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}{PSTYLE "Title " 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 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 {EXCHG {PARA 18 "" 0 "" {TEXT -1 26 "Interpolazione polinomial e" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 18 "Background teorico" }} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 "I metodi usati sono descritti nel libro di J. Stoer e R. Bulirsch \"Int roduction to numerical analysis\", 2nd Edition, Springer-Verlag (1993) ." }}{PARA 0 "" 0 "" {TEXT -1 37 "Il problema quello di trovare, dati " }{TEXT 266 3 "n+1" }{TEXT -1 7 " punti " }{TEXT 267 1 "(" } {XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 0 "" }{TEXT 268 1 " ," }{XPPEDIT 18 0 "y[1];" "6#&%\"yG6#\"\"\"" }{TEXT -1 0 "" }{TEXT 269 7 "),...,(" }{XPPEDIT 18 0 "x[n+1];" "6#&%\"xG6#,&%\"nG\"\"\"\"\" \"F(" }{TEXT -1 0 "" }{TEXT 271 1 "," }{TEXT -1 0 "" }{XPPEDIT 18 0 "y [n+1];" "6#&%\"yG6#,&%\"nG\"\"\"\"\"\"F(" }{TEXT -1 0 "" }{TEXT 270 2 ") " }{TEXT -1 13 "un polinomio " }{XPPEDIT 18 0 "P[n](t);" "6#-&%\"PG 6#%\"nG6#%\"tG" }{TEXT -1 28 " di grado minore o uguale a " }{TEXT 272 1 "n" }{TEXT -1 10 " tale che " }{XPPEDIT 18 0 "P[n](x[i]) = y[i]; " "6#/-&%\"PG6#%\"nG6#&%\"xG6#%\"iG&%\"yG6#F-" }{TEXT -1 5 " per " } {TEXT 273 9 "i=1...n+1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 144 "Si pu dimostrare con vari metodi che, qualunque sia la scelta del le ascisse (purch diverse fra loro) tale problema ha sempre un'unica s oluzione." }}{PARA 0 "" 0 "" {TEXT -1 77 "Il problema che ci poniamo q ua di trovare tale polinomio in modo efficiente." }}{PARA 0 "" 0 "" {TEXT -1 59 "L'idea di base di scrivere un generico polinomio di grad o " }{TEXT 258 1 "n" }{TEXT -1 12 " nella forma" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }{TEXT 259 13 "P(t) = ((...(" }{XPPEDIT 18 0 "a[n+1];" " 6#&%\"aG6#,&%\"nG\"\"\"\"\"\"F(" }{TEXT -1 0 "" }{TEXT 260 3 "(t-" } {XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG" }{TEXT -1 0 "" }{TEXT 261 2 ") +" }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 0 "" }{TEXT 262 4 ")(t-" }{XPPEDIT 18 0 "x[n-1];" "6#&%\"xG6#,&%\"nG\"\"\"\"\"\"!\"\" " }{TEXT -1 0 "" }{TEXT 263 6 ")+...+" }{XPPEDIT 18 0 "a[2];" "6#&%\"a G6#\"\"#" }{TEXT -1 0 "" }{TEXT 278 4 ")(t-" }{XPPEDIT 18 0 "x[1];" "6 #&%\"xG6#\"\"\"" }{TEXT -1 0 "" }{TEXT 264 2 ")+" }{XPPEDIT 18 0 "a[1] ;" "6#&%\"aG6#\"\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "dov e " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 0 "" }{TEXT 265 5 ",...," }{XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 "x[n+1];" "6#&%\"xG6#,&%\"nG\"\"\"\"\"\"F(" }{TEXT -1 81 " sono i punti su cui si vuole compiere l'interpolazione che sup porremo ordinati: " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 0 "" }{TEXT 275 1 "<" }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" } {TEXT -1 1 "<" }{TEXT 277 4 "...<" }{TEXT -1 0 "" }{TEXT 276 0 "" } {XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG" }{TEXT -1 1 "<" }{XPPEDIT 18 0 "x[n+1];" "6#&%\"xG6#,&%\"nG\"\"\"\"\"\"F(" }{TEXT 274 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 60 "Scritto in questo modo il problema molto pi semplice perch " }{XPPEDIT 18 0 "P(x[1]) = a[1];" "6#/-%\"PG6#&%\" xG6#\"\"\"&%\"aG6#\"\"\"" }{TEXT -1 57 " e quindi dalla condizione del ll'interpolazione si trova " }{XPPEDIT 18 0 "a[1] = y[1];" "6#/&%\"aG6 #\"\"\"&%\"yG6#\"\"\"" }{TEXT -1 14 "; poi abbiamo " }{XPPEDIT 18 0 "P (x[2]) = a[2](x[2]-x[1])+a[1];" "6#/-%\"PG6#&%\"xG6#\"\"#,&-&%\"aG6#\" \"#6#,&&F(6#\"\"#\"\"\"&F(6#\"\"\"!\"\"F6&F.6#\"\"\"F6" }{TEXT -1 24 " ; ponendo cio' uguale a " }{XPPEDIT 18 0 "y[2];" "6#&%\"yG6#\"\"#" } {TEXT -1 18 " e ricordando che " }{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\" \"\"" }{TEXT -1 58 " e' stato trovato al passo precedente si trova fac ilmente " }{XPPEDIT 18 0 "a[2];" "6#&%\"aG6#\"\"#" }{TEXT -1 14 "; e c osi' via." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 37 "Definizione ricors iva di un polinomio" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 194 "Per scrive re un programma dobbiamo precisare cosa vuol dire \"e cosi' via\" nell a frase precedente. In realt dobbiamo prima di tutto precisare cosa vo gliono dire quei puntini nella definizione di " }{TEXT 256 6 "P(t). " }}{PARA 0 "" 0 "" {TEXT -1 62 "Cominciamo dal caso di un polinomio scr itto nel modo \"normale\"" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "P(t) = a [1]+a[2];" "6#/-%\"PG6#%\"tG,&&%\"aG6#\"\"\"\"\"\"&F*6#\"\"#F-" } {TEXT -1 0 "" }{TEXT 257 4 "t + " }{XPPEDIT 18 0 "a[3];" "6#&%\"aG6#\" \"$" }{TEXT -1 1 " " }{XPPEDIT 18 0 "t^2;" "6#*$%\"tG\"\"#" }{TEXT -1 5 "+...+" }{XPPEDIT 18 0 "a[n+1];" "6#&%\"aG6#,&%\"nG\"\"\"\"\"\"F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "t^n;" "6#)%\"tG%\"nG" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 122 "(Notate che, siccome difficile, almeno per me, far partire gli indici degi array da 0, ho fatto partire i co efficienti da" }{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"\"" }{TEXT -1 75 " anche se, dal punto di vista matematico, sarebbe piu' naturale parti re da " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }{TEXT -1 2 ")." }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "La regola di Horner" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Dal punto di vista computazionale, la pre cedente definizione di " }{TEXT 279 4 "P(t)" }{TEXT -1 65 " non conve niente. E' molto migliore la definizione (equivalente)" }}{PARA 258 " " 0 "" {TEXT -1 1 " " }{TEXT 280 13 "P(t) = ((...(" }{XPPEDIT 18 0 "a[ n+1];" "6#&%\"aG6#,&%\"nG\"\"\"\"\"\"F(" }{TEXT -1 1 " " }{TEXT 281 4 "t + " }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 0 "" }{TEXT 282 1 ")" }{TEXT -1 0 "" }{TEXT 283 7 "t +...+" }{XPPEDIT 18 0 "a[2]; " "6#&%\"aG6#\"\"#" }{TEXT -1 0 "" }{TEXT 284 3 ")t+" }{XPPEDIT 18 0 " a[1];" "6#&%\"aG6#\"\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 285 11 "Esercizio: " }{TEXT -1 21 "per alcuni valori di " } {TEXT 286 1 "n" }{TEXT -1 69 ", calcolate il numero di operazioni nece ssarie per la valutazione di " }{TEXT 287 4 "P(t)" }{TEXT -1 82 " seco ndo la definizione originale e secondo quest'ultima (detta schema di H orner)." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "Funzioni ricorsive" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 193 "Senza volere entrare in definiz ioni rigorose, si intende per definizione di funzione ricorsiva, una f unzione definita sui naturali (o su qualche altro insieme con la stess a struttura) tale che " }{TEXT 288 4 "f(n)" }{TEXT -1 22 " sia definit o tramite " }{TEXT 289 6 "f(n-1)" }{TEXT -1 42 ". Naturalmente anche \+ necessario definire " }{TEXT 290 5 "f(1) " }{TEXT -1 3 "(o " }{TEXT 291 4 "f(0)" }{TEXT -1 40 ") in modo da avere un punto di partenza." } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "l'esempio pi classico quello de l fattoriale. Eccone una definizione in modo ricorsivo:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "fatt := n -> if n=0 then 1 else fat t(n-1)*n fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fattGR6#%\"nG6\"6$% )operatorG%&arrowGF(@%/9$\"\"!\"\"\"*&-F$6#,&F.F0!\"\"F0F0F.F0F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "l'istruzione \"fi\" quella che chiude le istruzione subordinate a \"if\"" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "fatt(4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#C" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "fatt(20);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"4++kw\"3?!HV#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "fatt (-5);" }}{PARA 8 "" 1 "" {TEXT -1 45 "Error, (in fatt) too many levels of recursion" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "Nell'ultimo caso Maple protesta perch per calcolare fatt(-5) c alcola f(-6), poi f(-7) e cos via, senza mai giungere a una fine." }}} }{SECT 0 {PARA 4 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 37 "La regola di Horner in modo ricorsivo" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 59 "pol := i -> if i=0 then a[n+1] else pol(i-1) *t+a[n+1-i] fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$polGR6#%\"iG6\"6 $%)operatorG%&arrowGF(@%/9$\"\"!