5.6 Diferenciación numérica – Tablas con diferencias divididas

Diferencias Divididas [ hacia adelante ] [ centradas ] [hacia atrás] ..

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Referencia: Chapra Fig.23.1 p669, Burden 4.1 p167, Rodríguez 8.2,3,4,6 p324

Diferencias divididas hacia adelante

Primera derivada

$f'(x_i) = \frac{f(x_{i+1})-f(x_i)}{h} + O(h)$ $f'(x_i) = \frac{-f(x_{i+2})+4f(x_{i+1})-3f(x_i)}{2h} + O(h^2)$

Segunda derivada

$f''(x_i) = \frac{f(x_{i+2})-2f(x_{i+1})+f(x_i)}{h^2} + O(h)$ $f''(x_i) = \frac{-f(x_{i+3})+4f(x_{i+2})-5f(x_{i+1})+2f(x_i)}{h^2}$ $+ O(h^2)$

Tercera derivada

$f'''(x_i) = \frac{f(x_{i+3})-3f(x_{i+2})+3f(x_{i+1})-f(x_i)}{h^3}$ $+ O(h)$ $f'''(x_i) = \frac{-3f(x_{i+4})+14f(x_{i+3})-24f(x_{i+2})+18f(x_{i+1})-5f(x_i)}{2h^3}$ $+ O(h^2)$

Cuarta derivada

$f''''(x_i) = \frac{f(x_{i+4})-4f(x_{i+3})+6f(x_{i+2})-4f(x_{i+1})+f(x_i)}{h^3}$ $+ O(h)$

Diferencias Divididas [ hacia adelante ] [ centradas ] [hacia atrás]

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Diferencias divididas centradas

Primera derivada

$f'(x_i) = \frac{f(x_{i+1})-f(x_{i-1})}{2h} + O(h^2)$ $f'(x_i) = \frac{-f(x_{i+2})+8f(x_{i+1})-8f(x_{i-1}) +f(x_{i-2})}{12h}$ $+ O(h^4)$

Segunda derivada

$f''(x_i) = \frac{f(x_{i+1})-2f(x_{i})+f(x_{i-1})}{h^2} + O(h^2)$ $f''(x_i) = \frac{-f(x_{i+2})+16f(x_{i+1})-30f(x_{i})+16f(x_{i-1})-f(x_{i-2})}{12h^2}$ $+ O(h^4)$

Tercera derivada

$f'''(x_i) = \frac{f(x_{i+2})-2f(x_{i+1})+2f(x_{i-1})-f(x_{i-2})}{2h^3}$ $+ O(h^2)$ $f'''(x_i) = \frac{-f(x_{i+3})+8f(x_{i+2})-13f(x_{i+1})+13f(x_{i-1})-8f(x_{i-2})+f(x_{i-3})}{8h^3}$ $+ O(h^4)$

Diferencias Divididas [ hacia adelante ] [ centradas ] [hacia atrás]

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Diferencias divididas hacia atrás

Primera derivada

$f'(x_i) = \frac{f(x_{i})-f(x_{i-1})}{h} + O(h)$ $f'(x_i) = \frac{3f(x_{i})-4f(x_{i-1})+f(x_{i-2})}{2h}$ $+ O(h^2)$

Segunda derivada

$f''(x_i) = \frac{f(x_{i})-2f(x_{i-1})+f(x_{i-2})}{h^2} + O(h)$ $f''(x_i) = \frac{2f(x_{i})-5f(x_{i-1})+4f(x_{i-2})-f(x_{i-3})}{h^2}$ $+ O(h^2)$

Tercera derivada

$f'''(x_i) = \frac{f(x_{i})-3f(x_{i-1})+3f(x_{i-2})-f(x_{i-3})}{h^3} + O(h)$ $f'''(x_i) = \frac{5f(x_{i})-18f(x_{i-1})+24f(x_{i-2})-14f(x_{i-3})+3f(x_{i-4})}{2h^3}$ $+ O(h^2)$

Diferencias Divididas [ hacia adelante ] [ centradas ] [hacia atrás]