Optimization (:mod:`scipy.optimize`)
====================================
.. sectionauthor:: Travis E. Oliphant
.. sectionauthor:: Pauli Virtanen
.. currentmodule:: scipy.optimize
The :mod:`scipy.optimize` package provides several commonly used
optimization algorithms. An detailed listing is available:
:mod:`scipy.optimize` (can also be found by ``help(scipy.optimize)``).
The module contains:
1. Unconstrained and constrained minimization and least-squares algorithms
(e.g., :func:`fmin`: Nelder-Mead simplex, :func:`fmin_bfgs`:
BFGS, :func:`fmin_ncg`: Newton Conjugate Gradient,
:func:`leastsq`: Levenberg-Marquardt, :func:`fmin_cobyla`: COBYLA).
2. Global (brute-force) optimization routines (e.g., :func:`anneal`)
3. Curve fitting (:func:`curve_fit`)
4. Scalar function minimizers and root finders (e.g., Brent's method
:func:`fminbound`, and :func:`newton`)
5. Multivariate equation system solvers (:func:`fsolve`)
6. Large-scale multivariate equation system solvers (e.g. :func:`newton_krylov`)
Below, several examples demonstrate their basic usage.
Nelder-Mead Simplex algorithm (:func:`fmin`)
--------------------------------------------
The simplex algorithm is probably the simplest way to minimize a
fairly well-behaved function. The simplex algorithm requires only
function evaluations and is a good choice for simple minimization
problems. However, because it does not use any gradient evaluations,
it may take longer to find the minimum. To demonstrate the
minimization function consider the problem of minimizing the
Rosenbrock function of :math:`N` variables:
.. math::
:nowrap:
\[ f\left(\mathbf{x}\right)=\sum_{i=1}^{N-1}100\left(x_{i}-x_{i-1}^{2}\right)^{2}+\left(1-x_{i-1}\right)^{2}.\]
The minimum value of this function is 0 which is achieved when :math:`x_{i}=1.` This minimum can be found using the :obj:`fmin` routine as shown in the example below:
>>> from scipy.optimize import fmin
>>> def rosen(x):
... """The Rosenbrock function"""
... return sum(100.0*(x[1:]-x[:-1]**2.0)**2.0 + (1-x[:-1])**2.0)
>>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
>>> xopt = fmin(rosen, x0, xtol=1e-8)
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 339
Function evaluations: 571
>>> print xopt
[ 1. 1. 1. 1. 1.]
Another optimization algorithm that needs only function calls to find
the minimum is Powell's method available as :func:`fmin_powell`.
Broyden-Fletcher-Goldfarb-Shanno algorithm (:func:`fmin_bfgs`)
--------------------------------------------------------------
In order to converge more quickly to the solution, this routine uses
the gradient of the objective function. If the gradient is not given
by the user, then it is estimated using first-differences. The
Broyden-Fletcher-Goldfarb-Shanno (BFGS) method typically requires
fewer function calls than the simplex algorithm even when the gradient
must be estimated.
To demonstrate this algorithm, the Rosenbrock function is again used.
The gradient of the Rosenbrock function is the vector:
.. math::
:nowrap:
\begin{eqnarray*} \frac{\partial f}{\partial x_{j}} & = & \sum_{i=1}^{N}200\left(x_{i}-x_{i-1}^{2}\right)\left(\delta_{i,j}-2x_{i-1}\delta_{i-1,j}\right)-2\left(1-x_{i-1}\right)\delta_{i-1,j}.\\ & = & 200\left(x_{j}-x_{j-1}^{2}\right)-400x_{j}\left(x_{j+1}-x_{j}^{2}\right)-2\left(1-x_{j}\right).\end{eqnarray*}
This expression is valid for the interior derivatives. Special cases
are
.. math::
:nowrap:
\begin{eqnarray*} \frac{\partial f}{\partial x_{0}} & = & -400x_{0}\left(x_{1}-x_{0}^{2}\right)-2\left(1-x_{0}\right),\\ \frac{\partial f}{\partial x_{N-1}} & = & 200\left(x_{N-1}-x_{N-2}^{2}\right).\end{eqnarray*}
A Python function which computes this gradient is constructed by the
code-segment:
>>> def rosen_der(x):
... xm = x[1:-1]
... xm_m1 = x[:-2]
... xm_p1 = x[2:]
... der = zeros_like(x)
... der[1:-1] = 200*(xm-xm_m1**2) - 400*(xm_p1 - xm**2)*xm - 2*(1-xm)
... der[0] = -400*x[0]*(x[1]-x[0]**2) - 2*(1-x[0])
... der[-1] = 200*(x[-1]-x[-2]**2)
... return der
The calling signature for the BFGS minimization algorithm is similar
to :obj:`fmin` with the addition of the *fprime* argument. An example
usage of :obj:`fmin_bfgs` is shown in the following example which
minimizes the Rosenbrock function.
>>> from scipy.optimize import fmin_bfgs
>>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
>>> xopt = fmin_bfgs(rosen, x0, fprime=rosen_der)
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 53
Function evaluations: 65
Gradient evaluations: 65
>>> print xopt
[ 1. 1. 1. 1. 1.]
Newton-Conjugate-Gradient (:func:`fmin_ncg`)
--------------------------------------------
The method which requires the fewest function calls and is therefore
often the fastest method to minimize functions of many variables is
:obj:`fmin_ncg`. This method is a modified Newton's method and uses a
conjugate gradient algorithm to (approximately) invert the local
Hessian. Newton's method is based on fitting the function locally to
a quadratic form:
.. math::
:nowrap:
\[ f\left(\mathbf{x}\right)\approx f\left(\mathbf{x}_{0}\right)+\nabla f\left(\mathbf{x}_{0}\right)\cdot\left(\mathbf{x}-\mathbf{x}_{0}\right)+\frac{1}{2}\left(\mathbf{x}-\mathbf{x}_{0}\right)^{T}\mathbf{H}\left(\mathbf{x}_{0}\right)\left(\mathbf{x}-\mathbf{x}_{0}\right).\]
where :math:`\mathbf{H}\left(\mathbf{x}_{0}\right)` is a matrix of second-derivatives (the Hessian). If the Hessian is
positive definite then the local minimum of this function can be found
by setting the gradient of the quadratic form to zero, resulting in
.. math::
:nowrap:
\[ \mathbf{x}_{\textrm{opt}}=\mathbf{x}_{0}-\mathbf{H}^{-1}\nabla f.\]
The inverse of the Hessian is evaluted using the conjugate-gradient
method. An example of employing this method to minimizing the
Rosenbrock function is given below. To take full advantage of the
NewtonCG method, a function which computes the Hessian must be
provided. The Hessian matrix itself does not need to be constructed,
only a vector which is the product of the Hessian with an arbitrary
vector needs to be available to the minimization routine. As a result,
the user can provide either a function to compute the Hessian matrix,
or a function to compute the product of the Hessian with an arbitrary
vector.
Full Hessian example:
^^^^^^^^^^^^^^^^^^^^^
The Hessian of the Rosenbrock function is
.. math::
:nowrap:
\begin{eqnarray*} H_{ij}=\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}} & = & 200\left(\delta_{i,j}-2x_{i-1}\delta_{i-1,j}\right)-400x_{i}\left(\delta_{i+1,j}-2x_{i}\delta_{i,j}\right)-400\delta_{i,j}\left(x_{i+1}-x_{i}^{2}\right)+2\delta_{i,j},\\ & = & \left(202+1200x_{i}^{2}-400x_{i+1}\right)\delta_{i,j}-400x_{i}\delta_{i+1,j}-400x_{i-1}\delta_{i-1,j},\end{eqnarray*}
if :math:`i,j\in\left[1,N-2\right]` with :math:`i,j\in\left[0,N-1\right]` defining the :math:`N\times N` matrix. Other non-zero entries of the matrix are
.. math::
:nowrap:
\begin{eqnarray*} \frac{\partial^{2}f}{\partial x_{0}^{2}} & = & 1200x_{0}^{2}-400x_{1}+2,\\ \frac{\partial^{2}f}{\partial x_{0}\partial x_{1}}=\frac{\partial^{2}f}{\partial x_{1}\partial x_{0}} & = & -400x_{0},\\ \frac{\partial^{2}f}{\partial x_{N-1}\partial x_{N-2}}=\frac{\partial^{2}f}{\partial x_{N-2}\partial x_{N-1}} & = & -400x_{N-2},\\ \frac{\partial^{2}f}{\partial x_{N-1}^{2}} & = & 200.\end{eqnarray*}