&%\"aG6#,&%\"nG\"\"\"F5F5,&*&-F$6#,&F. F5!\"\"F5F5%\"tGF5F5&F16#,(F4F5F5F5F.F;F5F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "fissato n, prescriviamo cosa bisogna fare al passo \+ i della valutazione di \"pol\": al passo 0 calcoliamo " }{XPPEDIT 18 0 "a[n+1];" "6#&%\"aG6#,&%\"nG\"\"\"\"\"\"F(" }{TEXT -1 76 " . Ai pass i successivi moltiplichiamo il risultato del passo precedente per " } {TEXT 292 2 "t " }{TEXT -1 55 "e poi aggiungiamo il coefficiente neces sario. Al passo " }{TEXT 293 1 "n" }{TEXT -1 35 " troveremo il valore \+ del polinomio." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "Per vedere se f unziona, dobbiamo prima di tutto fissare " }{TEXT 294 1 "n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "n := 5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "pol(0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"aG6#\"\"'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "pol(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&*&,&*&&%\"aG6#\"\"'\"\"\"%\"tGF-F-&F*6#\"\"&F-F-F .\"\"\"F-&F*6#\"\"%F-F-F.F2F-&F*6#\"\"$F-" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 7 "pol(5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&*&, &*&,&*&,&*&&%\"aG6#\"\"'\"\"\"%\"tGF1F1&F.6#\"\"&F1F1F2\"\"\"F1&F.6#\" \"%F1F1F2F6F1&F.6#\"\"$F1F1F2F6F1&F.6#\"\"#F1F1F2F6F1&F.6#F1F1" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "per ora questa una scrittura solt anto formale e NON una funzione di t." }}{PARA 0 "" 0 "" {TEXT -1 57 "Per renderla una funzione di t, c' l'istruzione \"unapply\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "P := unapply(pol(5),'t');" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGR6#%\"tG6\"6$%)operatorG%&arrow GF(,&*&,&*&,&*&,&*&,&*&&%\"aG6#\"\"'\"\"\"9$F:F:&F76#\"\"&F:F:F;\"\"\" F:&F76#\"\"%F:F:F;F?F:&F76#\"\"$F:F:F;F?F:&F76#\"\"#F:F:F;F?F:&F76#F:F :F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "P(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.&%\"aG6#\"\"'\"$V#&F%6#\"\"&\"#\")&F%6#\" \"%\"#F&F%6#\"\"$\"\"*&F%6#\"\"#F3&F%6#\"\"\"F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "a questo punto, se vogliamo possiamo dare dei valori ai coefficienti e calcolare P." }}{PARA 0 "" 0 "" {TEXT -1 10 "Ad ese mpio" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "a := array(1..n+1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG-%&arrayG6$;\"\"\"\"\"'7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "a[1]:=1; a[2]:=-1; a[3]:= 0; a[4]:=4; a[5]:=-1; a[6]:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&% \"aG6#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"#!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"$\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6# \"\"'\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "P(3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"$o#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "P(-3.2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!)#RFn&! \"&" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 53 "Polinomi nella forma piu ' comoda per l'interpolazione" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "C ome detto, vogliamo ora passare alla forma" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }{TEXT 295 13 "P(t) = ((...