Module FirstDerivatives

Automatic differentiation for functions with any number of variables

Instances of the class DerivVar represent the values of a function and its partial derivatives with respect to a list of variables. All common mathematical operations and functions are available for these numbers. There is no restriction on the type of the numbers fed into the code; it works for real and complex numbers as well as for any Python type that implements the necessary operations.

This module is as far as possible compatible with the n-th order derivatives module Derivatives. If only first-order derivatives are required, this module is faster than the general one.

Example:

 print sin(DerivVar(2))

produces the output:

 (0.909297426826, [-0.416146836547])

The first number is the value of sin(2); the number in the following list is the value of the derivative of sin(x) at x=2, i.e. cos(2).

When there is more than one variable, DerivVar must be called with an integer second argument that specifies the number of the variable.

Example:

   >>>x = DerivVar(7., 0)
   >>>y = DerivVar(42., 1)
   >>>z = DerivVar(pi, 2)
   >>>print (sqrt(pow(x,2)+pow(y,2)+pow(z,2)))

   produces the output

   >>>(42.6950770511, [0.163953328662, 0.98371997197, 0.0735820818365])

The numbers in the list are the partial derivatives with respect to x, y, and z, respectively.

Note: It doesn't make sense to use DerivVar with different values for the same variable index in one calculation, but there is no check for this. I.e.:

   >>>print DerivVar(3, 0)+DerivVar(5, 0)

   produces

   >>>(8, [2])

but this result is meaningless.