DIFFERENTIATED DIFFERENCES

Ano: 1984 | Volume: 6 | Número: 6

Autores: Alcibiades Gazzoni, Alsimar T. Ferreira Gazzoni

Palavras-chave: there are no notes

Resumo Inglês:

Given a sequence of real numbers, not necessarily distinct, τ = (τ _i ) ⁿ _{i = 1} , it is said that a polynomial P interpolates a function f into τ when, for every τ _i of τ, which occurs m times, If P ^(j-1) (τ _i ) = f ^(i-1) (τ _i ), j = 1, ..., m where P ^(j-1) f ^(j-1) represent, respectively, the derivative of order j-1 of P and f. Define the kth difference of f at points τ _i , ..., τ _{i + K}of τ, as the leading coefficient of the degree polynomial at maximum K that interpolates f into τ _i , ..., τ _{i + K}. Considering the definition of kth difference divided, initially, in the present work, the uniqueness of the interpolation polynomial is studied. The following is the definition and some properties of the kth divided difference, and finally, if f is of class C ^k in [a, b], where a = min {τ _j , .. ., τ _{j + k} } b = max {τ _j , ..., τ _{j + k} }, then the previous definition of k-th divided difference is equal to iNT (0-1) int (0-t ₁ ) int ... (0- _tk-1 ) f ^(k) [t _k (τ _{i + k} - τ _{i + k-1} ) + ... + t ₁ (τ _{i + 1} - τ_i ) + τ _i ] dt _k ... dt ₁