ZERO -- Zeros of Functions and Systems

The following table shows performance of Newtons method for the problem f1=x**2-y-2=0 and f2=x*y+1=0 with initial guess -4,4

x1	x2	f1	f2	||inv(jacob)||
-400	400	100	-150	.33333
-2.4722	1.7778	2.3341	-3.3951	.48011
-1.8176	.87522	.42851	-.59082	.60279
-1.6346	.63831	.33506E-01	-.43366E-01	.65321
-1.6182	.61819	.26912E-03	-.33013E-03	.65832

See also the picture. The solutions of the system occurwhere the three colors corresponding to the surfaces z=f1(x,y), z=f2(x,y) and z=0(red , dark blue and light blue) meet. The iterates are represented by vertical bars.

Single Variable

		f is a function of one real variable (Brent-Decker method), using function values only, zerofinder , needs inclusion by change of sign on interval
	SERVER	server routines for brent.shar (C)
	DFZERO	SLATEC root finder Java version
	polyzero	f is a polynomial of one complex variable and has real coefficients (least squares method, interactive/file input, f77/C)
	MultRoot	f polynomial with real (inexact) coefficients, especially for multiple roots (Matlab)
	companion C-version	f polynomial with real coefficients, QR for eigenvalues of companion matrix
	Aberth C-version	f is a polynomial with complex coefficients
	Muller C-version	f is a general function of one complex variable
	ZERO	search for a zero in a given interval
	RPOLY	find all zeroes of a real polynomial(Jenkins-Traub method)f90-version
	Madsen	find all zeroes of a real polynomial, C++ version
	CPOLY	find all zeroes of a complex polynomial(Jenkins-Traub method)	find all zeroes of a complex polynomial(several methods, root refinement, Windows version)
	RROOT	safely enclosing zeroes of a continuous function in an interval

Multiple Variable

	KINSOL	part of the SUNDIALS suite for ODE/DAE (C, Fortran interface)
	ALIAS	Solve systems of equations or inequalities with interval arithmetic (C++)
	box_zeros	f depends on one or more complex variables, zeros sought in box (based on TOMS666)
	HYBRJ	f not too nonlinear, exact Jacobian (Powell's hybrid method)
	HYBRD f90 version	not too nonlinear, Powells hybrid method, finite difference Jacobian
	DogLeg	Powell's dogleg method (Matlab)
	csolve	Robust solver by C. Sims (Matlab)
	BROYDEN-MEX	Matlab interface to quasi-Newton Fortran code
	STRSCNE	Trust region method for bound-constrained nonlinear systems (Matlab)
	CRBond's code	real/complex one/multidimensional rootfinders and LS solvers (C/C++)
	Filtrane	part of Galahad, nonlinear constraint solver, filter method, f90
	PITCON	f not too nonlinear, no good initial guess available, continuation method
	ALCON1	Continuation method for systems of algebraic equations f(x,tau)=0. Optional computation of turning and (simple) bifurcation points (by Deuflhard and Kunkel)
	ALCON2	Continuation method for systems of algebraic equations f(x,tau)=0. Optional computation of turning and (simple) bifurcation points (by Deulfhard, Fiedler and Kunkel). Optional automatic construction of completebifurcation diagrams.
	ALCON_S	Continuation method for systems of algebraic equations f(x,tau)=0. Large and sparse systems. (from elib/codelib, author Klein-Robbenhaar)
	LOCA	Library of continuation algorithms; uses bordering algorithm to permit very large dimensions; distributed memory parallel (C)
	Continuation Toolbox	Includes continuation of limit cycles and codim 1 bifurcations (Matlab)
	MATDS	Another Matlab continuation toolbox (for dynamical systems)
	nleq1	f strongly nonlinear, damped Newtons methodMatlab version
	nleq1s	damped Newtons method, large and sparse problems, direct linear algebra
	nleq2	damped Newtons method, provision for singular Jacobian
	NLEQ-paper	PDF file on above nleq* codes
	SNES	part of PETSC, serial and MPI parallel versions
	PEQN/PEQL	inexact Newton and inverse column update methods, for large/sparse problems
	giant	damped Newtons method, large and sparse problems, iterative linear algebra
	nnes	zeros sought in box: l,see also users' guide
	DAFNE	differential equations approach with second order system inspired by mechanics
	CHABIS	characteristic bisection method
	EPSILON	Maple-based package for operations on polynomials including solving systems
	HOMPACK	globally convergent continuation method for polynomial systems
	HOMPACK90	f90 version of HOMPACK
	POLSYS_PLP	more sophisticated version of HOMPACK90
	PHoM	Polyhedral Homotopy Continuation Method for Polynomial Systems (C++), module CMPS also in Matlab
	TENSOLVE	A Software Package for Solving Systems of Nonlinear Equations and Nonlinear Least-squares Problems Using Tensor Methods	A general-purpose solver for polynomial systems by homotopy continuation
	INTOPT_90	f90, see under Interval software

(Constrained) minimization approach

Rootfinders based on Newtons method or its modification will run intoconsiderable trouble when presented with a problem where singularities ofthe Jacobian occur. In this case the use of unconstrained orconstrained minimization might help,since there exist minimization methods which are rather robust if the constraints gradients are linearly dependent. To use an unconstrained minimizer, choose e.g. f(x)=||F(X)||^2 (theEuclidean length squared) and minimize this with some code proposed inthe unconstrained minimization section. To use a constrained minimizer, choose simplyf==0 as an objective function and minimize this under the constraint F(x)=0.The following code works often successfully in this situation:

LANCELOT A or LANCELOT B

augmented Lagrangian based method

Date 02-07-2006

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