{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 "" 1 24 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 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 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 295 "" 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 "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 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 P lot" -1 13 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 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 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 "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 49 "Aula Pr\341tica sobre o M \351todo dos Quadrados M\355nimos" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 69 "Aspetos te\363ricos devem ser consultados na bibliografia da disciplina." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 16 "I. Caso \+ discreto" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "Dada uma tabela com os valores de uma fun\347\343o f em m pontos \+ diferentes: x1, ... , xm, e f(x1), ... , f(xm) e n fun\347\365es " }}{PARA 0 "" 0 "" {TEXT -1 52 "previamente selecionadas g1,..., gn tal que m > n. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 "Deseja-se computar coeficientes alpha1,..., alphan, tais que a fun\347\343o phi(x) = alpha1 g1(x) + ... + alphan gn(x) se " }} {PARA 0 "" 0 "" {TEXT -1 35 "aproxime o m\341ximo possivel de f(x)." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "O m\351 todo dos quadrados m\355nimos seleciona tais coeficientes de forma tal que seja minimizada a suma dos quadrados" }}{PARA 0 "" 0 "" {TEXT -1 69 "das dist\342ncias entre phi(x) e f(x) nos pontos x = xi, para i=1, ...,m." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "Para isto \351 necess\341rio resolver o sistema linear A x = b, \+ onde A \351 uma matriz nxn com componentes Aij da forma:" }}{PARA 0 " " 0 "" {XPPEDIT 18 0 "Sum(gi(xk)*gj(xk),k = 1 .. m);" "6#-%$SumG6$*&-% #giG6#%#xkG\"\"\"-%#gjG6#F*F+/%\"kG;F+%\"mG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "x \351 o vetor [ alpha1,...,alphan] de coeficientes-incognitas e b \351 o n-vetor com c omponentes bi da forma: " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "Sum(gi(xk)* f(xk),k = 1 .. m);" "6#-%$SumG6$*&-%#giG6#%#xkG\"\"\"-%\"fG6#F*F+/%\"k G;F+%\"mG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 39 "Especificar-se-\341 um procedimento geral " } {TEXT 258 9 "cdquamin " }{TEXT -1 58 "que implementa o m\351todo dos q uadrados m\355nimos para o caso " }}{PARA 0 "" 0 "" {TEXT -1 59 "discr eto com um vetor de fun\347\365es escolhidas g de tamanho n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Dito procedimen to ter\341 como par\342metros o seguinte:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "bin: m-vector de n\363s x1,...,xm ;" }}{PARA 0 "" 0 "" {TEXT -1 63 "fin: m-vector de valores da fun \347\343o f nos n\363s f(x1),...,f(xm);" }}{PARA 0 "" 0 "" {TEXT -1 20 "m: n\372mero de pontos;" }}{PARA 0 "" 0 "" {TEXT -1 36 "g: n-vec tor de fun\347\365es escolhidas;" }}{PARA 0 "" 0 "" {TEXT -1 34 "n: \+ n\372mero de fun\347\365es escolhidas;" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 123 "O procedimento inicialmente constroi r\341 o sistema linear A x = b e logo utilizar\341 um m\351todo de sol u\347\343o de sistemas lineares " }}{PARA 0 "" 0 "" {TEXT -1 1 "(" } {TEXT 259 13 "pvtriangGauss" }{TEXT -1 104 ") para estabelecer os coef icientes alpha1, ... , alphan do ajuste otimal de f com as fun\347\365 es g1,..., gn." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "Inicialmente, s\343o incluidos alguns dos m\351todos pre viamente implementados para solu\347\343o de sistemas lineares." }} {PARA 0 "" 0 "" {TEXT -1 45 "Logo disto ser\341 incluido o novo proced imento " }{TEXT 260 9 "cdquamin." }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "intercambiar := proc(A,i,j, n); A1:=A; for k from 1 to n do aux := A1[i,k]; A1[i,k]:=A1[j,k]; A1[j ,k]:=aux; end do; return A1; end proc:" }}{PARA 7 "" 1 "" {TEXT -1 71 "Warning, `A1` is implicitly declared local to procedure `intercambiar `\n" }}{PARA 7 "" 1 "" {TEXT -1 70 "Warning, `k` is implicitly declare d local to procedure `intercambiar`\n" }}{PARA 7 "" 1 "" {TEXT -1 72 " Warning, `aux` is implicitly declared local to procedure `intercambiar `\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 610 "pvtriangGauss := pr oc(Ain,bin,n) x := vector(n); A := Ain; b := bin; for i from 1 to n-1 do maior:=i; for l from i+1 to n do if abs(A[maior,i]) < abs(A[l,i] ) then maior := l; end if; end do; if maior <> i then A := intercambi ar(A,i,maior,n); aux:=b[i]; b[i]:=b[maior]; b[maior]:=aux; end if; \+ for j from i+1 to n do mult := A[j,i]/A[i,i]; A[j,i] := 0; for k from \+ i+1 to n do A[j,k]:= A[j,k]-mult * A[i,k]; end do; b[j]:=b[j]-b[i]*mul t; end do; end do; for i from n by -1 to 1 do x[i]:= b[i]; for k f rom i+1 to n do x[i] := x[i] - x[k]*A[i,k]; end do; x[i] := x[i]/A[i,i ]; end do; print(x); end proc:" }}{PARA 7 "" 1 "" {TEXT -1 71 "Warnin g, `x` is implicitly declared local to procedure `pvtriangGauss`\n" }} {PARA 7 "" 1 "" {TEXT -1 71 "Warning, `A` is implicitly declared local to procedure `pvtriangGauss`\n" }}{PARA 7 "" 1 "" {TEXT -1 71 "Warnin g, `b` is implicitly declared local to procedure `pvtriangGauss`\n" }} {PARA 7 "" 1 "" {TEXT -1 71 "Warning, `i` is implicitly declared local to procedure `pvtriangGauss`\n" }}{PARA 7 "" 1 "" {TEXT -1 75 "Warnin g, `maior` is implicitly declared local to procedure `pvtriangGauss`\n " }}{PARA 7 "" 1 "" {TEXT -1 71 "Warning, `l` is implicitly declared l ocal to procedure `pvtriangGauss`\n" }}{PARA 7 "" 1 "" {TEXT -1 73 "Wa rning, `aux` is implicitly declared local to procedure `pvtriangGauss` \n" }}{PARA 7 "" 1 "" {TEXT -1 71 "Warning, `j` is implicitly declared local to procedure `pvtriangGauss`\n" }}{PARA 7 "" 1 "" {TEXT -1 74 " Warning, `mult` is implicitly declared local to procedure `pvtriangGau ss`\n" }}{PARA 7 "" 1 "" {TEXT -1 71 "Warning, `k` is implicitly decla red local to procedure `pvtriangGauss`\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 38 "Segue a especifica\347 \343o do procedimento " }{TEXT 261 8 "cdquamin" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 416 "cdquamin := proc(xin, fin, m, g, n) A := matrix(n,n); b := vect or(n); for i to n do for j to n do A[i,j] := sum('g[i](xin[k])*g[j](x in[k])','k'=1..m); end do; b[i] := sum('g[i](xin[k])*fin[k]','k'=1..m) ; end do; print(`Sistema linear`);print(`A = `,evalm(A));print(`b = \+ `,evalm(convert(convert(b,Vector[column]),Matrix)));print(`Solu\347 \343o do sistema linear - coeficientes alpha - `);pvtriangGauss(A,b,n) ;end proc;" }}{PARA 7 "" 1 "" {TEXT -1 66 "Warning, `A` is implicitly \+ declared local to procedure `cdquamin`\n" }}{PARA 7 "" 1 "" {TEXT -1 66 "Warning, `b` is implicitly declared local to procedure `cdquamin` \n" }}{PARA 7 "" 1 "" {TEXT -1 66 "Warning, `i` is implicitly declared local to procedure `cdquamin`\n" }}{PARA 7 "" 1 "" {TEXT -1 66 "Warni ng, `j` is implicitly declared local to procedure `cdquamin`\n" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%)cdquaminGf*6'%$xinG%$finG%\"mG%\"gG %\"nG6&%\"AG%\"bG%\"iG%\"jG6\"F1C*>8$-%'matrixG6$9(F8>8%-%'vectorG6#F8 ?(8&\"\"\"F@F8%%trueGC$?(8'F@F@F8FA>&F46$F?FD-%$sumG6$.*&-&9'6#F?6#&9$ 6#%\"kGF@-&FO6#FDFQF@/.FU;F@9&>&F:FP-FI6$.*&FMF@&9%FTF@FY-%&printG6#%/ Sistema~linearG-F`o6$%'A~=~~~G-%&evalmG6#F4-F`o6$%'b~=~~~G-Fgo6#-%(con vertG6$-F_p6$F:&%'VectorG6#%'columnG%'MatrixG-F`o6#%RSolu|by|^yo~do~si stema~linear~-~coeficientes~alpha~-~G-%.pvtriangGaussG6%F4F:F8F1F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Para utilizar o procedimento " }{TEXT 266 8 "cdquamin" }{TEXT -1 44 " ser\341 necess\341rio fornecer como argumento os " }{TEXT 267 1 "m" } {TEXT -1 26 " n\363s e valores, o pr\363prio " }{TEXT 268 1 "m" } {TEXT -1 13 ", o vetor de " }{TEXT 269 1 "n" }{TEXT -1 8 " fun\347\365 es" }}{PARA 0 "" 0 "" {TEXT -1 25 "selecionadas e o pr\363prio " } {TEXT 270 1 "n" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 262 9 "Exemplo " }{TEXT -1 39 "Ajuste os 11 pon tos dados pelos n\363s " }{XPPEDIT 18 0 "[-1, -.75, -.6, -.5, -.3, \+ 0, .2, .4, .5, .7, 1];" "6#7-,$\"\"\"!\"\",$-%&FloatG6$\"#v!\"#F&,$-F) 6$\"\"'F&F&,$-F)6$\"\"&F&F&,$-F)6$\"\"$F&F&\"\"!-F)6$\"\"#F&-F)6$\"\"% F&-F)6$F4F&-F)6$\"\"(F&F%" }{TEXT -1 11 " e valores " }}{PARA 0 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "[2.05, 1.153, .45, .4, .5, 0, .2, .6, .512, 1.2, 2.05];" "6#7--%&FloatG6$\"$0#!\"#-F%6$\"%`6!\"$-F%6$\"#XF( -F%6$\"\"%!\"\"-F%6$\"\"&F3\"\"!-F%6$\"\"#F3-F%6$\"\"'F3-F%6$\"$7&F,-F %6$\"#7F3-F%6$F'F(" }}{PARA 0 "" 0 "" {TEXT -1 29 " com um polin\364mi o quadr\341tico." }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 " " {TEXT -1 128 "Primeiro definimos o vetor de fun\347\365es selecionad as para realizar o ajuste quadr\341tico phi(x) := alpha1 + alpha2 x + alpha3 x^2: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "g[1] := x -> 1; g[2] := x -> x; g[3] := x -> x^2 ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"gG6#\"\"\"F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"gG6#\"\"#f*6#%\"xG6\"6$%)operatorG%&arrowGF +9$F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"gG6#\"\"$f*6#%\"xG6 \"6$%)operatorG%&arrowGF+*$)9$\"\"#\"\"\"F+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Logo pode ser utiliz ado o procedimento:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "cdquamin([-1, -.75, -.6, -.5, -.3, 0, .2 , .4, .5, .7, 1],[2.05, 1.153, .45, .4, .5, 0, .2, .6, .512, 1.2, 2.05 ],11,g,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%/Sistema~linearG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%'A~=~~~G-%'matrixG6#7%7%\"#6$!