Culegere de probleme de Analiz˘a numeric˘a

6.1. Interpolare polinomială 79
cu
Dacă f ∈ C m+1 [a,b] atunci
(Rmf)(x) =
b
ϕm(x;s) = 1
(x−s)
m!
m + −
a
ϕm(x,s)f (m+1) (s)ds
m
k=0
lk(x)(xk −s) m +
Dacă lm(x,·) păstrează semn constant pe[a,b] atunci
(Rmf)(x) =
(Nmf)(x) = f(x0)+
Pentru noduri echidistante
1
x
(m+1)!
m+1 −
m
k=0
lk(x)x m+1
k
f (m+1) (ξ)
ξ ∈ [a,b]
m
(x−x0)...(x−xi−1)[x0,...,xi;f]
i=0
f = Nmf +Rmf formula de int.Newton
(Rmf)(x) = u(x)[x,x0,...,xm;f] x ∈ [a,b]
(Lmf)(x0 +th) = t[m+1]
m!
xi = x0 +ih, i = 0,m
m
(−1) m−i
m 1
i t−i f(xi)
i=0
(Rmf)(x0 +th) = hm+1 t [m+1]
(m+1)! f(m+1) (ξ)
(Nmf)(x0 +th) = (Nmf)(t) =
m
k=0
t
∆
k
k hf(x0) (Formula Gregory-Newton, formula lui Newton cu diferent¸e progresive)
(Nmf)(x) = (Nmf)(x0 +th) = f(xn)+
=
k=0
m
t+k −1
∇
k
k hf(xm) =
k=1
m
(−1) k
−t
∇
k
k h(xm)