Geodesic
A general Approach to Methods with a Sparse Jacobian for Solving Nonlinear Systems of Equations
Serdica Mathematical Journal, Tome 33 (2007) no. 4, pp. 433-448.

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Here we give methodological survey of contemporary methods for solving nonlinear systems of equations in Rn. The reason of this review is that many authors in present days rediscovered such classical methods. In particular, we consider Newton’s-type algorithms with sparse Jacobian. Method for which the inverse matrix of the Jacobian is replaced by the inverse matrix of the Vandermondian is proposed. A number of illustrative numerical examples are displayed. We demonstrate Herzberger’s model with fixed-point relations to the some discrete versions of Halley’s and Euler-Chebyshev’s methods for solving such kind of systems.
Keywords: Nonlinear Systems of Equations, Numerical Solution, Halley’s and Euler-Chebyshev’s Methods, Fixed-Point Relations 
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     title = {A general {Approach} to {Methods} with a {Sparse} {Jacobian} for {Solving} {Nonlinear} {Systems} of {Equations}},
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Kyurkchiev, Nikolay; Iliev, Anton. A general Approach to Methods with a Sparse Jacobian for Solving Nonlinear Systems of Equations. Serdica Mathematical Journal, Tome 33 (2007) no. 4, pp. 433-448. http://geodesic.mathdoc.fr/item/SMJ2_2007_33_4_a3/