Question

The second divided difference f[xo, x1, x2]

can be written as 

f[xo, x1, x2] = af(x0) + bf(x1) + cf(x2). 

Find the expressions for a, b, c. 

Answer 1

The first divided difference: $f[x_i,x_{i+1}]=\frac{f[x_{i+1}]-f[x_i]}{x_{i+1}-x_i}$ , where $f[x_i]=f(x_i)$ .

The second divided difference: $f[x_0,x_1,x_2]=\frac{f[x_1,x_2]-f[x_0,x_1]}{x_2-x_0}=\frac{\frac{f[x_{2}]-f[x_1]}{x_{2}-x_1} -\frac{f[x_{1}]-f[x_0]}{x_{1}-x_0} }{x_2-x_0}=\\ =\frac{f[x_0]}{(x_1-x_0)(x_2-x_0) }+\frac{f[x_1](x_0-x_2)}{ (x_1-x_0)(x_2-x_1)(x_2-x_0) }+ \frac{f[x_2]}{(x_2-x_1)(x_2-x_0) }=\\ = \frac{f(x_0)}{(x_0-x_1)(x_0-x_2) }+ \frac{f(x_1)}{(x_1-x_0)(x_1-x_2) }+ \frac{f(x_2)}{(x_2-x_0)(x_2-x_1) }$

Answer: $a=\frac{1}{(x_0-x_1)(x_0-x_2)},\quad b=\frac{1}{(x_1-x_0)(x_1-x_2)},\quad c=\frac{1}{(x_2-x_0)(x_2-x_1)}.$