For example, the Hessian when :math:`N=5` is
.. math::
:nowrap:
\[ \mathbf{H}=\left[\begin{array}{ccccc} 1200x_{0}^{2}-400x_{1}+2 & -400x_{0} & 0 & 0 & 0\\ -400x_{0} & 202+1200x_{1}^{2}-400x_{2} & -400x_{1} & 0 & 0\\ 0 & -400x_{1} & 202+1200x_{2}^{2}-400x_{3} & -400x_{2} & 0\\ 0 & & -400x_{2} & 202+1200x_{3}^{2}-400x_{4} & -400x_{3}\\ 0 & 0 & 0 & -400x_{3} & 200\end{array}\right].\]
The code which computes this Hessian along with the code to minimize
the function using :obj:`fmin_ncg` is shown in the following example:
>>> from scipy.optimize import fmin_ncg
>>> def rosen_hess(x):
... x = asarray(x)
... H = diag(-400*x[:-1],1) - diag(400*x[:-1],-1)
... diagonal = zeros_like(x)
... diagonal[0] = 1200*x[0]-400*x[1]+2
... diagonal[-1] = 200
... diagonal[1:-1] = 202 + 1200*x[1:-1]**2 - 400*x[2:]
... H = H + diag(diagonal)
... return H
>>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
>>> xopt = fmin_ncg(rosen, x0, rosen_der, fhess=rosen_hess, avextol=1e-8)
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 23
Function evaluations: 26
Gradient evaluations: 23
Hessian evaluations: 23
>>> print xopt
[ 1. 1. 1. 1. 1.]
Hessian product example:
^^^^^^^^^^^^^^^^^^^^^^^^
For larger minimization problems, storing the entire Hessian matrix
can consume considerable time and memory. The Newton-CG algorithm only
needs the product of the Hessian times an arbitrary vector. As a
result, the user can supply code to compute this product rather than
the full Hessian by setting the *fhess_p* keyword to the desired
function. The *fhess_p* function should take the minimization vector as
the first argument and the arbitrary vector as the second
argument. Any extra arguments passed to the function to be minimized
will also be passed to this function. If possible, using Newton-CG
with the hessian product option is probably the fastest way to
minimize the function.
In this case, the product of the Rosenbrock Hessian with an arbitrary
vector is not difficult to compute. If :math:`\mathbf{p}` is the arbitrary vector, then :math:`\mathbf{H}\left(\mathbf{x}\right)\mathbf{p}` has elements:
.. math::
:nowrap:
\[ \mathbf{H}\left(\mathbf{x}\right)\mathbf{p}=\left[\begin{array}{c} \left(1200x_{0}^{2}-400x_{1}+2\right)p_{0}-400x_{0}p_{1}\\ \vdots\\ -400x_{i-1}p_{i-1}+\left(202+1200x_{i}^{2}-400x_{i+1}\right)p_{i}-400x_{i}p_{i+1}\\ \vdots\\ -400x_{N-2}p_{N-2}+200p_{N-1}\end{array}\right].\]
Code which makes use of the *fhess_p* keyword to minimize the
Rosenbrock function using :obj:`fmin_ncg` follows:
>>> from scipy.optimize import fmin_ncg
>>> def rosen_hess_p(x,p):
... x = asarray(x)
... Hp = zeros_like(x)
... Hp[0] = (1200*x[0]**2 - 400*x[1] + 2)*p[0] - 400*x[0]*p[1]
... Hp[1:-1] = -400*x[:-2]*p[:-2]+(202+1200*x[1:-1]**2-400*x[2:])*p[1:-1] \
... -400*x[1:-1]*p[2:]
... Hp[-1] = -400*x[-2]*p[-2] + 200*p[-1]
... return Hp
>>> x0 = [1.3, 0.7, 0.8, 1.9, 1.2]
>>> xopt = fmin_ncg(rosen, x0, rosen_der, fhess_p=rosen_hess_p, avextol=1e-8)
Optimization terminated successfully.
Current function value: 0.000000
Iterations: 22
Function evaluations: 25
Gradient evaluations: 22
Hessian evaluations: 54
>>> print xopt
[ 1. 1. 1. 1. 1.]
Least-square fitting (:func:`leastsq`)
--------------------------------------
All of the previously-explained minimization procedures can be used to
solve a least-squares problem provided the appropriate objective
function is constructed. For example, suppose it is desired to fit a
set of data :math:`\left\{\mathbf{x}_{i}, \mathbf{y}_{i}\right\}`
to a known model,
:math:`\mathbf{y}=\mathbf{f}\left(\mathbf{x},\mathbf{p}\right)`
where :math:`\mathbf{p}` is a vector of parameters for the model that
need to be found. A common method for determining which parameter
vector gives the best fit to the data is to minimize the sum of squares
of the residuals. The residual is usually defined for each observed
data-point as
.. math::
:nowrap:
\[ e_{i}\left(\mathbf{p},\mathbf{y}_{i},\mathbf{x}_{i}\right)=\left\Vert \mathbf{y}_{i}-\mathbf{f}\left(\mathbf{x}_{i},\mathbf{p}\right)\right\Vert .\]
An objective function to pass to any of the previous minization
algorithms to obtain a least-squares fit is.
.. math::
:nowrap:
\[ J\left(\mathbf{p}\right)=\sum_{i=0}^{N-1}e_{i}^{2}\left(\mathbf{p}\right).\]
The :obj:`leastsq` algorithm performs this squaring and summing of the
residuals automatically. It takes as an input argument the vector
function :math:`\mathbf{e}\left(\mathbf{p}\right)` and returns the
value of :math:`\mathbf{p}` which minimizes
:math:`J\left(\mathbf{p}\right)=\mathbf{e}^{T}\mathbf{e}`
directly. The user is also encouraged to provide the Jacobian matrix
of the function (with derivatives down the columns or across the
rows). If the Jacobian is not provided, it is estimated.
An example should clarify the usage. Suppose it is believed some
measured data follow a sinusoidal pattern
.. math::
:nowrap:
\[ y_{i}=A\sin\left(2\pi kx_{i}+\theta\right)\]
where the parameters :math:`A,` :math:`k` , and :math:`\theta` are unknown. The residual vector is
.. math::
:nowrap:
\[ e_{i}=\left|y_{i}-A\sin\left(2\pi kx_{i}+\theta\right)\right|.\]
By defining a function to compute the residuals and (selecting an
appropriate starting position), the least-squares fit routine can be
used to find the best-fit parameters :math:`\hat{A},\,\hat{k},\,\hat{\theta}`.
This is shown in the following example:
.. plot::
>>> from numpy import *
>>> x = arange(0,6e-2,6e-2/30)
>>> A,k,theta = 10, 1.0/3e-2, pi/6
>>> y_true = A*sin(2*pi*k*x+theta)
>>> y_meas = y_true + 2*random.randn(len(x))
>>> def residuals(p, y, x):
... A,k,theta = p
... err = y-A*sin(2*pi*k*x+theta)
... return err
>>> def peval(x, p):
... return p[0]*sin(2*pi*p[1]*x+p[2])
>>> p0 = [8, 1/2.3e-2, pi/3]
>>> print array(p0)
[ 8. 43.4783 1.0472]
>>> from scipy.optimize import leastsq
>>> plsq = leastsq(residuals, p0, args=(y_meas, x))
>>> print plsq[0]
[ 10.9437 33.3605 0.5834]
>>> print array([A, k, theta])
[ 10. 33.3333 0.5236]
>>> import matplotlib.pyplot as plt
>>> plt.plot(x,peval(x,plsq[0]),x,y_meas,'o',x,y_true)
>>> plt.title('Least-squares fit to noisy data')
>>> plt.legend(['Fit', 'Noisy', 'True'])
>>> plt.show()
.. :caption: Least-square fitting to noisy data using
.. :obj:`scipy.optimize.leastsq`
.. _tutorial-sqlsp:
Sequential Least-square fitting with constraints (:func:`fmin_slsqp`)
---------------------------------------------------------------------
This module implements the Sequential Least SQuares Programming optimization algorithm (SLSQP).