#N!\"#$ \"&D?%!\"%7%F*F-$!'v)\\#!\"'7%F-F1$\"*D1k%G!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%'b~=~~~G-%'matrixG6#7%7#$\"%:\"*!\"$7#$!&v3\"!\"&7#$\" )Dcve!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%RSolu|by|^yo~do~sistema~ linear~-~coeficientes~alpha~-~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%' vectorG6#7%$\"+Fc;T\"*!#6$\"+Y6=&p*F)$\"+3DvP>!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Temos ent\343o qu e o ajuste \351 dado pela fun\347\343o phi definida por:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "phi \+ := x -> .9141165627e-1+.9695181146e-1*x+1.937752508*x^2;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$phiGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,($\"+ Fc;T\"*!#6\"\"\"*&$\"+Y6=&p*F/F09$F0F0*&$\"+3DvP>!\"*F0)F4\"\"#F0F0F(F (F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "A seguir aparece o diagrama de dispers\343o dos pontos ajustado s e a fun\347\343o phi." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 165 "PLOT(POINTS([-1, 2.05],[-.75,1.153 ], [-.6, .45], [-.5, .4], [-.3,.5], [0,0], [.2,.2], [.4,.6], [.5,.512 ], [.7, 1.2],[1,2.05],SYMBOL(DIAMOND)));plot(phi(x),x=-1..1);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6#-%'POINTSG6.7$! \"\"$\"$0#!\"#7$$!#vF*$\"%`6!\"$7$$!\"'F'$\"#XF*7$$!\"&F'$\"\"%F'7$$F0 F'$\"\"&F'7$\"\"!F@7$$\"\"#F'FB7$F9$\"\"'F'7$F=$\"$7&F07$$\"\"(F'$\"#7 F'7$\"\"\"F(-%'SYMBOLG6#%(DIAMONDG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"\"\"\"!$\"3#****4GN7A$>!#<7 $$!3ommm;p0k&*!#=$\"3A``pn:\\!)**F17$$!3+nm;/siqmF1$\"3!zPm?Ic)*) ))F17$$!3[++](y$pZiF1$\"3uMeA6?:syF17$$!33LLL$yaE\"eF1$\"3%H?wI&\\k(*o F17$$!3hmmm\">s%HaF1$\"3DWWHDE0+hF17$$!3Q+++]$*4)*\\F1$\"3k%[jvvS-F&F1 7$$!39+++]_&\\c%F1$\"3%)e&\\Hw#e4XF17$$!31+++]1aZTF1$\"3k`)e[&\\MXQF17 $$!3umm;/#)[oPF1$\"3xFQCxYl+LF17$$!3hLLL$=exJ$F1$\"3vn`5A(Qas#F17$$!3* RLLLtIf$HF1$\"3E6]25_v*H#F17$$!3]++]PYx\"\\#F1$\"3Ac)4rctc(=F17$$!3EML LL7i)4#F1$\"3M[jNR&ySc\"F17$$!3c****\\P'psm\"F1$\"3cHTdwe7\"H\"F17$$!3 ')****\\74_c7F1$\"3Q)*y(=jN#)4\"F17$$!3)3LLL3x%z#)!#>$\"3wUT*okznm*Fer 7$$!3KMLL3s$QM%Fer$\"3CIPHWdl&3*Fer7$$!3]^omm;zr)*!#@$\"3/)ePXe$yJ\"*F er7$$\"3%pJL$ezw5VFer$\"3'y[n(>&*=>**Fer7$$\"3s*)***\\PQ#\\\")Fer$\"3( y$*He@6=7\"F17$$\"3GKLLe\"*[H7F1$\"3O#HK'zoBE8F17$$\"3I*******pvxl\"F1 $\"39e\\<#*zP2;F17$$\"3#z****\\_qn2#F1$\"3K![Uq>67&>F17$$\"3U)***\\i&p @[#F1$\"3-qI/M)['[BF17$$\"3B)****\\2'HKHF1$\"3-#HzpRdX'GF17$$\"3ElmmmZ vOLF1$\"3-s(p'>:5&R$F17$$\"3i******\\2goPF1$\"3+0;#yZ`:.%F17$$\"3UKL$e R<*fTF1$\"3W,g/'*=pqYF17$$\"3m******\\)Hxe%F1$\"382FRqVMPaF17$$\"3ckm; H!o-*\\F1$\"3Z'*fgfNZBiF17$$\"3y)***\\7k.6aF1$\"3;izEjHL7rF17$$\"3#emm mT9C#eF1$\"3WA24X**oZ!)F17$$\"33****\\i!*3`iF1$\"35skAcZ>(4*F17$$\"3%Q LLL$*zym'F1$\"3%>_mNz%f<5F-7$$\"3wKLL3N1#4(F1$\"3/BHnW*3[8\"F-7$$\"3Nm m;HYt7vF1$\"3ay3c\"RRzD\"F-7$$\"3Y*******p(G**yF1$\"3cgYE]*HrP\"F-7$$ \"3]mmmT6KU$)F1$\"35B?\\\"pd3_\"F-7$$\"3fKLLLbdQ()F1$\"3#fq@!4R&el\"F- 7$$\"3[++]i`1h\"*F1$\"3dW\")\\M4\\1=F-7$$\"3W++]P?Wl&*F1$\"31ny7l*[r&> F-7$$\"\"\"F*$\"3*)***Hd(f6E@F--%'COLOURG6&%$RGBG$\"#5F)$F*F*Fa[l-%+AX ESLABELSG6$Q\"x6\"Q!Ff[l-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 57 "1. Explique os detalhes da especifica\347\343o do procedimento " }{TEXT 271 8 "cdquamin" }{TEXT -1 57 ". Em particular , o que significam express\365es do estilo " }{TEXT 263 13 "g[j](xin[ k])," }}{PARA 256 "" 0 "" {TEXT 264 37 "sum('g[i](xin[k])*fin[k]','k'= 1..m), " }{TEXT -1 4 "etc?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 265 26 "2. Utilize o procedimento " }{TEXT 273 9 "cdquamin \+ " }{TEXT 272 96 "para computar um ajuste c\372bico dos mesmos 11 ponto s. Primeiro ter\341 que estender ou redefinir o" }}{PARA 256 "" 0 " " {TEXT -1 32 "vetor g de fun\347\365es selecionadas." }}{PARA 0 "" 0 "" {TEXT -1 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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "3. (Exerc\355cio 6.4 Ruggiero & Lopes, C\341lcu lo Num\351rico, Ed 2a, 1997) A tabela abaixo mostra as alturas e peso s " }}{PARA 0 "" 0 "" {TEXT -1 107 "de uma amostra de nove homens entr e as ideades de 25 a 29 anos, extra\355da ao acaso entre funcion\341ri os de uma" }}{PARA 0 "" 0 "" {TEXT -1 17 "grande ind\372stria:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Altura \+ 183 173 168 188 158 163 193 163 178 cm" }}{PARA 0 "" 0 "" {TEXT -1 57 "Peso 79 69 70 81 61 63 79 71 73 kg" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 105 "3.1 Co nstrua o diagrama de dispers\343o destes dados e observe que parece ex istir uma rela\347\343o linear entre a" }}{PARA 256 "" 0 "" {TEXT -1 16 "altura e o peso." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 66 "3.2 Utilize o m\351todo dos quadrados m\355nimos para ajustar uma reta, " }{TEXT 285 37 "peso(altura) = alp ha1 + alpha2 altura" }{TEXT -1 17 ", que descreva o " }}{PARA 256 "" 0 "" {TEXT -1 61 "comportamento do peso em fun\347\343o da altura. D efina a fun\347\343o " }{TEXT 286 4 "peso" }{TEXT -1 50 " e o gr\341fi co para comparar o ajuste com o diagrama" }}{PARA 256 "" 0 "" {TEXT -1 13 "de dispers\343o." }}{PARA 0 "" 0 "" {TEXT -1 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 0 "" }}{PARA 0 "" 0 "" {TEXT 274 102 " 3.3 Estime o peso de um funcion\341rio de 175 cm de altura e a altura \+ de um funcion\341rio de 80 kg de peso." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 54 "Para o segundo pode utilizar o comand o solve do Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 26 "3.4 Ajuste agora uma reta " }{TEXT 284 36 "altura(peso) : = alpha1 + alpha2 peso" }{TEXT -1 46 " e estime novamente a altura de \+ um funcion\341rio" }}{PARA 256 "" 0 "" {TEXT -1 85 "de 80 kg e o peso \+ de outro de 175 cm. Explique porque as estimativas s\343o diferentes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}{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 0 "" }}{PARA 0 " " 0 "" {TEXT 280 26 "II. Lineariza\347\343o de dados " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "4. (Exerc\355cio 6.7 Ruggiero & Lopes, C\341lculo Num \351rico, Ed 2a, 1997) O n\372mero de bact\351rias por unidade de" }} {PARA 0 "" 0 "" {TEXT -1 68 "volume existente em uma cultura ap\363s x horas \351 apresentado a seguir." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 66 "tempo 0 1 2 3 4 5 6 horas " }}{PARA 0 "" 0 "" {TEXT -1 89 "bact\351rias 32 47 65 92 132 190 275 n\372mero de bact\351rias por u nidade de volume" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 94 "4.1 Construa o diagrama de dispers\343o destes dados e \+ verifique que a curva para ajustar estes " }}{PARA 0 "" 0 "" {TEXT 276 62 "dados e do tipo exponencial: bact(t) := alpha1 exp(alpha2 t). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 120 "A lineariza\347\343o de bact(t) resulta em ln(bac t(t)) = ln(alpha1 exp(alpha2 t)) = ln(alpha1) + alpha2 t. Define-se, \+ ent\343o, " }}{PARA 0 "" 0 "" {TEXT -1 56 "phi(t) := a1 + a2 t, onde a 1 = ln(alpha1) e a2 = alpha2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 277 75 "4.2 Determine o ajuste phi(t) dos dados \+ linearizados: compute a fun\347\343o phi." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 63 "4.3 Verifique que a escolha de lineariz a\347\343o foi adequada com o " }{TEXT 283 20 "teste de alinhamento" } {TEXT -1 3 ": " }}{PARA 256 "" 0 "" {TEXT -1 98 "construa o diagrama \+ de dispers\343o dos dados linearizados e compare com o ajuste lineariz ado obtido." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 278 117 "4.4 Finalmente, com base em phi(t) determine o a juste exponencial (i.e., defina a fun\347\343o bact(t)) dos dados inic iais." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 69 "4.5 Estime o n\372mero de bact\351r ias por unidade de volume ap\363s 24 horas." }{TEXT -1 3 " " }} {PARA 257 "" 0 "" {TEXT 281 30 "Quantas horas s\343o necess\341rias " }{TEXT -1 2 "pa" }{TEXT 282 29 "ra atingir um n\372mero de 10^6 " } {TEXT -1 32 "bact\351rias por unidade de volume?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 287 19 "III. Caso Cont\355nuo" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 149 "Neste caso a fun\347\343o f a ser aj ustada \351 conhecida cont\355nua num intervalo [inf,sup]. Por simpl icidade trabalharemos com um ajuste linear para somente" }}{PARA 0 "" 0 "" {TEXT -1 64 "duas fun\347\365es cont\355nuas nesse intervalo g1 e g2 pr\351-selecionadas." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 87 "O sistema linear a ser resolvido \351 A x = b, ond e A \351 uma matriz 2x2 com componentes Aij" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "Int(g[i](x)*g[j](x),x = inf .. sup);" "6#-%$In tG6$*&-&%\"gG6#%\"iG6#%\"xG\"\"\"-&F)6#%\"jG6#F-F./F-;%$infG%$supG" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "e o vector b \351 um 2-ve ctor com componentes bi" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "Int(g[i](x)* f(x),x = inf .. sup);" "6#-%$IntG6$*&-&%\"gG6#%\"iG6#%\"xG\"\"\"-%\"fG 6#F-F./F-;%$infG%$supG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 29 "Definir-se-\341 um procedimento " } {TEXT 288 13 "ccquamin2gs, " }{TEXT -1 25 "que de maneira similar ao" }{TEXT 290 1 " " }{TEXT -1 13 "procedimento " }{TEXT 289 8 "cdquamin" }{TEXT -1 18 " do caso discreto " }}{PARA 0 "" 0 "" {TEXT -1 109 "esta belece o sistema linear A x = b e invoca um m\351todo de solu\347\343o de sistemas lineares para obter o ajuste. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "Dito procedimento utilizar\341 \+ o comando " }{TEXT 291 3 "int" }{TEXT -1 79 " do Maple para computar a s integrais envolvidas e ter\341 os seguintes par\342metros:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "inf: limite inferior do intervalo;" }}{PARA 0 "" 0 "" {TEXT -1 36 "sup: limite \+ superior do intervalo;" }}{PARA 0 "" 0 "" {TEXT -1 28 "f: fun\347 \343o a ser ajustada;" }}{PARA 0 "" 0 "" {TEXT -1 44 "g: 2-vector de fun\347\365es g1 e g2 escolhidas;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "ccquamin2gs := proc(inf, s up, f, g) A := matrix(2,2); b := vector(2); for i to 2 do for j to 2 \+ do A[i,j] := int(g[i](k)*g[j](k),k=inf..sup); end do; b[i] := int(g[i] (k)*f(k),k=inf..sup); end do; print(`Sistema linear`);print(`A = `,e valm(A));print(`b = `,evalm(convert(convert(b,Vector[column]),Matrix )));print(`Solu\347\343o do sistema linear - coeficientes alpha - `);p vtriangGauss(A,b,2);end proc;" }}{PARA 7 "" 1 "" {TEXT -1 76 "`Warning , \\`A\\` is implicitly declared local to procedure \\`ccquamin2gs\\` \\n`" }}{PARA 7 "" 1 "" {TEXT -1 76 "`Warning, \\`b\\` is implicitly d eclared local to procedure \\`ccquamin2gs\\`\\n`" }}{PARA 7 "" 1 "" {TEXT -1 76 "`Warning, \\`i\\` is implicitly declared local to procedu re \\`ccquamin2gs\\`\\n`" }}{PARA 7 "" 1 "" {TEXT -1 76 "`Warning, \\` j\\` is implicitly declared local to procedure \\`ccquamin2gs\\`\\n`" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,ccquamin2gsGf*6&%$infG%$supG%\"fG %\"gG6&%\"AG%\"bG%\"iG%\"jG6\"F0C*>8$-%'matrixG6$\"\"#F7>8%-%'vectorG6 #F7?(8&\"\"\"F?F7%%trueGC$?(8'F?F?F7F@>&F36$F>FC-%$intG6$*&-&9'6#F>6#% \"kGF?-&FM6#FCFOF?/FP;9$9%>&F9FN-FH6$*&FKF?-9&FOF?FT-%&printG6#%/Siste ma~linearG-Fjn6$%'A~=~~~G-%&evalmG6#F3-Fjn6$%'b~=~~~G-Fao6#-%(convertG 6$-Fio6$F9&%'VectorG6#%'columnG%'MatrixG-Fjn6#%RSolu|by|^yo~do~sistema ~linear~-~coeficientes~alpha~-~G-%.pvtriangGaussG6%F3F9F7F0F0F0" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 292 87 "5 .1 Utilize o procedimento ccquamin2gs para ajustar a fun\347\343o 4x^3 no intervalo [-1..2]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 293 60 "5.2 Grafique ambas a fun\347\343o 4x^3 e o ajuste nesse intervalo." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 64 "6. Ajuste a fun\347\343o e^x no intervalo [0..5] com fun \347\365es da forma " }{TEXT 294 20 "alpha1 + alpha2 x^n " }{TEXT -1 17 " para n=1,2,3. " }}{PARA 256 "" 0 "" {TEXT -1 93 "Desenhe as gr \341ficas da fun\347\343o e^x e de todos os ajustes num mesmo gr\341fi co e compare. Qual o " }}{PARA 0 "" 0 "" {TEXT 295 14 "melhor ajuste? " }}{PARA 0 "" 0 "" {TEXT -1 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 "" {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 "" }}}}{MARK "62 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }