Polinómio de Newton

O polinômio ${\displaystyle p_{2}(x)}$, que interpola ${\displaystyle f(x)}$ nos pontos dados[2]

${\displaystyle x_{0}=-1~\,f(x_{0})=4}$

${\displaystyle x_{1}=0~\,f(x_{1})=1}$

${\displaystyle x_{2}=2~\,f(x_{2})=-1}$

Na forma de Newton[3]

${\displaystyle {\begin{array}{c||ccc}x&-1&0&2\\\hline f(x)&4&1&-1\\\end{array}}}$ (Operador diferenças divididas )

${\displaystyle p_{2}(x)=f(x_{0})+(x-x_{0})f[x_{0},x_{1}]+(x-x_{0})(x-x_{1})f[x_{0},x_{1},x_{2}]}$

${\displaystyle {\begin{array}{c||ccc}x&{\mbox{Ordem 0}}&{\mbox{Ordem 1}}&{\mbox{Ordem 2}}\\\hline -1&4&&\\&&-3&\\0&1&&{\frac {2}{3}}\\&&-1&\\2&-1&&\\\end{array}}}$

${\displaystyle p_{2}(x)=4+(x-(-1))({\frac {(1-4)}{(0-(-1))}})+(x-(-1))(x-0)({\frac {{\frac {((-1)-1)}{(2-0)}}-{\frac {(1-4)}{(0-(-1))}}}{2-(-1)}})}$

${\displaystyle p_{2}(x)=4+(x+1)(-3)+(x+1)(x-0)({\frac {2}{3}})}$

• Amy Hui-Lin Chiu, Quartic Newton's method and matings of polynomials Thesis (Ph. D.)--Boston University, 1997. OCLC 37597068 (em inglês)
• Wilhelm Huisinga, Faber and Newton polynomial integrators for open system density matrix propagation , Berlin : ZIB, 1998. OCLC 76017284 (em inglês)