(" }{XPPEDIT 18 0 "a[n+1];" "6#&%\" aG6#,&%\"nG\"\"\"\"\"\"F(" }{TEXT -1 0 "" }{TEXT 296 3 "(t-" } {XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG" }{TEXT -1 0 "" }{TEXT 297 2 ") +" }{XPPEDIT 18 0 "a[n];" "6#&%\"aG6#%\"nG" }{TEXT -1 0 "" }{TEXT 298 4 ")(t-" }{XPPEDIT 18 0 "x[n-1];" "6#&%\"xG6#,&%\"nG\"\"\"\"\"\"!\"\" " }{TEXT -1 0 "" }{TEXT 299 6 ")+...+" }{XPPEDIT 18 0 "a[2];" "6#&%\"a G6#\"\"#" }{TEXT -1 0 "" }{TEXT 301 4 ")(t-" }{XPPEDIT 18 0 "x[1];" "6 #&%\"xG6#\"\"\"" }{TEXT -1 0 "" }{TEXT 300 2 ")+" }{XPPEDIT 18 0 "a[1] ;" "6#&%\"aG6#\"\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "de scritta all'inizio." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Usiamo lo \+ stesso metodo dello schema di Horner." }}{PARA 0 "" 0 "" {TEXT -1 75 " Prima di tutto, per usare una scrittura formale, \"deassegnamo\" i val ori di " }{TEXT 302 1 "n" }{TEXT -1 3 " e " }{TEXT 303 1 "a" }{TEXT -1 12 " usati prima" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "unas sign('n','a');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "a[2];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"aG6#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "e ora definiamo \"pol\" in modo simile a quanto fatto \+ per i polinomi \"normali\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "pol := i -> if i=0 then a[n+1] else pol(i-1)*(t-x[n+1-i])+a[n+1-i] fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$polGR6#%\"iG6\"6$%)operator G%&arrowGF(@%/9$\"\"!&%\"aG6#,&%\"nG\"\"\"F5F5,&*&-F$6#,&F.F5!\"\"F5F5 ,&%\"tGF5&%\"xG6#,(F4F5F5F5F.F;F;F5F5&F1F@F5F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "n:=5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"nG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "pol(5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&*&,&*&,&*&,&*&&%\"aG6#\"\"'\"\" \",&%\"tGF1&%\"xG6#\"\"&!\"\"F1F1&F.F6F1F1,&F3F1&F56#\"\"%F8F1F1&F.F " 0 "" {MPLTEXT 1 0 23 "P \+ := unapply(pol(5),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"PGR6#%\"t G6\"6$%)operatorG%&arrowGF(,&*&,&*&,&*&,&*&,&*&&%\"aG6#\"\"'\"\"\",&9$ F:&%\"xG6#\"\"&!\"\"F:F:&F7F?F:F:,&F6#\"\"%FAF:F:&F7FEF:F:,&F6#\"\"$FAF:F:&F7FJF:F:,&F6#\"\"#FAF:F:&F7FOF:F:,&F6#F:F AF:F:&F7FTF:F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Se ora vogl iamo cambiare " }{TEXT 304 1 "n" }{TEXT -1 39 ", dobbiamo prima di tut to, riassegnare " }{TEXT 305 1 "n" }{TEXT -1 19 " e poi ricalcolare " }{TEXT 306 3 "pol" }{TEXT -1 24 " con il nuovo argomento." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "n := 20;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "pol(20);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,&*&,&*&,&*&,&*&,&*&,&*&, &*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&&%\"aG6#\"#@\" \"\",&%\"tGFO&%\"xG6#\"#?!\"\"FOFO&FLFTFOFO,&FQFO&FS6#\"#>FVFOFO&FLFZF OFO,&FQFO&FS6#\"#=FVFOFO&FLFinFOFO,&FQFO&FS6#\"# " 0 "" {MPLTEXT 1 0 5 "P(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&*&,&*&,&*&,&*&&%\"aG6#\"\"'\"\"\",&\"\"$F 1&%\"xG6#\"\"&!\"\"F1F1&F.F6F1F1,&F3F1&F56#\"\"%F8F1F1&F.F " 0 "" {MPLTEXT 1 0 76 "pol := (i,n) -> if i=0 then a[n+1] else pol(i-1,n)*(t-x[n+1-i] )+a[n+1-i] fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$polGR6$%\"iG%\"nG 6\"6$%)operatorG%&arrowGF)@%/9$\"\"!&%\"aG6#,&9%\"\"\"F6F6,&*&-F$6$,&F /F6!\"\"F6F5F6,&%\"tGF6&%\"xG6#,(F5F6F6F6F/F " 0 "" {MPLTEXT 1 0 9 "pol(5,5);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,&*&,&*&,&*&,&*&,&*&&%\"aG6#\"\"'\"\"\",&%\"tGF1&%\"x G6#\"\"&!\"\"F1F1&F.F6F1F1,&F3F1&F56#\"\"%F8F1F1&F.F " 0 "" {MPLTEXT 1 0 11 "pol(20,20);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,& *&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&&%\"aG6#\"#@\"\"\",&%\"tGFO& %\"xG6#\"#?!\"\"FOFO&FLFTFOFO,&FQFO&FS6#\"#>FVFOFO&FLFZFOFO,&FQFO&FS6# \"#=FVFOFO&FLFinFOFO,&FQFO&FS6#\"# " 0 "" {MPLTEXT 1 0 25 "P := unapply(pol(n,n),t);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"PGR6#%\"tG6\"6$%)operatorG%&arrow GF(,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&*&,&* &,&*&,&*&,&*&&%\"aG6#\"#@\"\"\",&9$FX&%\"xG6#\"#?!