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Volume 06,Issue 01
   Volume & Issue: Volume 06,Issue 01



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

  Volume 06,Issue 01

 

Author :

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.

 

Keywords :

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

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