{VERSION 3 0 "IBM INTEL LINUX" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{PSTYLE " Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading \+ 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 40 "Mejor aproximaci\363n por m\355nimos cuadrados" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 16 "Regresi\363n lineal" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Sup\363ngase que se ha obtenido una colecci\363n de puntos en el plano, de forma experimental, sabiendo que est\341n " }} {PARA 0 "" 0 "" {TEXT -1 108 "sobre una recta desconocida. Sin embarg o, debido a errores de medici\363n cometidos, los puntos encontrados n o" }}{PARA 0 "" 0 "" {TEXT -1 60 "est\341n alineados. Por tanto, se bu sca la recta de ecuaci\363n " }{XPPEDIT 18 0 "r(x) := A*x+B;" "6#>-% \"rG6#%\"xG,&*&%\"AG\"\"\"F'F+F+%\"BGF+" }{TEXT -1 41 " que se ajuste mejor a la colecci\363n dada" }}{PARA 0 "" 0 "" {TEXT -1 10 "de punto s " }{XPPEDIT 18 0 "\{x[i], y[i]\};" "6#<$&%\"xG6#%\"iG&%\"yG6#F'" } {TEXT -1 4 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "N:=20;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "x:=[seq(i/2-7, i=1..N)];" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "y:=[seq(3*(i/2-7)+1+((-1)^i)/5,i=1. .N)];" }{TEXT -1 33 " " }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 42 "for n from 1 to N do l[n]:=[x[n],y[n]] od:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "l[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG76#!#8\"\"#! \"'#!#6F(!\"&#!\"*F(!\"%#!\"(F(!\"$#F,F(!\"##F2F(!\"\"#F6F(\"\"!#\"\" \"F(F:#\"\"$F(F(#\"\"&F(F<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG76 #!$(=\"#5#!#%)\"\"&#!$d\"F(#!#pF+#!$F\"F(#!#aF+#!#(*F(#!#RF+#!#nF(#!#C F+#!#PF(#!\"*F+#!\"(F(#\"\"'F+#\"#BF(#\"#@F+#\"#`F(#\"#OF+#\"#$)F(#\"# ^F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$#!#6\"\"##!$d\"\"#5" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Lo s puntos " }{XPPEDIT 18 0 "l[i] = \{x[i], y[i]\};" "6#/&%\"lG6#%\"iG<$ &%\"xG6#F'&%\"yG6#F'" }{TEXT -1 52 " est\341n \"aproximadamente\" alin eados. Los dibujamos: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 " plot([seq(l[j],j=1..N)],style=point,color=[blue]);" }}{PARA 13 "" 1 " " {INLPLOT "6&-%'CURVESG6$767$$!1+++++++l!#:$!1++++++q=!#97$$!\"'\"\"! $!1++++++!o\"F-7$$!1+++++++bF*$!1++++++q:F-7$$!\"&F1$!1++++++!Q\"F-7$$ !1+++++++XF*$!1++++++q7F-7$$!\"%F1$!1++++++!3\"F-7$$!1+++++++NF*$!1*** **********p*F*7$$!\"$F1$!1+++++++yF*7$$!1+++++++DF*$!1+++++++nF*7$$!\" #F1$!1+++++++[F*7$$!1+++++++:F*$!1+++++++PF*7$$!\"\"F1$!1+++++++=F*7$$ !1+++++++]!#;$!1+++++++qFco7$F1$\"1+++++++7F*7$$\"1+++++++]Fco$\"1++++ +++BF*7$$\"\"\"F1$\"1+++++++UF*7$$\"1+++++++:F*$\"1+++++++`F*7$$\"\"#F 1$\"1+++++++sF*7$$\"1+++++++DF*$\"1,++++++$)F*7$$\"\"$F1$\"1++++++?5F- -%'COLOURG6&%$RGBGF1F1$\"*++++\"!