root#

scipy.optimize.root(fun,x0,args=(),method='hybr',jac=None,tol=None,callback=None,options=None)[source]#

Find a root of a vector function.

Parameters:

funcallable

A vector function to find a root of.

Suppose the callable has signaturef0(x,*my_args,**my_kwargs), wheremy_args andmy_kwargs are required positional and keyword arguments.Rather than passingf0 as the callable, wrap it to acceptonlyx; e.g., passfun=lambdax:f0(x,*my_args,**my_kwargs) as thecallable, wheremy_args (tuple) andmy_kwargs (dict) have beengathered before invoking this function.

x0ndarray

Initial guess.

argstuple, optional

Extra arguments passed to the objective function and its Jacobian.

methodstr, optional

Type of solver. Should be one of

• ‘hybr’(see here)

• ‘lm’(see here)

• ‘broyden1’(see here)

• ‘broyden2’(see here)

• ‘anderson’(see here)

• ‘linearmixing’(see here)

• ‘diagbroyden’(see here)

• ‘excitingmixing’(see here)

• ‘krylov’(see here)

• ‘df-sane’(see here)

jacbool or callable, optional

Ifjac is a Boolean and is True,fun is assumed to return thevalue of Jacobian along with the objective function. If False, theJacobian will be estimated numerically.jac can also be a callable returning the Jacobian offun. Inthis case, it must accept the same arguments asfun.

tolfloat, optional

Tolerance for termination. For detailed control, use solver-specificoptions.

callbackfunction, optional

Optional callback function. It is called on every iteration ascallback(x,f) wherex is the current solution andfthe corresponding residual. For all methods but ‘hybr’ and ‘lm’.

optionsdict, optional

A dictionary of solver options. E.g.,xtol ormaxiter, seeshow_options() for details.

Returns:

solOptimizeResult

The solution represented as aOptimizeResult object.Important attributes are:x the solution array,success aBoolean flag indicating if the algorithm exited successfully andmessage which describes the cause of the termination. SeeOptimizeResult for a description of other attributes.

See also

show_options

Additional options accepted by the solvers

Notes

This section describes the available solvers that can be selected by the‘method’ parameter. The default method ishybr.

Methodhybr uses a modification of the Powell hybrid method asimplemented in MINPACK[1].

Methodlm solves the system of nonlinear equations in a least squaressense using a modification of the Levenberg-Marquardt algorithm asimplemented in MINPACK[1].

Methoddf-sane is a derivative-free spectral method.[3]

Methodsbroyden1,broyden2,anderson,linearmixing,diagbroyden,excitingmixing,krylov are inexact Newton methods,with backtracking or full line searches[2]. Each method correspondsto a particular Jacobian approximations.

• Methodbroyden1 uses Broyden’s first Jacobian approximation, it isknown as Broyden’s good method.

• Methodbroyden2 uses Broyden’s second Jacobian approximation, itis known as Broyden’s bad method.

• Methodanderson uses (extended) Anderson mixing.

• MethodKrylov uses Krylov approximation for inverse Jacobian. Itis suitable for large-scale problem.

• Methoddiagbroyden uses diagonal Broyden Jacobian approximation.

• Methodlinearmixing uses a scalar Jacobian approximation.

• Methodexcitingmixing uses a tuned diagonal Jacobianapproximation.

Warning

The algorithms implemented for methodsdiagbroyden,linearmixing andexcitingmixing may be useful for specificproblems, but whether they will work may depend strongly on theproblem.

Added in version 0.11.0.

References

[1](1,2)

More, Jorge J., Burton S. Garbow, and Kenneth E. Hillstrom.1980. User Guide for MINPACK-1.

[2]

C. T. Kelley. 1995. Iterative Methods for Linear and NonlinearEquations. Society for Industrial and Applied Mathematics.DOI:10.1137/1.9781611970944

[3]

W. La Cruz, J.M. Martinez, M. Raydan.Math. Comp. 75, 1429 (2006).

Examples

Try it in your browser!

The following functions define a system of nonlinear equations and itsjacobian.

>>>importnumpyasnp>>>deffun(x):...return[x[0]+0.5*(x[0]-x[1])**3-1.0,...0.5*(x[1]-x[0])**3+x[1]]

>>>defjac(x):...returnnp.array([[1+1.5*(x[0]-x[1])**2,...-1.5*(x[0]-x[1])**2],...[-1.5*(x[1]-x[0])**2,...1+1.5*(x[1]-x[0])**2]])

A solution can be obtained as follows.

>>>fromscipyimportoptimize>>>sol=optimize.root(fun,[0,0],jac=jac,method='hybr')>>>sol.xarray([ 0.8411639,  0.1588361])

Large problem

Suppose that we needed to solve the following integrodifferentialequation 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 ofthe square.

The solution can be found using themethod='krylov' solver:

>>>fromscipyimportoptimize>>># 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

>>>defresidual(P):...d2x=np.zeros_like(P)...d2y=np.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......returnd2x+d2y-10*np.cosh(P).mean()**2

>>>guess=np.zeros((nx,ny),float)>>>sol=optimize.root(residual,guess,method='krylov')>>>print('Residual:%g'%abs(residual(sol.x)).max())Residual: 5.7972e-06  # may vary

>>>importmatplotlib.pyplotasplt>>>x,y=np.mgrid[0:1:(nx*1j),0:1:(ny*1j)]>>>plt.pcolormesh(x,y,sol.x,shading='gouraud')>>>plt.colorbar()>>>plt.show()

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