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SOLVING MULTIPLE VARIABLE POLYNOMIAL EQUATION
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# 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