\")-%&STYLEG6#%&POINTG-%+AXESLABELSG6 $%!GFer-%%VIEWG6$%(DEFAULTGFir" 2 400 300 300 5 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 6144 4088 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "Busc amos la recta que mejor se aproxime, en el sentido de m\355nimos cuadr ados, a la colecci\363n dada. En primer lugar, determinamos los coefic ientes del sistema a resolver:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "c1:=0: c2:=0: c3:=0: c4:=0: " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "for s from 1 to N do c1:=c1+x[s]; c 2:=c2+x[s]^2; c3:=c3+y[s]; c4:=c4+x[s]*y[s]; od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "c1;c2;c3;c4;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "eq s:=\{N*B+c1*A =c3,c1*B+c2*A=c4 \};" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "solve(eqs);" }{TEXT -1 39 " \+ " }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$b%\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!#&)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"%(H\"\"\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$eqsG<$/,&%\"BG\"#?%\"AG!#N!#&)/,&F(F+F*#\"$b% \"\"##\"%(H\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%\"AG#\"%**>\"$l '/%\"BG#\"#'*\"#&*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Asignamos l os valores obtenidos a " }{XPPEDIT 18 0 "A;" "6#%\"AG" }{TEXT -1 3 " \+ y " }{XPPEDIT 18 0 "B;" "6#%\"BG" }{TEXT -1 2 ": " }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "coef:=solve(eqs):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 13 "assign(coef);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "B;A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#'*\"#&*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6##\"%**>\"$l'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }{TEXT -1 20 "La recta buscada es " } {XPPEDIT 18 0 "y = A*x+B;" "6#/%\"yG,&*&%\"AG\"\"\"%\"xGF(F(%\"BGF(" } {TEXT -1 32 " , cuya gr\341fica es la siguiente:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "plot([A*t+B,[seq(l[ j],j=1..N)]],t=-7..3,style=[line,point],color=[red,blue]);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6%7S7$$!\"(\"\"!$!1Uot%*y:.?!#97$$!1L LLe%G?y'!#:$!1)3N:KNw$>F-7$$!1mmT&esBf'F1$!1O&R4[C1)=F-7$$!1LL$3s%3zjF 1$!1b@6G)4l\"=F-7$$!1ML$e/$QkhF1$!1%\\Z<=q>v\"F-7$$!1nmT5=q]fF1$!1yg0 \"GPxo\"F-7$$!1ML3_>f_dF1$!1;:tf^=G;F-7$$!1++vo1YZbF1$!1@M\\(QAlc\"F-7 $$!1ML3-OJN`F1$!1w_Bg1v-:F-7$$!1++v$*o%Q7&F1$!1D_r\\M=R9F-7$$!1mmm\"RF j!\\F1$!1!)>Hgnzt8F-7$$!1LL$e4OZr%F1$!1oslGT?;8F-7$$!1+++v'\\!*\\%F1$! 1'G9iYo8D\"F-7$$!1+++DwZ#G%F1$!1I?J#emi=\"F-7$$!1+++D.xtSF1$!1mk3a)GN7 \"F-7$$!1LL3-TC%)QF1$!1xeZ')pbm5F-7$$!1mmm\"4z)eOF1$!1Z()\\a#>\"))**F1 7$$!1mmmm`'zY$F1$!1jcEE(HUT*F17$$!1++v=t)eC$F1$!1Z'3Zxfmu)F17$$!1nmm;1 J\\IF1$!1:b?_sub\")F17$$!1***\\(=[jLGF1$!1JHUgDU2vF17$$!1++Dc/EGEF1$!1 `;TQT1!*oF17$$!1mm;aQ(RT#F1$!1LU>!R:fC'F17$$!1mmTg=>yTa)4DF17$$!1(****\\PZhh*!#;$!1=v]X?5!)=F17$$!1&***\\(=_\"*e( Fht$!1`>.WUyq7F17$$!1+++D'>&Q`Fht$!1keM8rSUfFht7$$!1ummmhA;LFht$\"1'f? $GVPm8!#<7$$!1&*****\\i*p:\"Fht$\"1,\\&>.:ti'Fht7$$\"1@mm\"zpe*zFju$\" 12M6kL)3D\"F17$$\"17++]#\\'QHFht$\"114O*p))Q*=F17$$\"1HL$e9S8&\\Fht$\" 1_7D4m!*)\\#F17$$\"12+]i?