Analytical existence of solutions to a system of nonlinear equations with application

• Published in: Journal of computational and applied mathematics Vol. 234; no. 1; pp. 297 - 304
• Main Author: N’Guessan, Assi
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier B.V 01.05.2010; Elsevier
• Subjects: Calculus of variations and optimal control; Crash data; Exact sciences and technology; Existence; Mathematical analysis; Mathematics; Multinomial distribution; Newton–Raphson; Nonlinear algebraic and transcendental equations; Nonlinear system; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Road safety measure; Schur complement; Sciences and techniques of general use; Road safety measure; Nonlinear system; Newton–Raphson; Crash data; Multinomial distribution; Schur complement; Existence; Existence of solution; Optimization method; Iterative method; Non linear equation; Direct method; Numerical solution; Distribution function; Analytical solution; Newton-Raphson; Transcendental equation; Numerical linear algebra; Non linear system; Constrained optimization; Equation system; Case study; Numerical analysis; Linear system; Matrix inversion; Applied mathematics; Optimal solution; Algebraic equation
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2009.12.026

We study the existence of analytical solutions to a system of nonlinear equations under constraints linked to the analysis of a road safety measure without computing second derivatives. We formally demonstrate this existence of solutions by applying a matrix inversion principle through Schur complement to a subsystem of equations derived from the main system. The analytical results thus obtained are used to construct a simple iterative procedure to look for optimal solutions as well as an initial solution adapted to data of each case study. We illustrate our results with simulated accident data obtained from the setting-up of a road safety measure. The numerical solutions thus obtained are then compared to those given through a classic Newton–Raphson type approach directly applied to the main system.