Newton Divided Difference

Standard Form:

Specific Example:

$$P_3(x)=a_0+a_1(x-x_1)+a_2(x-x_1)(x-x_2)+a_3(x-x_1)(x-x_2)(x-x_3)$$

General Example:

$$P_n(x)=a_0+a_1(x-x_1)+\text{ ... }+a_n(x-x_1)...(x-x_n)$$

Nested Form:

Specific Example:

$$P_3(x)=a_0+(x-x_1)(a_1+(x-x_2)(a_2+a_3(x-x_3)))$$

General Example:

$$P_n(x)=a_0+(x-x_1)(a_1+(x-x_2)(a_2+(x-x_3)(a_3+\text{ ... }(x-x_n)(a_n))))$$ $$P_n(x)=a_0+(x-x_1)(a_1+(x-x_n)(a_n+...))$$

Table:

$x$	f[]	f[,]	f[,,]	f[,,,]
1	3
$\frac{3}{2}$	$\frac{13}{4}$	$F[x_1,x_2]$ = $\frac{F[x_2]-F[x_1]}{x_2-x_1}=\frac{13/4-3}{3/2-1}=1/2$
0	3	$F[x_2,x_3]$ = $\frac{F[x_3]-F[x_2]}{x_3-x_2}=\frac{3-13/4}{0-3/2}=1/6$	$F[x_1,x_2,x_3]$ = $\frac{F[x_2,x_3]-F[x_1,x_2]}{x_3-x_1}=\frac{1/6-1/2}{0-1}=1/3$
2	$\frac{5}{3}$	$F[x_3,x_4]$ = $\frac{F[x_4]-F[x_3]}{x_4-x_3}=\frac{5/3-3}{2-0}=2/3$	$F[x_2,x_3,x_4]$ = $\frac{F[x_3,x_4]-F[x_2,x_3]}{x_4-x_2}=\frac{2/3-1/6}{2-3/2}=-5/3$	$F[x_1,x_2,x_3,x_4]$ = $\frac{F[x_2,x_3,x_4]-F[x_1,x_2,x_3]}{x_4-x_1}=\frac{-5/3-1/3}{2-1}=-2$

$\therefore$

$a_n$	$F[]$	Value
$a_0$	F[x_1]	3
$a_1$	$F[x_1,x_2]$	$\frac{1}{2}$
$a_2$	$F[x_1,x_2,x_3]$	$\frac{1}{3}$
$a_3$	$F[x_1,x_2,x_3,x_4]$	$-2$