BroydenFirst#
- class scipy.optimize.BroydenFirst(alpha=None, reduction_method='restart', max_rank=None)[source]#
Find a root of a function, using Broyden’s first Jacobian approximation.
This method is also known as "Broyden’s good method".
- Parameters:
- %(params_basic)s
- %(broyden_params)s
- %(params_extra)s
Methods
aspreconditioner
matvec
rmatvec
rsolve
setup
solve
todense
update
See also
root
Interface to root finding algorithms for multivariate functions. See
method='broyden1'
in particular.
Notes
This algorithm implements the inverse Jacobian Quasi-Newton update
\[H_+ = H + (dx - H df) dx^\dagger H / ( dx^\dagger H df)\]which corresponds to Broyden’s first Jacobian update
\[J_+ = J + (df - J dx) dx^\dagger / dx^\dagger dx\]References
[1]B.A. van der Rotten, PhD thesis, "A limited memory Broyden method to solve high-dimensional systems of nonlinear equations". Mathematisch Instituut, Universiteit Leiden, The Netherlands (2003).
https://web.archive.org/web/20161022015821/http://www.math.leidenuniv.nl/scripties/Rotten.pdf
Examples
The following functions define a system of nonlinear equations
>>> def fun(x): ... return [x[0] + 0.5 * (x[0] - x[1])**3 - 1.0, ... 0.5 * (x[1] - x[0])**3 + x[1]]
A solution can be obtained as follows.
>>> from scipy import optimize >>> sol = optimize.broyden1(fun, [0, 0]) >>> sol array([0.84116396, 0.15883641])