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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT 256 48 "MAT1154 - Equa\347\365es Diferenciais e de Diferen\347as
 " }}{PARA 256 "" 0 "" {TEXT 257 32 "Instabilidade e erro de m\341quin
a." }}{PARA 256 "" 0 "" {TEXT -1 3 "por" }}{PARA 256 "" 0 "" {TEXT -1 
17 "George Svetlichny" }}{PARA 256 "" 0 "" {TEXT 258 57 "\005\005\005M
\311TODOS NUM\311RICOS DE SOLU\307\303O DE ODE DE \010PRIMEIRA ORDEM" 
}}{PARA 256 "" 0 "" {TEXT 259 69 "Baseado nos scripts do site da Maple
soft e adapta\347\365es do Prof. Grivet" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "restart:with( DEtools \+
):with( plots ):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "A e.d.o. " }
{TEXT 261 7 "y'=2y-t" }{TEXT -1 26 " demonstra o fen\364mento de " }
{TEXT 260 15 "instabilidade. " }{TEXT -1 34 "Definimos primeiro o lado
 direito:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fun:=2*y(t)-t;
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Definimos agora a equa\347
\343o." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eq:=D(y)(t)=fun;
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Achamos a solu\347\343o geral
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sol:=dsolve(eq,y(t));
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Definimos uma condi\347\343o \+
inical em " }{TEXT 262 4 "y(0)" }{TEXT -1 10 " generica:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ci:=y(0)=a;" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 42 "Resolvemos agora com esta condi\347\343o inical:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sola:=dsolve([eq,ci],y(t));
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "A solu\347\343o para " }
{TEXT 274 5 "a=1/4" }{TEXT -1 3 " \351 " }{TEXT 275 0 "" }{TEXT 276 8 
"inst\341vel" }{TEXT 277 0 "" }{TEXT -1 80 " no sentido que todas as s
olu\347\365es pr\363ximas se afastam dela exponencialmente com " }
{TEXT 278 2 "t " }{TEXT -1 10 "crescendo." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 25 "Pegamos a fun\347\343o solu\347\343o:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 24 "funsol:=subs(sola,y(t));" }{TEXT -1 0 "" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "Plotamos agora as solu\347\365es \+
para o valor " }{TEXT 272 5 "exato" }{TEXT -1 4 " de " }{TEXT 273 1 "a
" }{TEXT -1 74 " da solu\347\343o inst\341vel e tambem para um milles
\355mo menos e um mill\351simo mais." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 100 "plot([subs(a=0.249,funsol),subs(a=0.25,funsol),subs(
a=0.251,funsol)],t=0..5,colour=[blue,red,blue]);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 128 "Vamos resolver agora a mesma equa\347\343o numeri
camente usando o m\351todo de Euler. Definimos o procedimento de Euler
. Nesta defini\347\343o " }{TEXT 280 1 "f" }{TEXT -1 31 " \351 a fun
\347\343o do lado direito de " }{TEXT 281 9 "y'=f(t,y)" }{TEXT -1 2 ",
 " }{TEXT 282 8 "(t0,y0) " }{TEXT -1 22 "\351 a condi\347\343o inicial
, " }{TEXT 283 1 "h" }{TEXT -1 25 " \351 o tamanho do passo, e " }
{TEXT 284 1 "T" }{TEXT -1 5 "  (> " }{TEXT 285 3 "t0)" }{TEXT -1 65 " \+
\351 o ponto final do intervalo onde procuramos a solu\347\343o n\372m
erica." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Euler := proc( f,
 t0, y0, h, T )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "  local i, L, X \+
,N; N:=round((T-t0)/h);" }{TEXT -1 1 " " }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 27 "  X := evalf( [ t0, y0 ] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "  L := X;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "  for i from 1 to N
 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "    X := X + [ h, h*f(op(X))
 ];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "    L := L, X;" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 9 "  end do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
11 "  return L;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{MPLTEXT 1 0 9 "end \+
proc;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Definimos
 a fun\347\343o " }{TEXT 263 6 "2*y-t:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "func:=(t,y)->2*y-t;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 63 "Chamamos agora o Euler com as mesmas condi\347\365es iniciais, \+
usand " }{TEXT 288 5 "h=0,5" }{TEXT -1 15 " e ponto final " }{TEXT 
289 1 "5" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 
"aa:=Euler(func,0,0.249,0.5,5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 30 "bb:=Euler(func,0,0.250,0.5,5);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "cc:=Euler(func,0,0.251,0.5,5);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 28 "Plotamos os tr\352s resultados." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 46 "plot([[aa],[bb],[cc]],colour=[blue,red,blue]
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Refazemos tudo com " }
{TEXT 264 5 "h=0.1" }{TEXT -1 39 ": (Sem mostrar o resultado das conta
s)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "aa:=Euler(func,0,0.2
49,0.1,5):bb:=Euler(func,0,0.250,0.1,5):cc:=Euler(func,0,0.251,0.1,5):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([[aa],[bb],[cc]],c
olour=[blue,red,blue]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 231 "Note \+
que neste caso o Euler d\341 resultado bastante bom se come\347ar com \+
a condi\347\343o incial da solu\347\343o inst\341vel. Isto \351 porque
 a solu\347\343o inst\341vel \351 uma fun\347\343o linear. Vamos agora
 complicar um pouco a equa\347\343o e repetimos os passos acima:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "fun:=2*y(t)-sin(t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eq:=D(y)(t)=fun;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sol:=dsolve(eq,y(t));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ci:=y(0)=a;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 27 "sola:=dsolve([eq,ci],y(t));" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 32 "A solu\347\343o inst\341vel \351 aquela com " }
{TEXT 286 5 "a=1/5" }{TEXT -1 41 ". A solu\347\343o inst\341vel em nun
ca ultrapassa " }{TEXT 287 3 "2/5" }{TEXT -1 11 " em m\363dulo." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "funsol:=subs(sola,y(t));" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Plotamos as solu\347\365es exatas
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "plot([subs(a=0.199,fu
nsol),subs(a=0.200,funsol),subs(a=0.201,funsol)],t=0..4,colour=[blue,r
ed,blue]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Agora vamos calcula
r com m\351todo de Euler com as mesmas condi\347\365es iniciais." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "func:=(t,y)->2*y-sin(t);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "aa:=Euler(func,0,0.199,0.1,
4):bb:=Euler(func,0,0.200,0.1,4):cc:=Euler(func,0,0.201,0.1,4):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot([[aa],[bb],[cc]],colour=[blue,
red,blue]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 105 "Compare os dois gr\341ficos, o exato e o
 obtido pelo m\351todo de Euler. Mesmo come\347ando com a condi\347
\343o incial " }{TEXT 265 5 "exata" }{TEXT -1 43 " da solu\347\343o in
st\341vel, a solu\347\343o num\351rica \351 " }{TEXT 266 5 "muito" }
{TEXT -1 98 " diferente. Tudo isto \351 devido ao erro de arredondamen
to (erro da m\341quina) na avalia\347\343o da fun\347\343o " }{TEXT 
267 6 "sin(t)" }{TEXT -1 24 ". Mude agora o valor de " }{TEXT 268 7 "h
 (0.1)" }{TEXT -1 1 " " }{TEXT 269 2 ") " }{TEXT -1 5 "para " }{TEXT 
270 4 "0.01" }{TEXT -1 1 " " }{TEXT -1 73 "nas chamada da rotina de Eu
iler acima. O resultado melhora mas n\343o muito." }}}}{MARK "0 4 0" 
17 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }