4.6. Nonlinear solvers

This is a collection of general-purpose nonlinear multidimensional solvers. These solvers find x for which F(x) = 0. Both x and F can be multidimensional.

4.6.1. Routines

Large-scale nonlinear solvers:

newton_krylov(F, xin[, iter, rdiff, method, ...]) Find a root of a function, using Krylov approximation for inverse Jacobian.
anderson(F, xin[, iter, alpha, w0, M, ...]) Find a root of a function, using (extended) Anderson mixing.

General nonlinear solvers:

broyden1(F, xin[, iter, alpha, ...]) Find a root of a function, using Broyden’s first Jacobian approximation.
broyden2(F, xin[, iter, alpha, ...]) Find a root of a function, using Broyden’s second Jacobian approximation.

Simple iterations:

excitingmixing(F, xin[, iter, alpha, ...]) Find a root of a function, using a tuned diagonal Jacobian approximation.
linearmixing(F, xin[, iter, alpha, verbose, ...]) Find a root of a function, using a scalar Jacobian approximation.
diagbroyden(F, xin[, iter, alpha, verbose, ...]) Find a root of a function, using diagonal Broyden Jacobian approximation.

4.6.2. Examples

Small problem

>>> def F(x):
...    return np.cos(x) + x[::-1] - [1, 2, 3, 4]
>>> import scipy.optimize
>>> x = scipy.optimize.broyden1(F, [1,1,1,1], f_tol=1e-14)
>>> x
array([ 4.04674914,  3.91158389,  2.71791677,  1.61756251])
>>> np.cos(x) + x[::-1]
array([ 1.,  2.,  3.,  4.])

Large problem

Suppose that we needed to solve the following integrodifferential equation on the square \([0,1]\times[0,1]\):

\[\nabla^2 P = 10 \left(\int_0^1\int_0^1\cosh(P)\,dx\,dy\right)^2\]

with \(P(x,1) = 1\) and \(P=0\) elsewhere on the boundary of the square.

The solution can be found using the newton_krylov solver:

(Source code, png, hires.png, pdf)

_images/optimize-nonlin-1.png