trusspy.solvers.zerosearch module#

title: TrussPy - Truss Solver for Python author: Andreas Dutzler year: 2023

trusspy.solvers.zerosearch.newton(f, dfdx, x, nfev=8, ftol=1e-08, xtol=1e-08, verbose=0, *args)[source]#

Find the roots of a function using the Newton-Rhapson algorithm.

Find the roots of the (non-linear) equations f given a starting estimate x. The derivative dfdx`of `f w.r.t. x has to be provided by the user. Iteration loop stops after maximum number of function evaluations is reached (default: nfev=8). Default tolerance is 8 digits for vector norm of function residual and 8 digits for vector norm of incremental dx. If both tolerances are reached then the loop ends and the success flag becomes True. Return the roots of the (non-linear) equations defined by f(x) = 0 given a starting estimate.

Parameters:

  • f (ndarray) – 1D Vectorized function to minimize which depends on vector x

  • dfdx (ndarray) – 2D Derivative of function w.r.t. x

  • x (ndarray) – 1D Initial solution vector

  • nfev (int, optional) – Maximum number of newton iterations (default is 8)

  • ftol (float, optional) – Tolerance for residual of function: norm(f) (default is 1e-8)

  • xtol (float, optional) – Tolerance for residual of x: norm(x) (default is 1e-8)

  • verbose (int, optional) – Level of information during iterations (default is 0): verbose=0 … no information, verbose=1 … print iteration number, norm(f), norm(x)

  • s (arg) – pass arbitrary number of additional variables to function and derivative

Returns:

  • x (ndarray) – final solution

  • success (bool) – ‘True’ if valid solution found, ‘False’ if not converged

  • n (int) – number of function iterations

  • f_norm (float) – norm(f)

  • x_norm (float) – norm(x)

Examples

>>> import numpy as np
>>> def f(x, a):
...     return np.array([a*x[0]**3-1])
>>> def dfdx(x, a):
...     return np.array([a*3*x[0]**2])

>>> x0 = np.array([1.5])
>>> a = 2

>>> from trusspy.solvers import newton
>>> x,success,ntot,f_norm,x_norm = newton(f, dfdx, x0, 8, 1e-8, 1e-8, 0, a)

>>> x
array([0.79370053])
>>> success
True
>>> ntot
6
>>> f_norm <= 1e-8
True
>>> x_norm <= 1e-8
True