.. math::
:nowrap:
\begin{eqnarray*} \min F(x) \\ \text{subject to } & C_j(X) = 0 , &j = 1,...,\text{MEQ}\\
& C_j(x) \geq 0 , &j = \text{MEQ}+1,...,M\\
& XL \leq x \leq XU , &I = 1,...,N. \end{eqnarray*}
The following script shows examples for how constraints can be specified.
::
"""
This script tests fmin_slsqp using Example 14.4 from Numerical Methods for
Engineers by Steven Chapra and Raymond Canale. This example maximizes the
function f(x) = 2*x0*x1 + 2*x0 - x0**2 - 2*x1**2, which has a maximum
at x0=2, x1=1.
"""
from scipy.optimize import fmin_slsqp
from numpy import array
def testfunc(x, *args):
"""
Parameters
----------
d : list
A list of two elements, where d[0] represents x and
d[1] represents y in the following equation.
args : tuple
First element of args is a multiplier for f.
Since the objective function should be maximized, and the scipy
optimizers can only minimize functions, it is nessessary to
multiply the objective function by -1 to achieve the desired
solution.
Returns
-------
res : float
The result, equal to ``2*x*y + 2*x - x**2 - 2*y**2``.
"""
try:
sign = args[0]
except:
sign = 1.0
return sign*(2*x[0]*x[1] + 2*x[0] - x[0]**2 - 2*x[1]**2)
def testfunc_deriv(x,*args):
""" This is the derivative of testfunc, returning a numpy array
representing df/dx and df/dy """
try:
sign = args[0]
except:
sign = 1.0
dfdx0 = sign*(-2*x[0] + 2*x[1] + 2)
dfdx1 = sign*(2*x[0] - 4*x[1])
return array([ dfdx0, dfdx1 ])
def test_eqcons(x,*args):
""" Lefthandside of the equality constraint """
return array([ x[0]**3-x[1] ])
def test_ieqcons(x,*args):
""" Lefthandside of inequality constraint """
return array([ x[1]-1 ])
def test_fprime_eqcons(x,*args):
""" First derivative of equality constraint """
return array([ 3.0*(x[0]**2.0), -1.0 ])
def test_fprime_ieqcons(x,*args):
""" First derivative of inequality constraint """
return array([ 0.0, 1.0 ])
from time import time
print "Unbounded optimization."
print "Derivatives of objective function approximated."
t0 = time()
result = fmin_slsqp(testfunc, [-1.0,1.0], args=(-1.0,), iprint=2, full_output=1)
print "Elapsed time:", 1000*(time()-t0), "ms"
print "Results", result, "\n\n"
print "Unbounded optimization."
print "Derivatives of objective function provided."
t0 = time()
result = fmin_slsqp(testfunc, [-1.0,1.0], fprime=testfunc_deriv, args=(-1.0,),
iprint=2, full_output=1)
print "Elapsed time:", 1000*(time()-t0), "ms"
print "Results", result, "\n\n"
print "Bound optimization (equality constraints)."
print "Constraints implemented via lambda function."
print "Derivatives of objective function approximated."
print "Derivatives of constraints approximated."
t0 = time()
result = fmin_slsqp(testfunc, [-1.0,1.0], args=(-1.0,),
eqcons=[lambda x, args: x[0]-x[1] ], iprint=2, full_output=1)
print "Elapsed time:", 1000*(time()-t0), "ms"
print "Results", result, "\n\n"
print "Bound optimization (equality constraints)."
print "Constraints implemented via lambda."
print "Derivatives of objective function provided."
print "Derivatives of constraints approximated."
t0 = time()
result = fmin_slsqp(testfunc, [-1.0,1.0], fprime=testfunc_deriv, args=(-1.0,),
eqcons=[lambda x, args: x[0]-x[1] ], iprint=2, full_output=1)
print "Elapsed time:", 1000*(time()-t0), "ms"
print "Results", result, "\n\n"
print "Bound optimization (equality and inequality constraints)."
