f,pt,epsilon)\n  local x1,y1,gradx,grady,x2,y2,
n;\n  x1:=pt[1];\n  y1:=pt[2];\n  gradx:=eval(diff(f(x,y),x),\{x=x1,y=
y1\});\n  grady:=eval(diff(f(x,y),y),\{x=x1,y=y1\});\n  n:=evalf(sqrt(
gradx^2+grady^2));\n  x2:=x1-epsilon*gradx/n;\n  y2:=y1-epsilon*grady/
n;\n  RETURN([x2,y2]);\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*gra
dstep2GR6%%\"fG%#ptG%(epsilonG6\"F*F*C*>%#x1G&9%6#\"\"\">%#y1G&F/6#\"
\"#>%&gradxG-%%evalG6$-%%diffG6$-9$6$%\"xG%\"yGFB<$/FBF-/FCF3>%&gradyG
-F:6$-F=6$F?FCFD>%\"nG-%&evalfG6#-%%sqrtG6#,&*$)F8F6\"\"\"F1*$)FHF6FXF
1>%#x2G,&F-F1*&*&9&F1F8F1FXFN!\"\"!\"\">%#y2G,&F3F1*&*&FjnFXFHF1FXFNF[
oF\\o-%'RETURNG6#7$FfnF^oF*6)F-F3F8FHFfnF^oFNF*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 267 "findmin2:=proc(f,guess,epsilon)\n  local npt,
nv,opt,ov,i;\n  npt:=guess;\n  nv:=f(npt[1],npt[2]);\n  ov:=nv+1;\n  f
or i from 1 to 10000 while (nv<ov) do\n    opt:=npt;\n    ov:=nv;\n   \+
 npt:=gradstep2(f,opt,epsilon);\n    nv:=f(npt[1],npt[2]);\n  od;\n  R
ETURN(opt,ov,i-1);\nend;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)findmin
2GR6%%\"fG%&guessG%(epsilonG6'%$nptG%#nvG%$optG%#ovG%\"iG6\"F0C'>8$9%>
8%-9$6$&F36#\"\"\"&F36#\"\"#>8',&F6F<F<F<?(8(F<F<\"&++\"2F6FAC&>8&F3>F
AF6>F3-%*gradstep2G6%F8FI9&>F6F7-%'RETURNG6%FIFA,&FDF<!\"\"F<F0F0F0" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f:=(x,y)->x^2+y^2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6$%\"xG%\"yG6\"6$%)operatorG%&a
rrowGF),&*$)9$\"\"#\"\"\"\"\"\"*$)9%F1F2F3F)F)F)" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 21 "findmin2(f,[1,1],.1);" }}{PARA 8 "" 1 "" 
{TEXT -1 44 "Error, (in findmin2) cannot evaluate boolean" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "gradx;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%&gradxG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "trace(fin
dmin2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)findmin2G" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "findmin2(f,[1,1],.1);" }}{PARA 9 "
" 1 "" {TEXT -1 41 "\{--> enter findmin2, args = f, [1, 1], .1" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$nptG7$\"\"\"F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#nvG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ovG\"
\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$optG7$\"\"\"F&" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%#ovG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$nptG7$,&\"\"\"F'*$-%%sqrtG6#\"\"#\"\"\"$!+++++]!#6F&" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#nvG,$*$),&\"\"\"F)*$-%%sqrtG6#\"\"#\"\"\"$!++++
+]!#6F.F/F." }}{PARA 9 "" 1 "" {TEXT -1 67 "<-- ERROR in findmin2 (now
 at top level) = cannot evaluate boolean\}" }}{PARA 8 "" 1 "" {TEXT 
-1 44 "Error, (in findmin2) cannot evaluate boolean" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 21 "findmin2(f,[1,1],.1);" }}{PARA 8 "" 1 "" 
{TEXT -1 71 "Error, (in gradstep2) attempting to assign to `norm` whic
h is protected" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "findmin2(
f,[1,1],.1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7$$\"+b1005!#6F$$\"+QO
D??!#8\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "findmin2(f,[2
,1],1e-3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%7$$\"*V@-5'!#8$\"*H5,0$F
&$\"+Gue^Y!#=\"%PA" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{MARK "28 0 0" 37 }{VIEWOPTS 1 1 0 2 1 1805 }