taylor

Compute finite difference approximations to derivatives of multidimensional, multivariate functions with respect to multidimensional variables.

This package is similar in function to numdifftools in that its purpose is to compute derivative objects (gradients, Jacobians, Hessians, etc.) for problems like optimization and the solution of nonlinear equations. However, taylor more naturally handles array stuctured variables and has a much less sophisticated algorithm (resulting in poorer accuracy).

Neither of these programs should be confused with those like findiff and fastfd, which focus on the computation of finite differences over multidimensional grids likely representing physical systems.

Interface

defdiff(fun,x,order,args=(),mask=None,rule='forward',delta=None,
idx_order='natural'):
""" Computes the numerical derivative of function using finite differences ---Inputs--- fun : {function} function whose derivative is sought has function definition "def fun(x,*args): ... return val" x : {scalar, array} independent variable with respect to which the derivative will be computed order : {integer} order of desired derivative 1: first derivative (gradient), 2: second derivative (Hessian), ... args : {tuple} tuple of additional arguments to fun mask : {integer, array} array of same shape as the returned derivative where element = 1 -> this entry should be computed, element = 0 -> entry should not be computed rule : {string} finite difference rule choose from: {'forward','backward','central'} delta : {float or array} scalar/array of same shape as x that specifies the finite difference step size idx_order : {string} string indicating how indices of derivative object should be ordered when returned 'natural' : indices corresponding to elements of function output are ordered first (like in Jacobians) 'reversed' : indices corresponding to derivatives are ordered first ---Outputs--- derivative : {scalar or array} numerical derivative of input function to order specified """

Examples

The first example computes the first derivative of the matrix vector product f(A,x) = A x with respect to both the matrix A and vector x.

importnumpyasnpimporttaylorasta# both functions compute matrix vector product,# but have different first argumentsdefmatvec_vec(x,A):
returnA @ xdefmatvec_mat(A,x):
returnA @ xif (__name__=="__main__"):
# set matrix and vectorA=np.array([[1.0,2.0,3.0],
[2.0,4.0,5.0],
[3.0,5.0,6.0]])
x=np.array([1.0,2.0,3.0])
# derivative of matrix vector product with respect to vectorderiv_matvec_vec=ta.diff(matvec_vec,x,1,args=(A,))
print(f'df / dx :\n{deriv_matvec_vec}\n')
# derivative of matrix vector product with respect to matrixderiv_matvec_mat=ta.diff(matvec_mat,A,1,args=(x,))
print(f'df / dA :\n{deriv_matvec_mat}')

Namesake

The package is named after Brook Taylor, the namesake for Taylor series and the originator of finite differences.

Miscellaneous

TODO: contact this user (pbrod/numdifftools#48) to advertise package