print "Constraints implemented via lambda."
print "Derivatives of objective function provided."
print "Derivatives of constraints approximated."
t0 = time()
result = fmin_slsqp(testfunc,[-1.0,1.0], fprime=testfunc_deriv, args=(-1.0,),
eqcons=[lambda x, args: x[0]-x[1] ],
ieqcons=[lambda x, args: x[0]-.5], iprint=2, full_output=1)
print "Elapsed time:", 1000*(time()-t0), "ms"
print "Results", result, "\n\n"
print "Bound optimization (equality and inequality constraints)."
print "Constraints implemented via function."
print "Derivatives of objective function provided."
print "Derivatives of constraints approximated."
t0 = time()
result = fmin_slsqp(testfunc, [-1.0,1.0], fprime=testfunc_deriv, args=(-1.0,),
f_eqcons=test_eqcons, f_ieqcons=test_ieqcons,
iprint=2, full_output=1)
print "Elapsed time:", 1000*(time()-t0), "ms"
print "Results", result, "\n\n"
print "Bound optimization (equality and inequality constraints)."
print "Constraints implemented via function."
print "All derivatives provided."
t0 = time()
result = fmin_slsqp(testfunc,[-1.0,1.0], fprime=testfunc_deriv, args=(-1.0,),
f_eqcons=test_eqcons, fprime_eqcons=test_fprime_eqcons,
f_ieqcons=test_ieqcons, fprime_ieqcons=test_fprime_ieqcons,
iprint=2, full_output=1)
print "Elapsed time:", 1000*(time()-t0), "ms"
print "Results", result, "\n\n"
Scalar function minimizers
--------------------------
Often only the minimum of a scalar function is needed (a scalar
function is one that takes a scalar as input and returns a scalar
output). In these circumstances, other optimization techniques have
been developed that can work faster.
Unconstrained minimization (:func:`brent`)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
There are actually two methods that can be used to minimize a scalar
function (:obj:`brent` and :func:`golden`), but :obj:`golden` is
included only for academic purposes and should rarely be used. The
brent method uses Brent's algorithm for locating a minimum. Optimally
a bracket should be given which contains the minimum desired. A
bracket is a triple :math:`\left(a,b,c\right)` such that
:math:`f\left(a\right)>f\left(b\right)>> from scipy.special import j1
>>> from scipy.optimize import fminbound
>>> xmin = fminbound(j1, 4, 7)
>>> print xmin
5.33144184241
Root finding
------------
Sets of equations
^^^^^^^^^^^^^^^^^
To find the roots of a polynomial, the command :obj:`roots
` is useful. To find a root of a set of non-linear
equations, the command :obj:`fsolve` is needed. For example, the
following example finds the roots of the single-variable
transcendental equation
.. math::
:nowrap:
\[ x+2\cos\left(x\right)=0,\]
and the set of non-linear equations
.. math::
:nowrap:
\begin{eqnarray*}
x_{0}\cos\left(x_{1}\right) & = & 4,\\
x_{0}x_{1}-x_{1} & = & 5.
\end{eqnarray*}
The results are :math:`x=-1.0299` and :math:`x_{0}=6.5041,\, x_{1}=0.9084` .
>>> def func(x):
... return x + 2*cos(x)
>>> def func2(x):
... out = [x[0]*cos(x[1]) - 4]
... out.append(x[1]*x[0] - x[1] - 5)
... return out
>>> from scipy.optimize import fsolve
>>> x0 = fsolve(func, 0.3)
>>> print x0
-1.02986652932
>>> x02 = fsolve(func2, [1, 1])
>>> print x02
[ 6.50409711 0.90841421]
Scalar function root finding
^^^^^^^^^^^^^^^^^^^^^^^^^^^^
If one has a single-variable equation, there are four different root
finder algorithms that can be tried. Each of these root finding
algorithms requires the endpoints of an interval where a root is
suspected (because the function changes signs). In general
:obj:`brentq` is the best choice, but the other methods may be useful
in certain circumstances or for academic purposes.
Fixed-point solving
^^^^^^^^^^^^^^^^^^^
A problem closely related to finding the zeros of a function is the
problem of finding a fixed-point of a function. A fixed point of a
function is the point at which evaluation of the function returns the
point: :math:`g\left(x\right)=x.` Clearly the fixed point of :math:`g`
is the root of :math:`f\left(x\right)=g\left(x\right)-x.`
Equivalently, the root of :math:`f` is the fixed_point of
:math:`g\left(x\right)=f\left(x\right)+x.` The routine
:obj:`fixed_point` provides a simple iterative method using Aitkens
sequence acceleration to estimate the fixed point of :math:`g` given a
starting point.
Root finding: Large problems
----------------------------
The :obj:`fsolve` function cannot deal with a very large number of
variables (*N*), as it needs to calculate and invert a dense *N x N*
Jacobian matrix on every Newton step. This becomes rather inefficent
when *N* grows.
Consider for instance the following problem: we need to solve the
following integrodifferential equation on the square
:math:`[0,1]\times[0,1]`:
.. math::
(\partial_x^2 + \partial_y^2) P + 5 \left(\int_0^1\int_0^1\cosh(P)\,dx\,dy\right)^2 = 0
with the boundary condition :math:`P(x,1) = 1` on the upper edge and
:math:`P=0` elsewhere on the boundary of the square. This can be done
by approximating the continuous function *P* by its values on a grid,
:math:`P_{n,m}\approx{}P(n h, m h)`, with a small grid spacing
*h*. The derivatives and integrals can then be approximated; for
instance :math:`\partial_x^2 P(x,y)\approx{}(P(x+h,y) - 2 P(x,y) +
P(x-h,y))/h^2`. The problem is then equivalent to finding the root of
some function *residual(P)*, where *P* is a vector of length
:math:`N_x N_y`.
Now, because :math:`N_x N_y` can be large, :obj:`fsolve` will take a
long time to solve this problem. The solution can however be found
using one of the large-scale solvers in :mod:`scipy.optimize`, for
example :obj:`newton_krylov`, :obj:`broyden2`, or
:obj:`anderson`. These use what is known as the inexact Newton method,
which instead of computing the Jacobian matrix exactly, forms an
approximation for it.
The problem we have can now be solved as follows:
.. plot::
import numpy as np
from scipy.optimize import newton_krylov
from numpy import cosh, zeros_like, mgrid, zeros
# parameters
nx, ny = 75, 75
hx, hy = 1./(nx-1), 1./(ny-1)
P_left, P_right = 0, 0
P_top, P_bottom = 1, 0
def residual(P):
d2x = zeros_like(P)
d2y = zeros_like(P)
d2x[1:-1] = (P[2:] - 2*P[1:-1] + P[:-2]) / hx/hx
d2x[0] = (P[1] - 2*P[0] + P_left)/hx/hx
d2x[-1] = (P_right - 2*P[-1] + P[-2])/hx/hx
d2y[:,1:-1] = (P[:,2:] - 2*P[:,1:-1] + P[:,:-2])/hy/hy
d2y[:,0] = (P[:,1] - 2*P[:,0] + P_bottom)/hy/hy
d2y[:,-1] = (P_top - 2*P[:,-1] + P[:,-2])/hy/hy
return d2x + d2y + 5*cosh(P).mean()**2
# solve
guess = zeros((nx, ny), float)
sol = newton_krylov(residual, guess, verbose=1)
#sol = broyden2(residual, guess, max_rank=50, verbose=1)
#sol = anderson(residual, guess, M=10, verbose=1)
print 'Residual', abs(residual(sol)).max()
# visualize
import matplotlib.pyplot as plt
x, y = mgrid[0:1:(nx*1j), 0:1:(ny*1j)]
plt.pcolor(x, y, sol)
plt.colorbar()
plt.show()
Still too slow? Preconditioning.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
When looking for the zero of the functions :math:`f_i({\bf x}) = 0`,
*i = 1, 2, ..., N*, the :obj:`newton_krylov` solver spends most of its
time inverting the Jacobian matrix,
.. math:: J_{ij} = \frac{\partial f_i}{\partial x_j} .