=bqFht$\"15L2`YKJJF17$$\"1HLL$3s?6*Fht$\"12NS '))G'\\PF17$$\"1++DJXaE6F1$\"1SiC!H'F1 7$$\"1,++]Qk\\>F1$\"13@%o/&=roF17$$\"1NL$3dg6<#F1$\"1>cuSw1PvF17$$\"1o mmmxGpBF1$\"1Xpxq4kK\")F17$$\"1++D\"oK0e#F1$\"1*pH1OYww)F17$$\"1,+v=5s #y#F1$\"1I3eVvUv$*F17$$\"\"$F*$\"1Vr&G9dG+\"F--%'COLOURG6&%$RGBG$\"*++ ++\"!\")F*F*-%&STYLEG6#%%LINEG-F$6%767$$!1+++++++lF1$!1++++++q=F-7$$! \"'F*$!1++++++!o\"F-7$$!1+++++++bF1$!1++++++q:F-7$$!\"&F*$!1++++++!Q\" F-7$$!1+++++++XF1$!1++++++q7F-7$$!\"%F*$!1++++++!3\"F-7$$!1+++++++NF1$ !1*************p*F17$$!\"$F*$!1+++++++yF17$$!1+++++++DF1$!1+++++++nF17 $$!\"#F*$!1+++++++[F17$$!1+++++++:F1$!1+++++++PF17$$!\"\"F*$!1+++++++= F17$$!1+++++++]Fht$!1+++++++qFht7$F*$\"1+++++++7F17$$\"1+++++++]Fht$\" 1+++++++BF17$$\"\"\"F*$\"1+++++++UF17$$\"1+++++++:F1$\"1+++++++`F17$$ \"\"#F*$\"1+++++++sF17$$\"1+++++++DF1$\"1,++++++$)F17$Fgz$\"1++++++?5F --F\\[l6&F^[lF*F*F_[l-Fc[l6#%&POINTG-%+AXESLABELSG6$Q\"t6\"%!G-%%VIEWG 6$;F(Fgz%(DEFAULTG" 2 400 300 300 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2138 1 0 0 0 0 0 0 }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "Ajuste de gr \341ficas" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 146 "Lo hecho en el apartado anterior para ajustar los puntos a una recta puede repetirse para otro tipo de curvas. Veamos otro eje mplo con una c\372bica:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "N:=8; #(pero ahora daremos los puntos uno a uno)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "x:=[seq(i-4.3, i=1..N)];" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "y:=[-2,-3,-1,-1,1,2,3,3];" }{TEXT -1 33 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "for n from 1 to N do l[n]:=[x[n],y[n]] od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "l[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"NG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG7*$!#L!\"\"$!#BF($ !#8F($!\"$F($\"\"(F($\"#%\"yG7*!\"#!\"$!\"\"F(\"\"\"\"\"#\"\"$F+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#7$$!#8!\"\"F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Dibujamos los puntos " } {XPPEDIT 18 0 "l[i] = \{x[i], y[i]\};" "6#/&%\"lG6#%\"iG<$&%\"xG6#F'&% \"yG6#F'" }{TEXT -1 2 ": " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot([seq(l[j],j=1..N)],style=point,color=[blue]);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7*7$$!1+++++++L!#:$!\"#\"\"!7$$!1+++++++B F*$!\"$F-7$$!1+++++++8F*$!\"\"F-7$$!1+++++++I!#;F67$$\"1+++++++qF;$\" \"\"F-7$$\"1+++++++ " 0 "" {MPLTEXT 1 0 13 "with(linalg) :" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "A:=array(1..4,1..4): " } {TEXT -1 0 "" }{MPLTEXT 1 0 310 " \+ for i to 4 do for j to 4 do \+ S:=0; \+ for k to N do S:=S+(x[k])^(i+j-2) od; A[i,j]:=S ; od; od; \+ " }} {PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "pr int(A);" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for no rm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7&7&\"\")$\"#;!\"\"$\"%K U!\"#$\"&k_#!\"$7&F)F,F/$\"(Gf)R!\"%7&F,F/F3$\")ci=R!\"&7&F/F3F7$\"+7N t@W!