{VERSION 5 0 "IBM INTEL LINUX" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Out put" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 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 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 27 "Kriittiset pisteet, min/m ax" }}{PARA 19 "" 0 "" {TEXT -1 7 "12.3.02" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart:" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, th e name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "with(linalg): with(LinearAlgebra): with(plots):" }} {PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra ce have been redefined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 64 "Warning, the assigned name GramSchmidt now has a global binding\n " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "setoptions3d(axes=boxed ,orientation=[-30,50]):" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 4 "Esim" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f:=x*y*exp(-(x^2+y^2)/2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG*(%\"xG\"\"\"%\"yGF'-%$expG6 #,&*$)F&\"\"#F'#!\"\"F/*&#F'F/F'*$)F(F/F'F'F1F'" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 28 "f1:=diff(f,x);f2:=diff(f,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G,&*&%\"yG\"\"\"-%$expG6#,&*$)%\"xG\"\"#F(#!\" \"F0*&#F(F0F(*$)F'F0F(F(F2F(F(*(F.F(F'F(F)F(F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G,&*&%\"xG\"\"\"-%$expG6#,&*$)F'\"\"#F(#!\"\"F/*&# F(F/F(*$)%\"yGF/F(F(F1F(F(*(F'F(F5F(F)F(F1" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 16 "f11:=diff(f1,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$f11G,&*(%\"xG\"\"\"%\"yGF(-%$expG6#,&*$)F'\"\"#F(#!\"\"F0*&#F(F0F( *$)F)F0F(F(F2F(!\"$*()F'\"\"$F(F)F(F*F(F(" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "f12:=diff(f1,y): f12:=simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$f12G,*-%$expG6#,&*$)%\"xG\"\"#\"\"\"#!\"\"F-*&# F.F-F.*$)%\"yGF-F.F.F0F.*&F4F.F&F.F0*&F+F.F&F.F0*(F+F.F4F.F&F.F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,-%$expG6#,&*$)%\"xG\"\"#\"\"\"#!\"\"F+*&#F,F+F,*$) %\"yGF+F,F,F.F,,&F3F,F,F.F,,&F3F,F,F,F,,&F*F,F,F.F,,&F*F,F,F,F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f22:=diff(f2,y);factor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$f22G,&*(%\"xG\"\"\"%\"yGF(-%$expG 6#,&*$)F'\"\"#F(#!\"\"F0*&#F(F0F(*$)F)F0F(F(F2F(!\"$*(F'F()F)\"\"$F(F* F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#**%\"xG\"\"\"%\"yGF%-%$expG6#, &*$)F$\"\"#F%#!\"\"F-*&#F%F-F%*$)F&F-F%F%F/F%,&!\"$F%F2F%F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(\{f1=0,f2=0\},\{x,y\}) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'<$/%\"xG\"\"!/%\"yGF&<$/F%\"\"\"/ F(F+<$F*/F(!\"\"<$/F%F/F,<$F1F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "H:=<|>:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "H00:=subs(x=0,y=0,H);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$H00G-%'RTABLEG6$\"*)[%fO\"-%'MATRIXG6#7$7$\"\"!\"\"\"7$F/F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Eigenvalues(H00);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"*kZlM\"-%'MATRIXG6#7$7# \"\"\"7#!