# Diferencias divididas de Newton

## Fórmula recursiva

\begin{align*} f[x_{0}] &= f(x_{0}) \\ f[x_{0}, x_{1}] &= \frac{f[x_{1}] - f[x_{0}]}{x_{1} - x_{0}} \\ f[x_{0}, x_{1}, x_{2}] &= \frac{f[x_{1}, x_{2}] - f[x_{0}, x_{1}]}{x_{2} - x_{0}} \\ f[x_{0}, x_{1}, x_{2}, x_{3}] &= \frac{f[x_{1}, x_{2}, x_{3}] - f[x_{0}, x_{1}, x_{2}]}{x_{3} - x_{0}} \\ f[x_{0}, x_{1}, x_{2}, x_{3}, x_{4}] &= \frac{f[x_{1}, x_{2}, x_{3}, x_{4}] - f[x_{0}, x_{1}, x_{2}, x_{3}]}{x_{4} - x_{0}} \\ f[x_{0}, x_{1}, \cdots, x_{n-1}, x_{n}] &= \frac{f[x_{1}, \cdots, x_{n-1}, x_{n}] - f[x_{0}, x_{1}, \cdots, x_{n-1}]}{x_{n} - x_{0}} \end{align*}

## Tabla de diferencias

\begin{equation*} \begin{array}{c|ccccc} x & f & f' & f'' & f''' & f^{(4)} \\ \hline x_{0} & f[x_{0}] & & & & \\ & & f[x_{0}, x_{1}] & & & \\ x_{1} & f[x_{1}] & & f[x_{0}, x_{1}, x_{2}] & & \\ & & f[x_{1}, x_{2}] & & f[x_{0}, x_{1}, x_{2}, x_{3}] & \\ x_{2} & f[x_{2}] & & f[x_{1}, x_{2}, x_{3}] & & f[x_{0}, x_{1}, x_{2}, x_{3}, x_{4}] \\ & & f[x_{2}, x_{3}] & & f[x_{1}, x_{2}, x_{3}, x_{4}] & \\ x_{3} & f[x_{3}] & & f[x_{2}, x_{3}, x_{4}] & & \\ & & f[x_{3}, x_{4}] & & & \\ x_{4} & f[x_{4}] & & & & \end{array} \end{equation*}

## Ejemplo

Calcular las derivadas en los puntos $x = [1, 4, 6, 7, 11]$ de

\begin{equation*} f(x) = \ln{x} \end{equation*}

\begin{align*} f[x_{0}] &= f(x_{0}) = f(1) = 0 \\ f[x_{1}] &= f(x_{1}) = f(4) = 1.386294 \\ f[x_{2}] &= f(x_{2}) = f(6) = 1.791759 \\ f[x_{3}] &= f(x_{3}) = f(7) = 1.945910 \\ f[x_{4}] &= f(x_{4}) = f(11) = 2.397895 \end{align*}

\begin{align*} f[x_{0}, x_{1}] &= \frac{f[x_{1}] - f[x_{0}]}{x_{1} - x_{0}} = \frac{1.386294 - 0}{4 - 1} = 0.462098 \\ f[x_{1}, x_{2}] &= \frac{f[x_{2}] - f[x_{1}]}{x_{2} - x_{1}} = \frac{1.791759 - 1.386294}{6 - 4} = 0.202732 \\ f[x_{2}, x_{3}] &= \frac{f[x_{3}] - f[x_{2}]}{x_{3} - x_{2}} = \frac{1.945910 - 1.791759}{7 - 6} = 0.154151 \\ f[x_{3}, x_{4}] &= \frac{f[x_{4}] - f[x_{3}]}{x_{4} - x_{3}} = \frac{2.397895 - 1.945910}{11 - 7} = 0.112996 \end{align*}

\begin{align*} f[x_{0}, x_{1}, x_{2}] &= \frac{f[x_{1}, x_{2}] - f[x_{0}, x_{1}]}{x_{2} - x_{0}} = \frac{0.202732 - 0.462098}{6 - 1} = -0.051873 \\ f[x_{1}, x_{2}, x_{3}] &= \frac{f[x_{2}, x_{3}] - f[x_{1}, x_{2}]}{x_{3} - x_{1}} = \frac{0.154151 - 0.202732}{7 - 4} = -0.016194 \\ f[x_{2}, x_{3}, x_{4}] &= \frac{f[x_{3}, x_{4}] - f[x_{2}, x_{3}]}{x_{4} - x_{2}} = \frac{0.112996 - 0.154151}{11 - 6} = -0.008231 \end{align*}