5 diferenciacion - Diferenciacion numerica Dado un set de...

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Diferenciacion numerica. Dado un set de datos experimentales (x 0 ,y 0 ), (x 1 ,y 1 ), …, (x n ,y n ), que nos representan una funcion, y=f(x), queremos encontrar el valor de la derivada df ( x ) dx Diferencias divididas. Cuando los datos experimentales son equidistantes en x, i.e. que la distancia entre dos valores de x consecutivos es constante, se puede hacer uso de diferencias divididas para encontrar la derivada en alguno de los valores de x experimentales. Si expandimos f(x) en serie de Taylor, tenemos: f ( x i + h )= f ( x i )+ h f ' ( x i )+ h 2 2 ! f ' ' ( x i )+ h 3 3 ! f ' ' ' ( x i )+… ( 1 ) f ( x i h )= f ( x i )− h f ' ( x i )+ h 2 2 ! f ' ' ( x i )− h 3 3 ! f ' ' ' ( x i )+… ( 2 ) f ( x i + 2 h )= f ( x i )+ 2 h f ' ( x i )+ 4 h 2 2 ! f ' ' ( x i )+ 8 h 3 3 ! f ' ' ' ( x i )+… ( 3 ) f ( x i 2 h )= f ( x i )− 2 h f ' ( x i )+ 4 h 2 2 ! f ' ' ( x i )− 8 h 3 3 ! f ' ' ' ( x i )+… ( 4 ) donde h = x i + 1 x i . A partir de estas, podemos encontrar las expresiones para las diferentes diferencias: -Centrales. Las diferencias centrales estiman la derivada en el punto x i utilizando utilizando el mismo numero de puntos anteriores y posteriores a x i .
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