Fórmulas de Diferenciación por diferencias divididas

Referencia: Chapra Fig.23.1 pag.669 pdf.693, Burden 4.1 p167, Rodriguez 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 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 atras

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)