Newton's Interpolating Polynomials

1 min read Last updated Mon Jun 01 2026 03:58:53 GMT+0000 (Coordinated Universal Time)

Suppose that $P_n (x)$ is the $n$ -th interpolating polynomial that agrees with the function $f$ at the distinct numbers $x_0, x_1,\dots,x_n$ . $P_n$ has the form:

P_n (x) = a_0 + a_1(x - x_0) + a_2(x - x_0)(x - x_1) + \dots + a_n(x - x_0)\dots(x-x_{n-1})

for appropriate constants $a_0, a_1,\dots,a_n$ . The constants can be found by setting $x$ to the known data points $x_0,x_1,\dots,x_n$ .

Divided-difference notation

Divided differences are defined with respect to a set of distinct numbers.

Zeroth divided difference

f[x_i] = f (x_i)

First divided difference

f[x_i,x_{i+1}] = \frac{f(x_{i+1}) - f(x_i)}{x_{i+1} - x_i}

Second divided difference

f[x_i,x_{i+1},x_{i+2}] = \frac{f[x_{i+1} , x_{i+2}] - f [x_i,x_{i+1}]}{x_{i+2} - x_i}

k-th divided difference

f[x_i,x_{i+1},x_{i+2},\dots,x_{i+k}] = \frac{f[x_{i+1},x_{i+2},\dots,x_{i+k}]-f [x_i,x_{i+1},x_{i+2},\dots,x_{i+k-1}]} {x_{i+k} - x_i}

Now $P_n(x)$ can be rewritten in a form called Newton’s Divided Difference:

P_n(x)=f[x_0] + \sum_{k=1}^n f[x_0,x_1,\dots,x_k](x-x_0)(x-x_1)\dots(x - x_{k-1})

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