Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method

ABSTRACT

In this paper a new ODE numerical integration method was successfully applied to solving nonlinear equations. The method is of same simplicity as fixed point iteration, but the efficiency has been significantly improved, so it is especially suitable for large scale systems. For Brown’s equations, an existing article reported that when the dimension of the equation N = 40, the subroutines they used could not give a solution, as compared with our method, we can easily solve this equation even when N = 100. Other two large equations have the dimension of N = 1000, all the existing available methods have great difficulties to handle them, however, our method proposed in this paper can deal with those tough equations without any difficulties. The sigularity and choosing initial values problems were also mentioned in this paper.

In this paper a new ODE numerical integration method was successfully applied to solving nonlinear equations. The method is of same simplicity as fixed point iteration, but the efficiency has been significantly improved, so it is especially suitable for large scale systems. For Brown’s equations, an existing article reported that when the dimension of the equation N = 40, the subroutines they used could not give a solution, as compared with our method, we can easily solve this equation even when N = 100. Other two large equations have the dimension of N = 1000, all the existing available methods have great difficulties to handle them, however, our method proposed in this paper can deal with those tough equations without any difficulties. The sigularity and choosing initial values problems were also mentioned in this paper.

KEYWORDS

Nonlinear Equations, Ordinary Differential Equations, Numerical Integration, Fixed Point Iteration, Newton’s Method, Stiff, Ill-Conditioned

Nonlinear Equations, Ordinary Differential Equations, Numerical Integration, Fixed Point Iteration, Newton’s Method, Stiff, Ill-Conditioned

Cite this paper

nullT. Han and Y. Han, "Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method,"*Applied Mathematics*, Vol. 1 No. 3, 2010, pp. 222-229. doi: 10.4236/am.2010.13027.

nullT. Han and Y. Han, "Solving Large Scale Nonlinear Equations by a New ODE Numerical Integration Method,"

References

[1] P. Deuflhard, “Newton Methods for Nonlinear Problems Affine Invariance and Adaptive Algorithms,” Springer- Verlag, Berlin, Heidelberg, 2004.

[2] P. T. Boggs, “The Solution of Nonlinear Systems of Equations by A-Stable Integration Technique,” SIAM Journal on Numerical Analysis, Vol. 8, No. 4, 1971, pp. 767-785.

[3] J. M. Ortega and W. C. Rheinboldt, “Iterative Solution of Nonlinear Equations in Several Variables,” Academic Press, New York, 1970.

[4] T. M. Han and Y. H. Han, “Solving Implicit Equations Arising from Adams-Moulton Methods,” BIT, Vol. 42, No. 2, 2002, pp. 336-350.

[5] K. M. Brown, “Computer Oriented Algorithms for Solving Systems of Simultaneous Nonlinear Algebraic Equations,” In: G. D. Byrne and C. A. Hall, Eds., Numerical Solution of Systems of Nonlinear Algebraic Equations, Academic Press, New York, 1973, pp. 281-348.

[6] J. J. Moré, B. S. Garbow and K. E. Hillstrom, “Testing Unconstrained Optimization Software,” ACM Transactions on Mathematical Software, Vol. 7, No. 1, 1981, pp. 17-41.

[7] J. J. Moré and M. Y. Cosnard, “Numerical Solution of Nonlinear Equations,” ACM Transactions on Mathematical Software, Vol. 5, No. 1, 1979, pp. 64-85.

[1] P. Deuflhard, “Newton Methods for Nonlinear Problems Affine Invariance and Adaptive Algorithms,” Springer- Verlag, Berlin, Heidelberg, 2004.

[2] P. T. Boggs, “The Solution of Nonlinear Systems of Equations by A-Stable Integration Technique,” SIAM Journal on Numerical Analysis, Vol. 8, No. 4, 1971, pp. 767-785.

[3] J. M. Ortega and W. C. Rheinboldt, “Iterative Solution of Nonlinear Equations in Several Variables,” Academic Press, New York, 1970.

[4] T. M. Han and Y. H. Han, “Solving Implicit Equations Arising from Adams-Moulton Methods,” BIT, Vol. 42, No. 2, 2002, pp. 336-350.

[5] K. M. Brown, “Computer Oriented Algorithms for Solving Systems of Simultaneous Nonlinear Algebraic Equations,” In: G. D. Byrne and C. A. Hall, Eds., Numerical Solution of Systems of Nonlinear Algebraic Equations, Academic Press, New York, 1973, pp. 281-348.

[6] J. J. Moré, B. S. Garbow and K. E. Hillstrom, “Testing Unconstrained Optimization Software,” ACM Transactions on Mathematical Software, Vol. 7, No. 1, 1981, pp. 17-41.

[7] J. J. Moré and M. Y. Cosnard, “Numerical Solution of Nonlinear Equations,” ACM Transactions on Mathematical Software, Vol. 5, No. 1, 1979, pp. 64-85.