\"'" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "(Obs\351rvese la simetr\355a del sistema. Caracter\355st icas de este tipo pueden ser aprovechadas en su resoluci\363n.) " }} {PARA 0 "" 0 "" {TEXT -1 59 "Obtenemos el vector de t\351rminos indepe ndientes del sistema:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "b:=array(1..4):" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 1 " " }{TEXT -1 1 " " }{MPLTEXT 1 0 130 " for j to 4 do s:=0: \+ for k to N do s:=s+(y[k]*(x[k])^(j-1)) od: b[j]:=s: od: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7&\"\"#$\"$%Q!\"\"$\"%yH!\"#$\"'w " 0 "" {MPLTEXT 1 0 17 "t:=linsolve(A,b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG-%' vectorG6#7&$!+OLLt:!#5$\"+<.Y-9!\"*$\"+9dG9d!#6$!+fbbbbF1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "Por tan to, el polinomio ser\341:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "f:=proc(x::uneval)local s,k; s:=0; \+ for k to 4 do s:=s+t[k]*x^(k-1) od: end:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "f(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*$!+OLLt:!#5 \"\"\"%\"xG$\"+<.Y-9!\"**$)F(\"\"#\"\"\"$\"+9dG9d!#6*$)F(\"\"$F/$!+fbb bbF2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "plot([f(t),[seq(l[j],j=1..N)]],t=-7..3,style=[line,po int],color=[red,blue]);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6%7U 7$$!\"(\"\"!$\"1+IY,+5)=\"!#97$$!1mm;HU,\"*o!#:$\"18T9_t526F-7$$!1LLLe %G?y'F1$\"1Ye44)y*G5F-7$$!1+](=_+so'F1$\"1\"\\2#Q<'Hj*F17$$!1mmT&esBf' F1$\"1^8g-w?(**)F17$$!1LL$3s%3zjF1$\"1PRn3>zUwF17$$!1ML$e/$QkhF1$\"1Ul 6qwL#Q'F17$$!1nmT5=q]fF1$\"1RLE.`:F_F17$$!1ML3_>f_dF1$\"1`W\"3C&zTUF17 $$!1++vo1YZbF1$\"1U<5gH^0LF17$$!1ML3-OJN`F1$\"1'RBB5!3CCF17$$!1++v$*o% Q7&F1$\"14=S&o_-j\"F17$$!1mmm\"RFj!\\F1$\"1\"ojR17p)*)!#;7$$!1LL$e4OZr %F1$\"1K)z))>!=IKFeo7$$!1+++v'\\!*\\%F1$!1]kELYC6DFeo7$$!1+++DwZ#G%F1$ !1:jQB5$3_(Feo7$$!1+++D.xtSF1$!1w@5?vQm6F17$$!1LL3-TC%)QF1$!1pAZ*=xp[ \"F17$$!1mmm\"4z)eOF1$!1YeJn<]-=F17$$!1mmmm`'zY$F1$!1L+'*\\^j;?F17$$!1 ++v=t)eC$F1$!1Wkx\"[Hw?#F17$$!1nmm;1J\\IF1$!1C\\tb4NFBF17$$!1***\\(=[j LGF1$!1AZuC3`3CF17$$!1++Dc/EGEF1$!1g@w&o.+W#F17$$!1mm;aQ(RT#F1$!1`V#p' eNGCF17$$!1mmTg=>%zPT!Q#F17$$!1LL$e*e$\\+#F1$!1r.\\q@t\"H#F 17$$!1LL3-;Y%y\"F1$!1\"p6\\CGB;#F17$$!1++D\"3QDf\"F1$!1LsKOP\\@?F17$$! 1LL$3Ub_Q\"F1$!1_BpotwU=F17$$!1+++]@6r6F1$!1_cAMv;K;F17$$!1(****\\PZhh *Feo$!19m:O#>PS\"F17$$!1&***\\(=_\"*e(Feo$!1jv0vp[k6F17$$!1+++D'>&Q`Fe o$!1)zx_O8I\"))Feo7$$!1ummmhA;LFeo$!1;ELZf5ThFeo7$$!1&*****\\i*p:\"Feo $!1I#3[%[Y(=$Feo7$$\"1@mm\"zpe*z!#<$!1E1/U'\\d[%F\\w7$$\"17++]#\\'QHFe o$\"1ur)y\"QD$e#Feo7$$\"1HL$e9S8&\\Feo$\"1p!pjAyLW&Feo7$$\"12+]i?=bqFe o$\"1.GO\"z81T)Feo7$$\"1HLL$3s?6*Feo$\"1#>7!eB,E6F17$$\"1++DJXaE6F1$\" 19rn/Ip:9F17$$\"1ommm*RRL\"F1$\"1W`+HsF$o\"F17$$\"1om;a<.Y:F1$\"1$HU3Y ,A%>F17$$\"1NLe9tOcF1$\"1?b%yJfCQ#F1 7$$\"1NL$3dg6<#F1$\"17O))GdS)e#F17$$\"1ommmxGpBF1$\"1[v*[iztu#F17$$\"1 ++D\"oK0e#F1$\"1[etk!4w)GF17$$\"1,+v=5s#y#F1$\"1+(f[_$p!*HF17$$\"\"$F* $\"1++tILLkIF1-%'COLOURG6&%$RGBG$\"*++++\"!\")F*F*-%&STYLEG6#%%LINEG-F $6%7*7$$!1+++++++LF1$!\"#F*7$$!1+++++++BF1$!\"$F*7$$!1+++++++8F1$!