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Manuscript Title: CANM, a program for numerical solution of a system of nonlinear equations using the continuous analog of Newton's method.
Authors: A. Abrashkevich, I.V. Puzynin
Program title: CANM
Catalogue identifier: ADSN
Journal reference: Comput. Phys. Commun. 156(2004)154
Programming language: Fortran 90.
Computer: IBM RS/6000 Model 320H, SGI Origin2000, SGI Octane, HP 9000/755, Intel Pentium IV PC.
Operating system: IRIX64 6.1/4/5, AIX 3.4, HP-UX 9.01, Linux 2.4.7.
RAM: 10K words
Word size: 64
Keywords: Nonlinear equations, Newton's method, Continuous analog of Newton's method, Continuous parameter, Evolutionary differential equation, Iteration, Iterative parameter, General purpose, Fitting.
Classification: 4.3, 4.9.

Nature of problem:
A system of nonlinear simultaneous equations Fi(x1,x2,...,xn)=0, 1<=i<=n, is numerically solved. It can be written in vector form as F(X) = 0, X ∈ Rn, where F:Rn -> Rn is a twice continuously differentiable function with domain and range in n-dimensional Euclidean space. The solutions of such systems of equations are often the last step in the solution of practical problems arising in physics and engineering. The purpose of this paper is to present the iterative procedure for finding zeroes of a system of n nonlinear equations in n variables using the continuous analog of Newton's method (CANM).

Solution method:
A system of n nonlinear simultaneous equations is solved by the iterative CANM procedure. In this approach, the solution of a system F(X) = 0 is reduced to a solution of evolutionary differential equations d/dt F(X(t)) = -F(X(t)), X(0) = X0 with respect to an additional continuous parameter t, 0 <= t < ∞. The resulting Cauchy problem is solved by Euler's method with variable step length τ. This iterative procedure with step τ specified in different ways, yields a variety of iterative Newton schemes that are frequently superior to the classical scheme.

The computer memory requirements depend on the number of nonlinear simultaneous equations to be solved. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements. The user must also supply subroutine FCN which evaluates the nonlinear functions. The user has the option of either providing a subroutinev JAC which calculates the Jacobian matrix or allowing the program to calculate it by the forward-difference approximation.

Unusual features:
The program uses routines dgeco and dges1 from the LINPACK Program Library.

Running time:
The running time depends critically upon the number of nonlinear equations to be solved. The test run which accompanies this paper took 0.06 seconds on the SGI Origin 2000.