If you have an approximation for the inverse matrix
:math:`M\approx{}J^{-1}`, you can use it for *preconditioning* the
linear inversion problem. The idea is that instead of solving
:math:`J{\bf s}={\bf y}` one solves :math:`MJ{\bf s}=M{\bf y}`: since
matrix :math:`MJ` is "closer" to the identity matrix than :math:`J`
is, the equation should be easier for the Krylov method to deal with.
The matrix *M* can be passed to :obj:`newton_krylov` as the *inner_M*
parameter. It can be a (sparse) matrix or a
:obj:`scipy.sparse.linalg.LinearOperator` instance.
For the problem in the previous section, we note that the function to
solve consists of two parts: the first one is application of the
Laplace operator, :math:`[\partial_x^2 + \partial_y^2] P`, and the second
is the integral. We can actually easily compute the Jacobian corresponding
to the Laplace operator part: we know that in one dimension
.. math::
\partial_x^2 \approx \frac{1}{h_x^2} \begin{pmatrix}
-2 & 1 & 0 & 0 \cdots \\
1 & -2 & 1 & 0 \cdots \\
0 & 1 & -2 & 1 \cdots \\
\ldots
\end{pmatrix}
= h_x^{-2} L
so that the whole 2-D operator is represented by
.. math::
J_1 = \partial_x^2 + \partial_y^2
\simeq
h_x^{-2} L \otimes I + h_y^{-2} I \otimes L
The matrix :math:`J_2` of the Jacobian corresponding to the integral
is more difficult to calculate, and since *all* of it entries are
nonzero, it will be difficult to invert. :math:`J_1` on the other hand
is a relatively simple matrix, and can be inverted by
:obj:`scipy.sparse.linalg.splu` (or the inverse can be approximated by
:obj:`scipy.sparse.linalg.spilu`). So we are content to take
:math:`M\approx{}J_1^{-1}` and hope for the best.
In the example below, we use the preconditioner :math:`M=J_1^{-1}`.
.. literalinclude:: examples/newton_krylov_preconditioning.py
Resulting run, first without preconditioning::
0: |F(x)| = 803.614; step 1; tol 0.000257947
1: |F(x)| = 345.912; step 1; tol 0.166755
2: |F(x)| = 139.159; step 1; tol 0.145657
3: |F(x)| = 27.3682; step 1; tol 0.0348109
4: |F(x)| = 1.03303; step 1; tol 0.00128227
5: |F(x)| = 0.0406634; step 1; tol 0.00139451
6: |F(x)| = 0.00344341; step 1; tol 0.00645373
7: |F(x)| = 0.000153671; step 1; tol 0.00179246
8: |F(x)| = 6.7424e-06; step 1; tol 0.00173256
Residual 3.57078908664e-07
Evaluations 317
and then with preconditioning::
0: |F(x)| = 136.993; step 1; tol 7.49599e-06
1: |F(x)| = 4.80983; step 1; tol 0.00110945
2: |F(x)| = 0.195942; step 1; tol 0.00149362
3: |F(x)| = 0.000563597; step 1; tol 7.44604e-06
4: |F(x)| = 1.00698e-09; step 1; tol 2.87308e-12
Residual 9.29603061195e-11
Evaluations 77
Using a preconditioner reduced the number of evaluations of the
*residual* function by a factor of *4*. For problems where the
residual is expensive to compute, good preconditioning can be crucial
--- it can even decide whether the problem is solvable in practice or
not.
Preconditioning is an art, science, and industry. Here, we were lucky
in making a simple choice that worked reasonably well, but there is a
lot more depth to this topic than is shown here.
.. rubric:: References
Some further reading and related software:
.. [KK] D.A. Knoll and D.E. Keyes, "Jacobian-free Newton-Krylov methods",
J. Comp. Phys. 193, 357 (2003).
.. [PP] PETSc http://www.mcs.anl.gov/petsc/ and its Python bindings
http://code.google.com/p/petsc4py/
.. [AMG] PyAMG (algebraic multigrid preconditioners/solvers)
http://code.google.com/p/pyamg/