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A Novel Method for Solving Ill-conditioned Systems of Linear Equations with Extreme Physical Property Contrasts

Cheng-Yu Ku1

Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung 20224, Taiwan.
Corresponding author. Email:

Computer Modeling in Engineering & Sciences 2013, 96(6), 409-434.


This paper proposes a novel method, named the dynamical Jacobianinverse free method (DJIFM), with the incorporation of a two-sided equilibrium algorithm for solving ill-conditioned systems of linear equations with extreme physical property contrasts. The DJIFM is based on the construction of a scalar homotopy function for transforming the vector function of linear or nonlinear algebraic equations into a time-dependent scalar function by introducing a fictitious time-like variable. The DJIFM demonstrated great numerical stability for solving linear or nonlinear algebraic equations, particularly for systems involving ill-conditioned Jacobian or poor initial values that cause convergence problems. With the incorporation of a newly developed two-sided equilibrium algorithm, the solution of layered problems with extreme contrasts in the physical property that are typically highly ill-conditioned can be solved. The proposed method was then adopted for the solution of several highly ill-conditioned numerical examples, including the linear Hilbert matrix, linear Vandermonde matrix, layered linear and nonlinear groundwater flow problems. The results revealed that using the DJIFM and the two-sided equilibrium algorithm can improve the convergence and increase the numerical stability for solving layered problems.


Cite This Article

Ku, C. (2013). A Novel Method for Solving Ill-conditioned Systems of Linear Equations with Extreme Physical Property Contrasts. CMES-Computer Modeling in Engineering & Sciences, 96(6), 409–434.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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