\"\"FXFX&FUFgnFXFX,& FZFX&Ffn6#\"#>FinFXFX&FUF]oFXFX,&FZFX&Ffn6#\"#=FinFXFX&FUFboFXFX,&FZFX &Ffn6#\"# " 0 "" {MPLTEXT 1 0 6 "n :=5 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "P := unapply(pol(n,n),t);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"PGR6#%\"tG6\"6$%)operatorG%&arrowGF(,&*&,&*&,&*&, &*&,&*&&%\"aG6#\"\"'\"\"\",&9$F:&%\"xG6#\"\"&!\"\"F:F:&F7F?F:F:,&F6#\"\"%FAF:F:&F7FEF:F:,&F6#\"\"$FAF:F:&F7FJF:F:,&F6#\"\" #FAF:F:&F7FOF:F:,&F6#F:FAF:F:&F7FTF:F(F(F(" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 25 "ora vediamo come rendere " }{TEXT 311 1 "P" } {TEXT -1 24 " una funzione \"numerica\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "Diamo dei valori a " }{TEXT 312 1 "x" }{TEXT -1 12 ". Ad \+ esempio" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "x:= seq(i,i=1..n );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG6'\"\"\"\"\"#\"\"$\"\"%\" \"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "e ad " }{TEXT 313 1 "a" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "a := seq(1,i=1..n+1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG6(\"\"\"F&F&F&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "P(3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "P(t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&*&,&*&,&*$),&%\"tG\"\"\"!\"%F.\" \"#\"\"\"F.F.F.F.,&F-F.!\"$F.F.F.F.F.F.,&F-F.!\"#F.F.F.F.F.F.,&F-F.!\" \"F.F.F.F.F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "Maple ha semplifi cato l'inizio dell'espressione " }{XPPEDIT 18 0 "t-5+1(t-4);" "6#,(%\" tG\"\"\"\"\"&!\"\"-\"\"\"6#,&F$F%\"\"%F'F%" }{TEXT -1 4 " in " } {XPPEDIT 18 0 "(t-4)^2;" "6#*$,&%\"tG\"\"\"\"\"%!\"\"\"\"#" }{TEXT -1 3 "..." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "plot(P(t),t=0..n+ 1);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'CURVESG6$7V7$\"\"!$!$+\"F(7$$\" 1+++]#HyI\"!#;$!1&3*z!G%pps!#97$$\"1++]([kdW#F.$!1#3!*\\rY:O&F17$$\"1+ ++v;\\DPF.$!1H$H'yAMfOF17$$\"1+++Dq0]\"FK$\"1!el8#\\@nRFK7$$\"1+++DM^I;FK$\"1h5BPS&y]$FK7$$ \"1+++0ytb4')>FK7$$\"1+++!y?#>@FK$\"13H#e$)*R>)*\\#FK$\" 1IND-5lI?FK7$$\"1++DEP/BEFK$\"1H*)\\SfMIDFK7$$\"1++](o:;v#FK$\"16\"*Qd '=:B$FK7$$\"1++v$)[opGFK$\"173r'*>58SFK7$$\"1++]i%Qq*HFK$\"1G$RT1Mj(\\ FK7$$\"1++vQIKHJFK$\"1=V%=#yu!3'FK7$$\"1++D^rZWKFK$\"1S!HOzX.6(FK7$$\" 1++]Zn%)oLFK$\"1!\\Yr6#*QG)FK7$$\"1+++5FL(\\$FK$\"1.-J*3jic*FK7$$\"1++ ]d6.BOFK$\"1\\+-X?'44\"F17$$\"1++vo3lWPFK$\"1>qYJM8L7F17$$\"1++]A))ozQ FK$\"1S]**G;K79F17$$\"1+++Ik-,SFK$\"1fW#>IYw7&FK$\"1m6Kp['G/)F17$$\"1++v)Q? QD&FK$\"1gwHbt=L**F17$$\"1+++5jyp`FK$\"1C'p#H+m/7!#87$$\"1++]Ujp-bFK$ \"1Ga(eEh()\\\"F_y7$$\"1+++gEd@cFK$\"1-Q%4*z9;=F_y7$$\"1+]PMh%\\o&FK$ \"1I!GCHs*3?F_y7$$\"1++v3'>$[dFK$\"1yX87(p(>AF_y7$$\"1+++5h(*3eFK$\"1C gxR'G&RCF_y7$$\"1++D6EjpeFK$\"1m)eA!\\1yEF_y7$$\"1,]i0j\"[$fFK$\"1NDUV TrcHF_y7$$\"\"'F($\"$E$F(-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELS G6$Q\"t6\"%!G-%%VIEWG6$;F(Fd[l%(DEFAULTG" 2 382 382 382 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 0 115 0 0 0 0 0 0 }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 3 "" 0 "" {TEXT -1 52 "Calcolo \+ dei coefficienti del polinomio interpolatore" }}{EXCHG {PARA 0 "" 0 " " {TEXT -1 121 "Il metodo piu' semplice e' tramite le cosiddette diffe renze divise (spiegate in dettaglio nel libro di Stoer e Bulirsch)." } }{PARA 0 "" 0 "" {TEXT -1 94 "Senza volere giustificare (ma non e' mol to difficile) tale metodo, l'algoritmo e' il seguente:" }}{PARA 0 "" 0 "" {TEXT -1 70 "si costruisce una matrice f. Sulla prima colonna, si mettono i valori " }{XPPEDIT 18 0 "y[1];" "6#&%\"yG6#\"\"\"" }{TEXT -1 5 ",...," }{XPPEDIT 18 0 "y[n+1];" "6#&%\"yG6#,&%\"nG\"\"\"\"\"\"F( " }{TEXT -1 169 "; sulle colonne successive si mettono le opportune \" differenze divise\" dei termini della colonna precedente. Alla fine gl i elementi sulla prima riga sono i coefficienti " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a[1];" "6#&%\"aG6#\"\"\"" }{TEXT -1 5 " ,...," }{XPPEDIT 18 0 "a[n+1];" "6#&%\"aG6#,&%\"nG\"\"\"\"\"\"F(" } {TEXT -1 29 " del polinomio interpolatore." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 314 10 "Esercizio:" }{TEXT -1 40 " giustificare questo \+ metodo, almeno per " }{TEXT 315 2 "n " }{TEXT -1 67 "piccolo. (Uno dei metodi consiste nel mostrare che i termini sulla " }{TEXT 316 1 "i" } {TEXT -1 72 "-esima riga sono i coefficienti del polinomio che interpo la i punti da (" }{XPPEDIT 18 0 "x[i],y[i];" "6$&%\"xG6#%\"iG&%\"yG6#F &" }{TEXT -1 9 ") in poi)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Cost ruiamo la matrice " }{TEXT 317 1 "f" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "f := array(1..n+1,1..n+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG-%&arrayG6%;\"\"\"\"\"'F(7\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 7 "f[1,1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&% \"fG6$\"\"\"F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "y := arra y (1..n+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"yG-%&arrayG6$;\"\" \"\"\"'7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "for i from 1 \+ to n+1 do f[i,1] := y[i] od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"f G6$\"\"\"F'&%\"yG6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6$\"\" #\"\"\"&%\"yG6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6$\"\"$\" \"\"&%\"yG6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6$\"\"%\"\"\" &%\"yG6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6$\"\"&\"\"\"&%\" yG6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"fG6$\"\"'\"\"\"&%\"yG6# F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "poiche' 'x' lo ho gia' asse gnato, lo devo \"deaasegnare\" per avere una formula generale" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " unassign('x');" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "adesso e' il passo fondamentale: la formu la delle differenze divise\"" }}{PARA 0 "" 0 "" {TEXT -1 235 "Vi e' un doppio \"loop\": per ogni colonna dalla seconda in poi calcolo gli el ementi di ogni riga. Il numero di righe diminuisce a ogni colonna (in \+ realta' la matrice e' triangolare, non nel senso che si usa nei corsi \+ di algebra lineare)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "for j from 2 to n+1 do for i from 1 to n+2-j do f[i,j] := (f[i+1,j-1] - f [i,j-1])/(x[i+j-1] - x[i]) od od;" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 "Alcuni valori della matrice" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f[2,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&&%\"yG6 #\"\"$\"\"\"&F&6#\"\"#!\"\"\"\"\",&&%\"xGF'F)&F1F+F-!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f[4,2];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&&%\"yG6#\"\"&\"\"\"&F&6#\"\"%!\"\"\"\"\",&&%\"xGF'F )&F1F+F-!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "f[3,3];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&,&&%\"yG6#\"\"&\"\"\"&F(6#\"\"%! \"\"\"\"\",&&%\"xGF)F+&F3F-F/!\"\"F+*&,&F,F+&F(6#\"\"$F/F0,&F4F+&F3F9F /F5F/F0,&F2F+F " 0 "" {MPLTEXT 1 0 7 "f[1,6 ];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,&*&,&*&,&*&,&*&,&&%\"yG6#\"\" '\"\"\"&F.6#\"\"&!\"\"\"\"\",&&%\"xGF/F1&F9F3F5!\"\"F1*&,&F2F1&F.6#\" \"%F5F6,&F:F1&F9F?F5F;F5F6,&F8F1FBF5F;F1*&,&FF1&F.6#\"\"$F5F6 ,&FBF1&F9FIF5F;F5F6,&F:F1FLF5F;F5F6,&F8F1FLF5F;F1*&,&FDF1*&,&FFF1*&,&F HF1&F.6#\"\"#F5F6,&FLF1&F9FVF5F;F5F6,&FBF1FYF5F;F5F6,&F:F1FYF5F;F5F6,& F8F1FYF5F;F1*&,&FOF1*&,&FQF1*&,&FSF1*&,&FUF1&F.6#F1F5F6,&FYF1&F9F`oF5F ;F5F6,&FLF1FboF5F;F5F6,&FBF1FboF5F;F5F6,&F:F1FboF5F;F5F6,&F8F1FboF5F; " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 23 "Poniamo i coefficienti " } {TEXT 319 1 "a" }{TEXT -1 38 " uguali alla prima riga della matrice " }{TEXT 318 1 "f" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "a:= array (1..n+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG-%&arrayG6$;\"\"\" \"\"'7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "for j from 1 to n+1 do a[j] := f[1,j] od:" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 35 " Calcolo del polinomio interpolatore" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Se adesso calcoliamo " }{TEXT 320 9 "pol(5,5) " }{TEXT -1 30 "v erranno usati i coefficienti " }{XPPEDIT 18 0 "a[i];" "6#&%\"aG6#%\"iG " }{TEXT -1 17 " appena calcolati" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "pol(5,5);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,&*&,&*&, &*&,&*&,&*&*&,&*&,&*&,&*&,&*&,&&%\"yG6#\"\"'\"\"\"&F86#\"\"&!\"\"\"\" \",&&%\"xGF9F;&FCF=F?!\"\"F;*&,&F " 0 "" {MPLTEXT 1 0 18 "x:= seq(i,i=0..5) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG6(\"\"!\"\"\"\"\"#\"\"$\"\" %\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "for i from 1 to 6 do y[i] := cos(x[i]) od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6# \"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"#-%$cosG6#\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"$-%$cosG6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"%-%$cosG6#\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"&-%$cosG6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"yG6#\"\"'-%$cosG6#\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "pol(5,5);" }}{PARA 12 "" 1 "" {XPPMATH 20 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tG#!$P\"\"#g*$)F%\"\"%\"\"\"#F$\"\")*$)F%\"\"&F,#!\"\"\"$?\"*&-%$cosG6 #F1F$F0F,#F$F4*&-F76#F+F$F0F,#F3\"#C*&-F76#\"\"$F$F0F,#F$\"#7*&-F76#\" \"#F$F0F,#F3FD*&-F76#F$F$F0F,#F$F>*$)F%FBF,#!#*&F6F,F%F$#F$F1*&F;F, F%F,#!\"&F+*&F@F,F%F,#\"#5FB*&FFF,F%F,FV*&FKF,F%F,F1*&F6F,)F%FHF,#FVFD *&F;F,FgnF,#\"#hF>*&F@F,FgnF,#!#8FH*&FFF,FgnF,#\"$2\"FD*&FKF,FgnF,#!#x FD*$FgnF,#\"#:F.*&F6F,FOF,#\"\"(F>*&F;F,FOF,#!#TF>*&F@F,FOF,#\"#\\FD*& FFF,FOF,#!#fFD*&FKF,FOF,#\"#rF>*&F6F,F*F,FI*&F;F,F*F,#\"#6F>*&F@F,F*F, F3*&FFF,F*F,#\"#8FD*&FKF,F*F,#!\"(FD" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Ho dovuto mettere " }{TEXT 322 1 "x" }{TEXT -1 72 " fra [] perc he' era una sequenza e dovevo trasformarlo in array, mentre " }{TEXT 323 1 "y" }{TEXT -1 30 " era a posto (almeno credo...)" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 27 "confronto col nostro metodo" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "polMaple := unapply(interp([x],y,t) ,t);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)polMapleGR6#%\"tG6\"6$%)ope ratorG%&arrowGF(,jn\"\"\"F-9$#!$P\"\"#g*&-%$cosG6#\"\"%F-)F.\"\"&\"\" \"#!\"\"\"#C*&-F46#\"\"$F-F7F9#F-\"#7*&-F46#\"\"#F-F7F9#F;FB*&-F46#F-F -F7F9#F-F<*&-F46#F8F-F.F-#F-F8*&F3F9F.F9#!\"&F6*&F>F9F.F9#\"#5F@*&FDF9 F.F9FR*&FIF9F.F9F8*&FMF9)F.FFF9#FRFB*&F3F9FYF9#\"#hF<*&F>F9FYF9#!#8FF* &FDF9FYF9#\"$2\"FB*&FIF9FYF9#!#xFB*&FMF9)F.F@F9#\"\"(F<*&F3F9FboF9#!#T F<*&F>F9FboF9#\"#\\FB*&FDF9FboF9#!#fFB*&FIF9FboF9#\"#rF<*&FMF9)F.F6F9F G*&F3F9FbpF9#\"#6F<*&F>F9FbpF9F;*&FDF9FbpF9#\"#8FB*&FIF9FbpF9#!\"(FB*& FMF9F7F9#F-\"$?\"*$FYF9#\"#:\"\")*$FbpF9#F-Fcq*$F7F9#F;F_q*$FboF9#!# " 0 "" {MPLTEXT 1 0 36 "plot([polMaple(t ),polin(t)],t=0..8);" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7Y7$ \"\"!$\"\"\"F(7$$\"1LLLLBxVN**F.7$$\"1nmm;$>5E$F.$\"1V UO%pkRe*F.7$$\"1LLLLAKn\\F.$\"10Z]q=)G*))F.7$$\"1LLLLc$\\o'F.$\"1,FEsL -=zF.7$$\"1mmm;bQ%R)F.$\"1GQ0fM\\6nF.7$$\"1LLL$Qk#z**F.$\"1:H*4RQ3U&F. 7$$\"1+++l9.i6!#:$\"1JKt\"GQ+&RF.7$$\"1LLL=\"\\&zkLd'!#<7$$\"1nmm'3Q\\n\"FM$!1bD]mFmr5F.7$$\" 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7$F^v$\"196,gdtp]F.7$Fcv$\"1ADkI(\\J'RF.7$Fhv$\"1W&>@<)3p>F.7$F]w$!1sq mGgH+8F.7$Fbw$!1%y3!41YxeF.7$Fgw$!1HLj8&fyB\"FM7$F\\x$!1RX'))e\\\\0#FM 7$Fax$!1))\\ZvCo7JFM7$Ffx$!1E,y2Vr8WFM7$F[y$!1y()e(p`W5&FM7$F`y$!1L`>y iFeeFM7$Fey$!1NFnv@?