\"\" F*7$$!1+++++++IFeoF`]l7$$\"1+++++++qFeo$\"\"\"F*7$$\"1+++++++ " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "N:=8;" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 24 "x:=[seq(i-4.3, i=1..N)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "y:=[-2,-3,-1,-1,1,2,3,3];" }{TEXT -1 33 " \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "for n from 1 to N do l[n]:=[x[n],y[n]] od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"NG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG7*$!#L!\"\"$!#BF($! #8F($!\"$F($\"\"(F($\"#%\"yG7*!\"#!\"$!\"\"F(\"\"\"\"\"#\"\"$F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 " Seguidamente se define la matriz (de " }{XPPEDIT 18 0 "N;" "6#%\"NG" }{TEXT -1 71 " filas y 4 columnas) del sistema cuy a pseudosoluci\363n se quiere obtener:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Q:=array(1..N,1..4): for i to N do for j to 4 do Q[i,j]:=x[i]^(j-1) od; od;" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(Q);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7*7&\"\"\"$!#L!\"\"$\"%*3\"!\"#$!&Pf$! \"$7&F($!#BF+$\"$H&F.$!&n@\"F17&F($!#8F+$\"$p\"F.$!%(>#F17&F($F1F+$\" \"*F.$!#FF17&F($\"\"(F+$\"#\\F.$\"$V$F17&F($\"#F17&F($\"#PF+$\"%p8F.$\"&`1&F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Finalmente, se utiliza la funci\363n \"" }{XPPEDIT 18 0 "leastsqrs();" "6#-%*leastsqrsG6\"" }{TEXT -1 46 "\" pa ra obtener directamente la pseudosoluci\363n:" }{MPLTEXT 1 0 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "leastsqrs(Q,y);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 86 "(Obtenidos ya los coeficientes del polino mio, se sigue como en el apartado anterior.) " }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definition for trace" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'vectorG6#7&$!+OLLt:!#5$\"+<.Y-9!\"*$\"+9dG9d!#6$!+fb bbbF/" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 22 "Polinomios ortogonale s" }}{SECT 0 {PARA 4 "" 0 "examples" {TEXT -1 41 " Utilizando funcion es internas del Maple" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "rest art;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "with(orthopoly);#(para pode r utilizar las funciones internas)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# 7(%\"GG%\"HG%\"LG%\"PG%\"TG%\"UG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 " Se puede utilizar toda una colecci\363n de familias de polinomio s ortogonales cl\341sicos ya implementados. Por ejemplo, los polinomio s de Tchebychev: " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "N:=10;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "T(N,x); #polinomio de Tchebychev de grado N" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,.*$)%\"xG\"#5\"\"\"\"$7&*$)F&\"\")F(!%!G\"*$)F&\"\"'F(\"%?6*$)F&\" \"%F(!$+%*$)F&\"\"#F(\"#]!\"\"\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Analicemos las ra\355ces de esta familia de polinomios:" } {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sol:=[evalf(sol ve(T(N,x),x))];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$solG7,$\"+5y1rq! #5$!+5y1rqF($\"+0M)o()*F($!+0M)o()*F($\"+ZYMk:F($!+ZYMk:F($\"+V_15*)F( $!+V_15*)F($\"+)*\\!*RXF($!+)*\\!*RXF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sol[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+0M)o()* !#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Dibujemos estas soluciones :" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "for i from \+ 1 to N do dib[i]:=[sol[i],0] od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot([seq(dib[j],j=1..N)],style=point,axes=BOX);" }}{PARA 0 "" 0 "" {TEXT -1 100 "Se puede observar que, al aumentar el grado del polinomi o, sus ra\355ces tienden a llenar el intervalo " }{XPPEDIT 18 0 "[-1, \+ 1];" "6#7$,$\"\"\"!\"\"\"\"\"" }{TEXT -1 32 " (son densas en este inte rvalo)." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\"\"7$$\"+5y1rq !#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\"#7$$!+5y1rq! #5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\"$7$$\"+0M)o() *!#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\"%7$$!+0M)o( )*!#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\"&7$$\"+ZYM k:!#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\"'7$$!+ZYMk :!#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\"(7$$\"+V_15 *)!#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\")7$$!+V_15 *)!#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\"*7$$\"+)* \\!*RX!#5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"#57$$!+) *\\!*RX!#5\"\"!" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6$7,7$$\"1++ +5y1rq!#;\"\"!7$$!1+++5y1rqF*F+7$$\"1+++0M)o()*F*F+7$$!1+++0M)o()*F*F+ 7$$\"1+++ZYMk:F*F+7$$!1+++ZYMk:F*F+7$$\"1+++V_15*)F*F+7$$!1+++V_15*)F* F+7$$\"1+++)*\\!*RXF*F+7$$!1+++)*\\!*RXF*F+-%'COLOURG6&%$RGBG$\"#5!\" \"F+F+-%+AXESLABELSG6$%!GFQ-%*AXESSTYLEG6#%$BOXG-%&STYLEG6#%&POINTG-%% VIEWG6$%(DEFAULTGFgn" 2 400 300 300 5 0 1 0 2 9 0 2 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 2213 400 0 0 0 0 0 0 }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 55 " Utilizando relaciones de recurrencia a tres t\351rminos" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 129 "Ahora se definir\341 una familia de polinomios ortogon ales a partir de su recurrencia a tres t\351rminos. Dicha recurrencia \+ es del tipo" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "P[n+1](x) = \{x-b[n+1] \}*P[n](x)-a[n+1]^2*P[n-1](x);" "6#/-&%\"PG6#,&%\"nG\"\"\"\"\"\"F*6#% \"xG,&*&<#,&F-F*&%\"bG6#,&F)F*\"\"\"F*!\"\"F*-&F&6#F)6#F-F*F**&&%\"aG6 #,&F)F*\"\"\"F*\"\"#-&F&6#,&F)F*\"\"\"F76#F-F*F7" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 7 "siendo " }{XPPEDIT 18 0 "b[n],a[n];" "6$&% \"bG6#%\"nG&%\"aG6#F&" }{TEXT -1 44 " n\372meros reales, con condicio nes iniciales " }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "P[-1](x) = 0,P[0](x ) = 1;" "6$/-&%\"PG6#,$\"\"\"!\"\"6#%\"xG\"\"!/-&F&6#F-6#F,\"\"\"" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "Definimos las su cesiones de la recurrencia: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "N:=10;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "a:=[seq( 2+(1/n^2) , n=1..N)];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "b:=(seq(1,n=1..N));" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"NG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG7,\"\"$#\"\"*\"\"%#\"#>F(#\"#L\"#;#\"#^\"#D#\"#t \"#O#\"#**\"#\\#\"$H\"\"#k#\"$j\"\"#\")#\"$,#\"$+\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"bG6,\"\"\"F&F&F&F&F&F&F&F&F&" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 51 "Definimos los pol inomios utilizando la recurrencia:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Q[1]:=0;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Q[2 ]:=1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "for n from 1 to N do Q[n+2 ]:=(x-b[n])*Q[n+1]-a[n]^2*Q[n] od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"QG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"QG6#\"\"# \"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Determinamos los ceros \+ de " }{XPPEDIT 18 0 "P[N] = Q[N+2];" "6#/&%\"PG6#%\"NG&%\"QG6#,&F'\"\" \"\"\"#F," }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "r:=RootOf(Q[N+2] ,x):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "q:=[evalf(allvalues(r))];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"qG7,$!+mA5JH!\"*$!+B%GrZ#F($!+(f J$>PF($\" +B%GrZ%F($\"+mA5J\\F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Dibujemo s estas soluciones:" }{MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "for i from 1 to N do dib[i]:=[q[i],0] od;" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 48 "plot([seq(dib[j],j=1..N)],style=point,axes=BOX);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\"\"7$$!+mA5JH!\"*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\"#7$$!+B%GrZ#!\"*\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\"$7$$!+(fJ$>&%$dibG6#\"\"%7$$!+!fALH(!#5\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\"&7$$\"+*Q8!pS!#5\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\"'7$$\"+h')4$f\"! \"*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\"(7$$\"+fALHF !\"*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\")7$$\"+(fJ$ >P!\"*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"\"*7$$\"+B% GrZ%!\"*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$dibG6#\"#57$$\"+m A5J\\!\"*\"\"!" }}{PARA 13 "" 1 "" {INLPLOT "6'-%'CURVESG6$7,7$$!1+++m A5JH!#:\"\"!7$$!1+++B%GrZ#F*F+7$$!1+++(fJ$>P F*F+7$$\"1+++B%GrZ%F*F+7$$\"1+++mA5J\\F*F+-%'COLOURG6&%$RGBG$\"#5!\"\" F+F+-%+AXESLABELSG6$%!GFR-%&STYLEG6#%&POINTG-%*AXESSTYLEG6#%$BOXG-%%VI EWG6$%(DEFAULTGFhn" 2 310 278 278 5 0 1 0 2 9 0 2 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 267 "Como ejercicio se pr opone al alumno modificar las sucesiones de coeficientes de la recurre ncia y analizar los ceros de los nuevos polinomios obtenidos para dife rentes grados. \277Existe alguna relaci\363n entre dichos coeficientes y el intervalo en el que est\341n los ceros? " }{MPLTEXT 1 0 0 "" }}} }}}{MARK "2 2 6 13" 0 }{VIEWOPTS 1 1 0 1 1 1803 }