Tema03 - 3 Splines c´ubiques 3.1 Definici´ o i...

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Unformatted text preview: 3. Splines c´ubiques 3.1 Definici´ o i construcci´ o Donats ( x i ,y i ) , i =1 ÷ n, a cada subinterval [ x i ,x i +1 ] , definim s i ( x ) = a x 3 + a 1 x 2 + a 2 x + a 3 , amb i =1 ÷ n- 1 . Condicions: 1. Interpolaci´ o als nodes i continu¨ ıtat de la funci´ o s i ( x i ) = y i , s i ( x i +1 ) = y i +1 , i =1 ÷ n- 1 , 2. Continu¨ ıtat de la primera derivada s i- 1 ( x i- ) = s i ( x i +) , i =2 ÷ n- 1 , 3. Continu¨ ıtat de la segona derivada s 00 i- 1 ( x i- ) = s 00 i ( x i +) , i =2 ÷ n- 1 . En principi es t´ e 4( n- 1) inc` ognites i 2( n- 1) + 2( n- 2) = 4 n- 6 equacions. Falten 2 condicions per determinar completament tots els coeficients i, per tant, el problema. Construcci´ o Sigui h i = x i +1- x i , w i = x- x i h i i ¯ w i = 1- w i = x- x i +1- h i . w i ¯ w i x i 1 x i +1 1 Fem interpolaci´ o lineal de la segona derivada de s i ( x ) que ´ es un segment: recta (polinomi interpolador de grau 1) que uneix ( x i ,τ i ) i ( x i +1 ,τ i +1 ) amb s 00 i ( x k ) = τ k , k = i,i + 1 . El resultat ´ es s 00 i ( x ) = τ i ¯ w i + τ i +1 w i . (3.1) Si s’integra dos cops, s’obt´ e s i ( x ): s i ( x )= Z s 00 i ( x ) dx + A = τ i Z ¯ w i dx + τ i +1 Z w i dx + A =- h i τ i 2 ¯ w 2 i + h i τ i +1 2 w 2 i + A . s i ( x ) = h 2 i [ τ i 6 ¯ w 3 i + τ i +1 6 w 3 i ] + A x + B (si σ i = τ i 6 , i =1 ÷ n ) = h 2 i [ σ i ¯ w 3 i + σ i +1 w 3 i ] + C w i + D ¯...
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Tema03 - 3 Splines c´ubiques 3.1 Definici´ o i...

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