/oFM7$Fjy$!1kr'43>F%yFM7$F_z$!1qEwyCZa))FM7$Fdz$!1 I2n9Fn[**FM7$Fiz$!1Q:q6&z57\"F][l7$F_[l$!1QmP39ld7F][l7$Fd[l$!1Y++&=i& )R\"F][l7$Fi[l$!1:p)[\"3$*\\:F][l7$F^\\l$!1FGb***=[s\"F][l7$Fc\\l$!1$Q 2`^%)H\">F][l-Fh\\l6&Fj\\lF(F[]lF(-%+AXESLABELSG6$Q\"t6\"%!G-%%VIEWG6$ ;F(Fc\\l%(DEFAULTG" 2 382 382 382 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "coincidono !!" }}}}} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 10 "Altri dati" }}{PARA 0 "" 0 "" {TEXT -1 36 "y(x) = polinomio + un piccolo errore" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 57 "richiamo il \"package\" stats per generare numeri \+ \"casuali\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(stats); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7*%&anovaG%)describeG%$fitG%+impor tdataG%'randomG%*statevalfG%*statplotsG%*transformG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 324 1 "w" }{TEXT -1 30 " sara' il vetto re degli errori" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "w:= stat s[random, normald](n+1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG6($! +8hn'=&!#5$!+y'\\Ee#F($!+<+ " 0 "" {MPLTEXT 1 0 64 "for i from 1 to n+1 do y[i] := -1 + 0.5*x[i]*x[i] + 0.1*w[i] od:" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "richiamo il \"package\" per fare piu' plot" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 54 "costruisco il vettore delle coppie di punti (i \"dati\")" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "pl := seq([x[i],y[i]],i=1..n +1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#plG6(7$\"\"!$!+hn'=0\"!\"*7 $\"\"\"$!+o\\Ee_!#57$\"\"#$\"+$)*HQY)F/7$\"\"$$\"+.1-UNF*7$\"\"%$\"+Oh RXpF*7$\"\"&$\"+gcwW6!\")" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 25 "e \+ faccio il plot dei dati" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "p l1:= pointplot([pl], color =red,view=[0..n,-2..12]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "pl1;" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'PO INTSG6(7$\"\"!$!+hn'=0\"!\"*7$$\"\"\"F'$!+o\\Ee_!#57$$\"\"#F'$\"+$)*HQ Y)F07$$\"\"$F'$\"+.1-UNF*7$$\"\"%F'$\"+OhRXpF*7$$\"\"&F'$\"+gcwW6!\")- %%VIEWG6$;F'FA;$!\"#F'$\"#7F'-%'COLOURG6&%$RGBG$\"*++++\"FEF'F'" 2 382 382 382 2 0 1 0 2 6 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 -26888 0 0 0 0 0 0 }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 37 "costruisco il polinomio interpolatore" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "polin := unapply(interp ([x] ,y, t),t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&polinGR6#%\"tG6\"6$%) operatorG%&arrowGF(,.*$)9$\"\"&\"\"\"$\"+^=#)\\\"!\"*F/$\"+tWO_&*F: $!+hn'=0\"FE\"\"\"F(F(F(" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "e n e faccio il plot" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "pl2 := p lot(polin(t),t=0..n, y =-2..12, color =black):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 4 "pl2;" }}{PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG 6#7S7$\"\"!$!1+++hn'=0\"!#:7$$\"1LLL3x&)*3\"!#;$!1#zq`X@pg*F/7$$\"1nm \"H2P\"Q?F/$!1n,IqIp))*)F/7$$\"1LL$eRwX5$F/$!1fs]10`K%)F/7$$\"1ML$3x%3 yTF/$!1'e8S0V$ozF/7$$\"1nm\"z%4\\Y_F/$!1TfAITj_vF/7$$\"1MLeR-/PiF/$!1; (fDUS2<(F/7$$\"1***\\il'pisF/$!1$y(3(fkeu'F/7$$\"1MLe*)>VB$)F/$!1T)39t wVC'F/7$$\"1++DJbw!Q*F/$!1XWBEfjacF/7$$\"1nm;/j$o/\"F+$!1+yh?#e.$\\F/7 $$\"1LL3_>jU6F+$!1zMNt\\`xTF/7$$\"1++]i^Z]7F+$!1Rb(Q%)*)o=$F/7$$\"1++] (=h(e8F+$!1%[5&R&)*z-#F/7$$\"1++]P[6j9F+$!1y_vp7&[Z(!#<7$$\"1L$e*[z(yb \"F+$\"1](*f1'fle&F]p7$$\"1nm;a/cq;F+$\"1`#RP)=k\"H#F/7$$\"1nmm;t,mW7RF/7$$\"1+]iSj0x=F+$\"1&RZ;6\\;(fF/7$$\"1nmm\"pW`(>F+$\"1 %p_mksl%zF/7$$\"1+]i!f#=$3#F+$\"1!Hx1V'=F5F+7$$\"1+](=xpe=#F+$\"1[f`%= HLE\"F+7$$\"1nm\"H28IH#F+$\"1'\\#y+:pxg#F+$\"1))3zPMMmBF +7$$\"1+]Pf4t.FF+$\"1'>\"3'[*oUEF+7$$\"1MLe*Gst!GF+$\"1qfJs,!*\\HF+7$$ \"1+++DRW9HF+$\"1!4M6UzeF$F+7$$\"1++DJE>>IF+$\"12n#3HlBg$F+7$$\"1+]i!R 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