{VERSION 5 0 "IBM INTEL NT" "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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 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 14 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 "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 Output" -1 11 1 {CSTYLE " " -1 -1 "Times" 1 14 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 14 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 "Map le Plot" -1 13 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 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 " Courier" 1 14 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 "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 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 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 260 1 {CSTYLE "" -1 -1 "T imes" 1 14 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 "Normal" -1 261 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 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart:with(linalg) ,with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected n ames norm and trace have been redefined and unprotected\n" }}{PARA 7 " " 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 268 34 "Este programa resuelve la \+ ecuacion" }{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 1 "-" } {XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 11 "u(x,y)=fen " }{TEXT 256 0 "" }{TEXT -1 0 "" }{XPPEDIT 18 0 "Omega;" "6#%&OmegaG" }{TEXT -1 3 " y " }}{PARA 258 "" 0 "" {TEXT -1 17 "u(x,y)=g(x,y) en " } {XPPEDIT 18 0 "Gamma" "6#%&GammaG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 267 5 "donde" }{TEXT -1 2 " " }{TEXT 260 0 "" }{TEXT -1 0 "" } {XPPEDIT 18 0 "Omega;" "6#%&OmegaG" }{TEXT -1 2 " " }{TEXT 266 22 "es un dominio de tipo " }{TEXT 262 15 "[a, b]x[c, d]" }{TEXT 264 13 ". el sugundo " }{TEXT 261 1 "f" }{TEXT 265 78 " miembro es una constant e. El metodo usado es triangulos con tres nodos. (P1)" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 39 "Nx:=4:Ny:=4:N:=Nx*Ny:NE:=(Nx-1)*(Ny-1):" } {TEXT -1 49 "N es el numero de nodos, NE el de los elementos. " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a:=0.:b:=1:c:=0.:d:=1:" }{TEXT -1 3 "El " }{MPLTEXT 1 0 0 "" }{TEXT -1 38 "dominio es rectangulo [a, b]x [c, d] y " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "hx:=(b-a)/(Nx-1):hy:=( d-c)/(Ny-1):" }{TEXT -1 33 " El paso en x es hx y en y es hy." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "tc:=matrix(N,3,0):" }{TEXT -1 121 "tc es la tabla delas coordenadas,\030 la primera columna es e l numero del nodo, la segunda y la tercera son las coordenadas" }} {PARA 0 "" 0 "" {TEXT -1 17 " de los nodos\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 119 "Costruccion de la tabla de conecciones cn. La tabl a tcn es una tabla intermedia que ayuda a sistematizar la numeracion. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "for i to N do tc[i,1]:=i od:fo r i to N do tc[i,3]:=hx*(i-1 mod Nx) od:for i to N do tc[i,2]:=hy*(iqu o(i-1,Nx)) od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 261 "tcn:=mat rix(NE,5,0):cn:=matrix(2*NE,4,0):for i to NE do tcn[i,1]:=i:od: for i \+ to NE do if (i mod (Nx-1)=0) then entier:=-1: else entier:=(i mod (Nx- 1)):end if; tcn[i,2]:=Nx*iquo(i,Nx-1)+entier: tcn[i,3]:=tcn[i,2]+1:tcn [i,4]:=tcn[i,3]+Nx:tcn[i,5]:=tcn[i,4]-1:od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 186 "for i to NE do cn[2*i-1,1]:=2*i-1:cn[2*i,1]:=2*i:cn[ 2*i-1,2]:= tcn[i,2]:cn[2*i-1,3]:= tcn[i,3]:cn[2*i-1,4]:= tcn[i,4]:cn[2 *i,2]:= tcn[i,2]: cn[2*i,3]:= tcn[i,4]:cn[2*i,4]:= tcn[i,5]: od:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 609 "malla:=plot([\nseq(\n[\n[tc [cn[i,2],2],tc[cn[i,2],3]],\n[tc[cn[i,3],2],tc[cn[i,3],3]],\n[tc[cn[i, 4],2],tc[cn[i,4],3]],\n[tc[cn[i,2],2],tc[cn[i,2],3]]\n]\n,i=1..2*NE\n) \n]):\nnumelts:=seq([\ntextplot([(tc[cn[i,2],2]+tc[cn[i,3],2]+tc[cn[i, 4],2])/3,\n(tc[cn[i,2],3]+tc[cn[i,3],3]+tc[cn[i,4],3])/3,i],align=\{AB OVE,RIGHT\},color=\nblack)\n],i=1..2*NE):\nnumnodos:=seq([\ntextplot([ tc[i,2]+.1*hx,tc[i,3]+hy*.1,i],align=\{ABOVE,RIGHT\},color=\nred)\n],i =1..N):\nnodos:=seq([\ntextplot([tc[i,2],tc[i,3],o],color=\nred)\n],i= 1..N):\n\nleyenda:=seq([\ntextplot([[b*(1.5),d,\"numeros rojos nodos\" ],[1.5*b,.8*d,\"numeros negos elementos\"]])\n],i=1..N):\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "display(malla,numelts,numnodos,nodos,leyend a);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6bq-%'CURVE SG6$7&7$$\"\"!F)F(7$F($\"3z*****HLLLL$!#=7$F+F+F'-%'COLOURG6&%$RGBG$\" #5!\"\"F(F(-F$6$7&F'F.7$F+F(F'-F06&F2F(F3F(-F$6$7&F*7$F($\"3e*****fmmm m'F-7$F+F@F*-F06&F2F3F3F(-F$6$7&F*FBF.F*-F06&F2F(F(F3-F$6$7&F?7$F($\"2 +++!**********!#<7$F+FNF?-F06&F2F3F(F3-F$6$7&F?FQFBF?-F06&F2F(F3F3-F$6 $7&F9F.7$F@F+F9F/-F$6$7&F9Ffn7$F@F(F9F:-F$6$7&F.FB7$F@F@F.FC-F$6$7&F.F ^oFfnF.FH-F$6$7&FBFQ7$F@FNFBFR-F$6$7&FBFeoF^oFBFW-F$6$7&FjnFfn7$FNF+Fj 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jqF[rF`w-F`q6'7$FhwF]xF]tFjqF[rF`w-F`q6'7$FhwFbxFatFjqF[rF`w-F`q6'7$F] xF]wFetFjqF[rF`w-F`q6'7$F]xFhwFitFjqF[rF`w-F`q6'7$F]xF]xF]uFjqF[rF`w-F `q6'7$F]xFbxFauFjqF[rF`w-F`q6'7$FbxF]wFeuFjqF[rF`w-F`q6'7$FbxFhwFiuFjq F[rF`w-F`q6'7$FbxF]xF]vFjqF[rF`w-F`q6'7$FbxFbxFavFjqF[rF`w-F`q6%F'Q\"o FiqF`w-F`q6%7$F($F^wFeqF[[lF`w-F`q6%7$F($\"+mmmmmFeqF[[lF`w-F`q6%7$F($ \"+**********FeqF[[lF`w-F`q6%7$F_[lF(F[[lF`w-F`q6%7$F_[lF_[lF[[lF`w-F` q6%7$F_[lFc[lF[[lF`w-F`q6%7$F_[lFh[lF[[lF`w-F`q6%7$Fc[lF(F[[lF`w-F`q6% 7$Fc[lF_[lF[[lF`w-F`q6%7$Fc[lFc[lF[[lF`w-F`q6%7$Fc[lFh[lF[[lF`w-F`q6%7 $Fh[lF(F[[lF`w-F`q6%7$Fh[lF_[lF[[lF`w-F`q6%7$Fh[lFc[lF[[lF`w-F`q6%7$Fh [lFh[lF[[lF`w-F`q6$7$$\"#:F5$\"\"\"F)Q4numeros~rojos~nodosFiq-F`q6$7$F a^l$\"\")F5Q8numeros~negos~elementosFiqF^^lFf^lF^^lFf^lF^^lFf^lF^^lFf^ lF^^lFf^lF^^lFf^lF^^lFf^lF^^lFf^lF^^lFf^lF^^lFf^lF^^lFf^lF^^lFf^lF^^lF f^lF^^lFf^lF^^lFf^l-%+AXESLABELSG6$Q!FiqF__l-%%VIEWG6$%(DEFAULTGFc_l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Cur ve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15 " "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "C urve 22" "Curve 23" "Curve 24" "Curve 25" "Curve 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32" "Curve 33" "Curve 34 " "Curve 35" "Curve 36" "Curve 37" "Curve 38" "Curve 39" "Curve 40" "C urve 41" "Curve 42" "Curve 43" "Curve 44" "Curve 45" "Curve 46" "Curve 47" "Curve 48" "Curve 49" "Curve 50" "Curve 51" "Curve 52" "Curve 53 " "Curve 54" "Curve 55" "Curve 56" "Curve 57" "Curve 58" "Curve 59" "C urve 60" "Curve 61" "Curve 62" "Curve 63" "Curve 64" "Curve 65" "Curve 66" "Curve 67" "Curve 68" "Curve 69" "Curve 70" "Curve 71" "Curve 72 " "Curve 73" "Curve 74" "Curve 75" "Curve 76" "Curve 77" "Curve 78" "C urve 79" "Curve 80" "Curve 81" "Curve 82" "Curve 83" "Curve 84" "Curve 85" "Curve 86" "Curve 87" "Curve 88" "Curve 89" "Curve 90" "Curve 91 " "Curve 92" "Curve 93" "Curve 94" "Curve 95" "Curve 96" "Curve 97" "C urve 98" "Curve 99" "Curve 100" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 23 "La solucion exacta es T" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "T:=( x,y)->3*x**2+10*y**2-x*y-x+y;ff:=-diff(T(x,y),x$2)-diff(T(x,y),y$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TGf*6$%\"xG%\"yG6\"6$%)operatorG %&arrowGF),,*$)9$\"\"#\"\"\"\"\"$*&\"#5F2)9%F1F2F2*&F0F2F7F2!\"\"F0F9F 7F2F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ffG!#E" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 204 "La funcion que calcula la integral de una funcion lineal (afin) sobre un triangulo. Los xi y los yi son las coordenadas de los vertices del triangulo. k=1,2,3 indica a que nodo esta asociada la funcion." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 631 "intl inealtriang:=proc(x1,y1,x2,y2,x3,y3,k)\nlocal cof,d,s,dd;\ncof:=vector (3): \nd:=x1*y2-x1*y3-x2*y1+x2*y3+x3*y1-x3*y2; \nif (k<2) then cof[1]: =(y2-y3)/d:cof[2]:=(x3-x2)/d:cof[3]:=(x2*y3-x3*y2)/d:\nelif (k>2) then cof[1]:=(y1-y2)/d:cof[2]:=(x2-x1)/d:cof[3]:=(x1*y2-x2*y1)/d:\nelse\nc of[1]:=(y3-y1)/d:cof[2]:=(x1-x3)/d:cof[3]:=(x3*y1-x1*y3)/d:\nend if:\n dd:=abs((x2-x1)*(y3-y1)-(x3-x1)*(y2-y1)): \ns := 1/6*cof[2]*y2+1/6* cof[2]*y1+1/6*cof[1]*x1+1/6*cof[1]*x2+1/6*cof[1]*x3+1/6*cof[2]*y3+1/2* cof[3]: \+ s:= dd*s:s;" }{TEXT -1 0 "" }{MPLTEXT 1 0 10 "\nend proc:" }}}{EXCHG {PARA 261 "" 0 "" {TEXT 259 51 "La construccion de la matriz y del seg undo miembro." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 663 "A:=matrix(N,N,0): \nF:=vector(N,0):FF:=vector(N,0):\nfor ie from 1 to 2*NE do mel:=matri x(3,3,[tc[cn[ie,2],2],tc[cn[ie,2],3],1,tc[cn[ie,3],2],tc[cn[ie,3],3],1 , tc[cn[ie,4],2],tc[cn[ie,4],3],1]);\nmelinv:=inverse(mel):\narea:=abs ((tc[cn[ie,4],2]-tc[cn[ie,2],2])*(tc[cn[ie,3],3]-tc[cn[ie,2],3])-(tc[c n[ie,4],3]-tc[cn[ie,2],3])*(tc[cn[ie,3],2]-tc[cn[ie,2],2]))/2;\nfor i \+ to 3 do for j to 3 do \nA[cn[ie,i+1],cn[ie,j+1]]:= A[cn[ie,i+1],cn [ie,j+1]]+area*(melinv[1,i]*melinv[1,j ]+melinv[2,i]*melinv[2,j ]) od \+ : ss:=intlinealtriang(tc[cn[ie,2],2],tc[cn[ie,2],3],tc[cn[ie,3],2], tc[cn[ie,3],3],tc[cn[ie,4],2],tc[cn[ie,4],3],i): F[cn[ie,i+1]]:=F[cn[i e,i+1]]+ff*ss: \nod:od:" }}}{EXCHG {PARA 260 "" 0 "" {TEXT 258 62 "Int roducccion de la condicion de dirichlet en toda la frontera" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "NF:=2*(Nx+Ny)-4:VNF:=vector(NF,0):icont: =1:for i to N do if ((i <= Nx) or (i mod(Nx)=0) or (i mod(Nx)=1) or (i > N-Nx)) then VNF[icont]:=i:icont:=icont+1:end if:od:\nprint(VNF);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7.\"\"\"\"\"#\"\"$\"\"%\" \"&\"\")\"\"*\"#7\"#8\"#9\"#:\"#;" }}}{EXCHG {PARA 259 "" 0 "" {TEXT 257 62 "Aplicacion de las condiciones de Dirichlet en toda la frontera " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "for i to NF do for j to N do A [VNF[i],j]:=0:od : A[VNF[i],VNF[i]]:=1:od: \+ for i to NF do x:=tc[VNF[i],2]: y:=tc[VNF[i],3]:F[ VNF[i]]:=T(x,y):od:unassign('x','y'): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 " La solucion \+ del sistema lineal obtenido." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sol cal:=linsolve(A,F);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'solcalG-%'ve ctorG6#72\"\"!$\"+WWWW9!\"*$\"+6666^F,$\"+++++6!\")F)$\"+KLLL8F,$\"+() ))))))[F,$\"+mmmm5F1$\"+kmmmm!#5$\"+))))))))=F,$\"+ILLL`F,$\"+*******4 \"F1$\"+********>F,$\"+5666JF,$\"+VWWWkF,$\"+++++7F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "La solucion exacta del problema" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 73 "solexacta:=vector(N,0):for i to N do solexacta [i]:=T(tc[i,2],tc[i,3]) od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "prin t(solexacta):" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#72$\"\"!F ($\"+WWWW9!\"*$\"+6666^F+$\"+++++6!\")F'$\"+LLLL8F+$\"+*)))))))[F+$\"+ mmmm5F0$\"+kmmmm!#5$\"+))))))))=F+$\"+LLLL`F+$\"+*******4\"F0$\"+***** ***>F+$\"+5666JF+$\"+VWWWkF+$\"+++++7F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Comparacion de la solucion exacta y la calculada." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "err:=evalm(solexacta-solcal);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Nerr:=norm(err);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$errG-%'vectorG6#72$\"\"!F*F)F)F)F)$\"\"\"!\"*$\"\" #F-F)F)F)$\"\"$F-F)F)F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%Nerr G$\"\"$!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "Fin del pro grama." }}}}{MARK "6 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }