Exact Solution of a Linear Difference Equation in a Finite Number of Steps
Keywords:
Linear difference equation, Exact iterative solution of a system of linear algebraic equations, Nilpotent matrix
Abstract
An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations is approximate. The nilpotency of the iteration matrix is the necessary and sufficient condition for getting an exact solution. The examples of iterative equations providing an exact solution to the simplest algebraic system are presented.
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References
[1] Strang, G., Linear Algebra and Its Applications, Academic Press, New York, (1976).
[2] Rice, J.R., Matrix Computations and Mathematical Software, McGraw-Hill, Inc., New York, (1981).
[3] Demmel, W.D., Applied Numerical Linear Algebra, Society for Industrial and Applied Mathematics, Philadelphia, (1997).
[4] Watkins, D.S., Fundamentals of Matrix Computations, John Wiley & Sons, Inc., New York, Second Edition, (2002).
[5] Iskhakov, A., Pospelov, V. and Skovpen, S., Non-Frobenius Spectrum-Transformation Method, Applied Mathematics, Vol. 3, No. 1, (2012), pp. 1471-1479.
[6] Iskhakov, A., Skovpen, S., A Direct Transformation of a Matrix Spectrum, Journal of Progressive Research in Mathematics, Vol. 5, No. 1, (2015), pp. 463-481.
[2] Rice, J.R., Matrix Computations and Mathematical Software, McGraw-Hill, Inc., New York, (1981).
[3] Demmel, W.D., Applied Numerical Linear Algebra, Society for Industrial and Applied Mathematics, Philadelphia, (1997).
[4] Watkins, D.S., Fundamentals of Matrix Computations, John Wiley & Sons, Inc., New York, Second Edition, (2002).
[5] Iskhakov, A., Pospelov, V. and Skovpen, S., Non-Frobenius Spectrum-Transformation Method, Applied Mathematics, Vol. 3, No. 1, (2012), pp. 1471-1479.
[6] Iskhakov, A., Skovpen, S., A Direct Transformation of a Matrix Spectrum, Journal of Progressive Research in Mathematics, Vol. 5, No. 1, (2015), pp. 463-481.
Published
2018-03-22
How to Cite
Iskhakov, A., & Skovpen, S. (2018). Exact Solution of a Linear Difference Equation in a Finite Number of Steps. Journal of Progressive Research in Mathematics, 13(2), 2259-2262. Retrieved from http://www.scitecresearch.com/journals/index.php/jprm/article/view/1459
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