Numerical differentiation formulas

by Trent Guidry4. September 2010 11:32

The numerical differentiation formulas shown below were calculated using the algorithm from Calculating derivatives of a function numerically.  In these equations, h is the distance between sequential x values.

Derivatives can be numerically calculated from sets of equally spaced function values. 

For example, for f=x2, consider the points below:

X  Y

0  0

1  1

2  4

3  9

For this series, there are 4 pairs for x and y sets.  To numerically calcuate the first derivative at x=2, the equation shown below for the first derivative of the third point in a four point series is used.

f1 = (2f3 + 3f2 - 6f1 + f0)/(6h) = (2*9 + 3*4 - 6*1 + 1*0)/(6*1)= 24/6 = 4

The analytic derivative of x2 is 2x.   At x = 2, this also gives 2*2 = 4.

Two points first derivatives.

Numerical Differentiation first order two points

Three points first derivatives.

Numerical Differentiation first order three points

Three points second derivatives.

Numerical Differentiation second order three points

Four points first derivatives.

Numerical Differentiation first order four points

Four points second derivatives.

Numerical Differentiation second order four points

Four points third derivatives.

Numerical Differentiation third order four points

Five points first derivatives.

Numerical Differentiation first order five points

Five points second derivatives.

Numerical Differentiation second order five points

Five points third derivatives.

Numerical Differentiation third order five points

Five points fourth derivatives.

Numerical Differentiation fourth order five points

Six points first derivatives.

Numerical Differentiation first order six points

Six points second derivatives.

Numerical Differentiation second order six points

Six points third derivatives.

Numerical Differentiation third order six points

Six points fourth derivatives.

Numerical Differentiation fourth order six points

Six points fifth derivatives.

Numerical Differentiation fifth order six points

Seven points first derivatives.

Numerical Differentiation first order seven points

Seven points second derivatives.

Numerical Differentiation second order seven points

Seven points third derivatives.

Numerical Differentiation third order seven points

Seven points fourth derivatives.

Numerical Differentiation fourth order seven points

Seven points fifth derivatives.

Numerical Differentiation fifth order seven points

Eight points  first derivatives.

Numerical Differentiation first order eight points

Eight points second derivatives.

Numerica -Differentiation second order eight points

Eight points third derivatives.

Numerical Differentiation third order eight points

Eight points fourth derivatives.

Numerical Differentiation fourth order eight points

Eight points fifth derivatives.

Numerical Differentiation fifth order eight points

Nine points first derivatives.

Numerical Differentiation first order nine points

Nine points second derivatives.

Numerical Differentiation second order nine points

Nine points third derivatives.

Numerical Differentiation third order nine points

Nine points fourth derivatives.

Numerical Differentiation fourth order nine points

Nine points fifth derivatives.

Numerical Differentiation fifth order nine points

Ten points first derivatives.

Numerical Differentiation first order ten points

Ten points second derivatives.

Numerical Differentiation second order ten points

Ten points third derivatives.

Numerical Differentiation third order ten points

Ten points fourth derivatives.

Numerical Differentiation fourth order ten points

Eleven points first derivatives.

Numerical Differentiation first order eleven points

Eleven points second derivatives.

Numerical Differentiation second order eleven points

Eleven points third derivatives.

Numerical Differentiation third order eleven points

Eleven points fourth derivatives.

Numerical Differentiation fourth order eleven points

Eleven points fifth derivatives.

Numerical Differentiation fifth order eleven points