;ò …Rã?c@sYdZdkZdfd„ƒYZd„Zd„Zd„Zd„Zdd d „ZdS( sæ This module provides automatic differentiation for functions with any number of variables up to any order. An instance of the class DerivVar represents the value of a function and the values of 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. If only first-order derivatives are required, the module FirstDerivatives should be used. It is compatible to this one, but significantly faster. 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. Higher-order derivatives are requested with an optional third argument to DerivVar. Example: >>>x = DerivVar(3., 0, 3) >>>y = DerivVar(5., 1, 3) >>>print sqrt(x*y) produces the output >>>(3.87298334621, >>> [0.645497224368, 0.387298334621], >>> [[-0.107582870728, 0.0645497224368], >>> [0.0645497224368, -0.0387298334621]], >>> [[[0.053791435364, -0.0107582870728], >>> [-0.0107582870728, -0.00645497224368]], >>> [[-0.0107582870728, -0.00645497224368], >>> [-0.00645497224368, 0.0116189500386]]]) The individual orders can be extracted by indexing: >>>print sqrt(x*y)[0] >>>3.87298334621 >>>print sqrt(x*y)[1] >>>[0.645497224368, 0.387298334621] An n-th order derivative is represented by a nested list of depth n. When variables with different differentiation orders are mixed, the result has the lower one of the two orders. An exception are zeroth-order variables, which are treated as constants. Caution: Higher-order derivatives are implemented by recursively using DerivVars to represent derivatives. This makes the code very slow for high orders. 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