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "H11:=subs (x=1,y=1,H),piste=[1,1];Hm11:=subs(x=-1,y=1,H),piste=[-1,1];Hm1m1:=sub s(x=-1,y=-1,H),piste=[-1,-1];H1m1:=subs(x=1,y=-1,H),piste=[1,-1];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$H11G6$-%'RTABLEG6$\"*g`$\\8-%'MATRI XG6#7$7$,$-%$expG6#!\"\"!\"#\"\"!7$F5F//%&pisteG7$\"\"\"F:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Hm11G6$-%'RTABLEG6$\"*WSPN\"-%'MATRIXG6#7 $7$,$-%$expG6#!\"\"\"\"#\"\"!7$F5F//%&pisteG7$F3\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&Hm1m1G6$-%'RTABLEG6$\"*S-JP\"-%'MATRIXG6#7$7$,$ -%$expG6#!\"\"!\"#\"\"!7$F5F//%&pisteG7$F3F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%H1m1G6$-%'RTABLEG6$\"*'Rau8-%'MATRIXG6#7$7$,$-%$expG 6#!\"\"\"\"#\"\"!7$F5F//%&pisteG7$\"\"\"F3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Pisteet (1,1) ja (-1,-1) ovat max (neg. def.), pisteet ( 1,-1) ja (-1,1) ovat min (pos. def.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot3d(f,x=-2..2,y=-2..2);" }}{PARA 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0 {PARA 3 "" 0 "" {TEXT -1 33 " KRP-yht\344l\366n ratkaisu numeerisesti" }}{PARA 0 "" 0 "" {TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "Todellisuudessa KRP-yht\344l\366s ysteemin ratkaisu onnistuu vain harvoin analyyttisesti. Numeerisen rat kaisijan " }{TEXT 0 6 "fsolve" }{TEXT -1 117 " k\344ytt\366 vaatii alk uarvojen tunnustelemista. 2 muuttujan tapauksessa piirtely on suureksi avuksi. T\344ss\344 v\344h\344n vihjett\344" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "#f:=(x,y)-> ...;g:=Vector(grad(f(x,y),[x,y]));" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "H:=Matrix(hessian(f(x,y),[x ,y])):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "KRPyht:=g[1]=0,g[ 2]=0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "display(implicitpl ot(g[1]=0,x=0.5..2,y=0.5..2),implicitplot(g[2]=0,x=0.5..2,y=0.5..2,col or=blue)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 71 "Kannattaa valita hi irell\344 alkuarvoja, jotka voi antaa fsolvelle tyylin:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "KRPa:=fsolve(\{KRPyht\},\{x=xa,y=ya \});" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "4 14 0 0" 23 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }{RTABLE_HANDLES 136594488 134654764 134935360 135374044 137310240 137454396 137384468 137382684 137293984 137294064 }{RTABLE M7R0 I6RTABLE_SAVE/136594488X,%)anythingG6"6"[gl!"%!!!#%"#"#""!-%$expG6#F'F(F'6" } {RTABLE M7R0 I6RTABLE_SAVE/134654764X*%)anythingG6"6"[gl!#%!!!"#"#"""!""6" } {RTABLE M7R0 I6RTABLE_SAVE/134935360X,%)anythingG6"6"[gl!"%!!!#%"#"#,$-%$expG6#!""!"#""!F-F' 6" } {RTABLE M7R0 I6RTABLE_SAVE/135374044X,%)anythingG6"6"[gl!"%!!!#%"#"#,$-%$expG6#!""""#""!F-F' 6" } {RTABLE M7R0 I6RTABLE_SAVE/137310240X,%)anythingG6"6"[gl!"%!!!#%"#"#,$-%$expG6#!""!"#""!F-F' 6" } {RTABLE M7R0 I6RTABLE_SAVE/137454396X,%)anythingG6"6"[gl!"%!!!#%"#"#,$-%$expG6#!""""#""!F-F' 6" } {RTABLE M7R0 I6RTABLE_SAVE/137384468X,%)anythingG6"6"[gl!"%!!!#*"$"$,$%"yG""#,$%"xGF)""!F*,$ %"zGF)F'F,F'F)6" } {RTABLE M7R0 I6RTABLE_SAVE/137382684X,%)anythingG6"6"[gl!"%!!!#*"$"$""#F'""!F'!""F'F(F'F'6" } {RTABLE M7R0 I6RTABLE_SAVE/137293984X*%)anythingG6"6"[gl!#%!!!"$"$""#,&#"""F'F**$"#TF)F),&F) F*F+#!""F'6" } {RTABLE M7R0 I6RTABLE_SAVE/137294064X*%)anythingG6"6"[gl!#%!!!"$"$$""#""!$"+=@c,P!"*$!+=@c,F F,6" }