Ill-Posed Algebraic Systems with Noise Data
Applied and Computational Mathematics
Volume 4, Issue 3, June 2015, Pages: 220-224
Received: May 31, 2015;
Accepted: Jun. 6, 2015;
Published: Jun. 19, 2015
Views 3423 Downloads 81
Vladimir V. Ternovski, Numerical Math and Cyber Departament, Lomonosov State University, Moscow, Russia
Mikhail M. Khapaev, Numerical Math and Cyber Departament, Lomonosov State University, Moscow, Russia
Alexander S. Grushicin, Information Systems Department, MATI Russian State Technological University, Moscow, Russia
Follow on us
Finding a numerical solution of linear algebraic equations is known to present an ill-posed in the sense that small perturbation in the right hand side may lead to large errors in the solution. It is important to verify the accuracy of an approximate solution by taking into account all possible errors in the elements of the matrix, and of the vector at the right hand side as well as roundoff errors. There may be computational difficulties with ill-posed systems as well. If to apply standard methods such as the method of Gauss elimination to such systems it may be not possible to obtain the correct solution though discrepancy can be less accuracy of data errors. Besides, a small discrepancy will not always guarantee proximity to a correct solution. Actually there is no need for preliminary assessment whether a given system of linear algebraic equations is inherently ill-conditioned or well-conditioned. In this paper we consider a new approach to the solution of algebraic systems, which is based on statistical effect in matrices of big order. It will be shown that the conditionality of the systems of equation may change with a high probability, if the matrix distorted by random noise. After applying some standard methods, we may introduce the received "chaotic" solution is used as a source of a priori information a more general variational problem.
Ill-Posed Problems, Condition Numbers, Random Matrix
To cite this article
Vladimir V. Ternovski,
Mikhail M. Khapaev,
Alexander S. Grushicin,
Ill-Posed Algebraic Systems with Noise Data, Applied and Computational Mathematics.
Vol. 4, No. 3,
2015, pp. 220-224.
A. Tikhonov and V. Arsenin, Solutions of ill-posed problems. Winston, Washington, DC(1977).
A.N. Tikhonov, A. S. Leonov, A. G. Yagola, Nonlinear Ill-Posed Problems (Applied Mathematical Sciences), Springer; Softcover reprint of the original 1st ed. 1998 edition (February 7, 2014).
Ivanov, Valentin K., Vladimir V. Vasin, and Vitalii P. Tanana. Theory of linear ill-posed problems and its applications. Vol. 36. Walter de Gruyter, 2002.
N. N. Kalitkin, L. F. Yuhno, L. V. Kuz’mina, Quantitative criterion of conditioning for systems of linear algebraic equations, Mathematical Models and Computer Simulations October 2011, Volume 3, Issue 5, pp 541-556
Hansen, P. C. (2007). Regularization tools version 4.0 for Matlab 7.3. Numerical Algorithms, 46(2), 189-194.
A. Bakushinsky and A. Goncharsky, Ill-posed problems: theory and applications. Springer Netherlands (October 9, 2012).
Terence Tao and Van Vu. Smooth analysis of the condition number and the least singular value. Mathematics of computation ,Volume 79, Number 272, October 2010, Pages 2333–2352
A. Edelman. Eigenvalues and condition numbers of random matrices. SIAM j. Matrix Anal. Appl., Vol.9, No. 4, October, 1988, Pages 543- 560.
David S. Watkins. Fundamentals of Matrix Computations, Third Edition John Wiley and Sons, July 2010, 644 pp.