Volume 06,Issue 01

On Ill-conditioned Linear System Due to the Levenberg-Marquardt Algorithm in Solving Nonlinear Equations

Authors

Seyyed Shahabeddin Azimi, Mansour Kalbasi

Abstract
In this paper, as an important problem, the nonlinear equations resulting from the finite volume approximation of the differential equations describing the nanofluid convective heat transfer in a tube are considered to obtain the temperature distribution. One of the numerical algorithms for solving nonlinear equations is the Levenberg-Marquardt in which a system of linear equations must be solved per iteration. In order to prevent the linear system from being ill-conditioned, the damping factor (?) of this algorithm should not be decreased unconditionally. Indeed, it was proposed that a minimum value for ? is set per iteration for obtaining the search direction, where the time complexity per iteration is O(n3) in which n is the number of unknowns (or equations) of the nonlinear system. However, in this case study, this linear system is ill-conditioned. It is shown that the ill conditioning of the linear system due to this case study can be prevented in all iterations by setting an approximate minimum value for ? in the first iteration. This can considerably reduce the computational cost, in other words, this paper introduces an algorithm with a low computational complexity.

Keyword: Damping Factor, Ill-conditioning, Levenberg-Marquardt Algorithm, Linear System, Nonlinear Equations.

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