# REVISED SOLUTION OF ILL-POSED ALGEBRAIC SYSTEMS FOR NOISE DATA

( Pp. 14-19)

More about authors

Ternovski Vladimir V.
docent fakulteta VMK

Lomonosov Moscow State University, Russia Khapaev Mikhail M. professor fakulteta VMK

Lomonosov Moscow State University, Russia

Lomonosov Moscow State University, Russia Khapaev Mikhail M. professor fakulteta VMK

Lomonosov Moscow State University, Russia

Abstract:

The problem of numerical solution of linear algebraic equations is known to be ill-posed in the sense that small perturbation in the right hand side may lead to large errors in the numerical solution. It is important to verify the accuracy of approximate solution by taking all possible errors of elements of a matrix, a vector of the right hand side, and roundoff errors into account. There are computational difficulties with ill-posed systems as well. If to apply standard methods, for example, a method of Gauss elimination, for such systems it isn't possible to catch the correct solution though discrepancy can be less accuracy of data-in and roundoff errors. The small discrepancy doesn't guarantee proximity to the correct solution. Actually there is no need for preliminary study of assessing whether a given system of linear algebraic equations is inherently ill-conditioned or well-conditioned. The new approach to the solution of algebraic systems based on statistical effect in matrixes of a big order is considered. The conditionality of the systems of equation changes with a high probability at a matrix distorted by random noise. After standard methods to be applied the received "chaotic" solution is used as a source of a priori information in more general variational problem

How to Cite:

Ternovski V.V., Khapaev M.M., (2015), REVISED SOLUTION OF ILL-POSED ALGEBRAIC SYSTEMS FOR NOISE DATA. Computational Nanotechnology, 2: 14-19.

Reference list:

A. Tikhonov and V. Arsenin, Solutions of ill-posed problems. Winston, Washington, DC(1977).

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

A. Bakushinsky and A. Goncharsky, Ill-posed problems: theory and applications. Springer Netherlands (1994).

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.

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

A. Bakushinsky and A. Goncharsky, Ill-posed problems: theory and applications. Springer Netherlands (1994).

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.

Keywords:

ill-posed problems, condition numbers, random matrix.