# Iterative method for calculation of weighted pseudoinverse matrices with mixed weights on the basis of their decompositions into matrix power series

 1Vareniuk, NA1Galba, EF1Sergienko, IVTukalevska, NI1V.M. Glushkov Institute of Cybernetics of the NAS of Ukraine, Kyiv Dopov. Nac. akad. nauk Ukr. 2020, 8:19-25 https://doi.org/10.15407/dopovidi2020.08.019 Section: Information Science and Cybernetics Language: Ukrainian Abstract:  The decompositions of weighted pseudoinverse matrices with mixed weights (one of the weighted matrices is positive definite, and an other one is nonsingular indefinite) into matrix power series with positive exponents are obtained and investigated. Iterative methods for calculation of weighted pseudoinverse matrices with mixed weights are built and investigated on the basis of obtained expansions of weighted pseudoinverse matrices. Weighted spectral decompositions of symmetrized matrices, properties of these matrices associated with weighted pseudoinverse matrices, and the representation of weighted pseudoinverse matrices with mixed weights in terms of the coefficients of characteristic polynomials of symmetrizable matrices are the mathematical apparatus for constructing and studying the iterative methods for calculating these weighted pseudoinverse matrices. The choice of the iterative parameter is substantiated that provides the convergence of iterative processes. The iterative processes of two types of weighted pseudoinverse matrices are considered. Keywords: iterative methods, matrix power series, weighted pseudoinverse matrices with indefinite and mixed weights
References:

1. Chipman, J. S. (1964). On least squares with insufficient observation. J. Amer. Statist. Assoc., 59, No. 308, pp. 1078-1111. https://doi.org/10.1080/01621459.1964.10480751
2. Ward, J. F., Boullion, T. L. & Lewis, T.O. (1971). Weighted pseudoinverses with singular weights. SIAM J. Appl. Math., 21, No. 3, pp. 480-482. https://doi.org/10.1137/0121051
3. Galba, E. F., Deineka, V. S. & Sergienko, I. V. (2009). Weighted pseudoinverses and weighted normal pseudoso lutions with singular weights. Comput. Math. Math. Phys., 49, No. 8, pp. 1281-1297. https://doi.org/10.1134/S0965542509080016
4. Sergienko, I. V., Galba, E. F. & Deineka, V. S. (2011). Existence and uniqueness of weighted pseudoinverse matrices and weighted normal pseudosolutions with singular weights. Ukr. Math. J., 63, Art. 98. https://doi.org/10.1007/s11253-011-0490-3
5. Sergienko, I. V., Galba, Y. F. & Deineka, V. S. (2011). Existence and uniqueness theorems in the theory of weighted pseudoinverses with singular weights. Cybern. Syst. Anal., 47, Iss. 1, pp. 11-28. https://doi.org/10.1007/s10559-011-9286-6
6. Sergienko, I. V. & Galba, E. F. (2016). Weighted pseudoinversion with singular weights. Cybern. Syst. Anal., 52, pp. 708-729. https://doi.org/10.1007/s10559-016-9873-7
7. Galba, E. F., Sergienko, I. V. (2018). Methods for Computing Weighted Pseudoinverses and Weighted Normal Pseudosolutions with Singular Weights. Cybern. Syst. Anal. 54, pp. 398-422. https://doi.org/10.1007/s10559-018-0042-z
8. Mitra, S. K. & Rao, C. R. (1974). Projections under seminorms and generalized Moore-Penroze inverses. Linear Algebra Appl., No. 9, pp. 155-167. https://doi.org/10.1016/0024-3795(74)90034-2
9. Rao, C. R. & Mitra, S. K. (1971). Generalized inverse of matrices and its applikations. New York: Wiley.
10. Varenyuk, N. A., Galba, E. F., Sergienko, I. V. & Khimich, A. N. (2018). Weighted Pseudoinversion with Indefinite Weights. Ukr. Math. J., 70, pp. 866-889. https://doi.org/10.1007/s11253-018-1539-3
11. Galba, E. F. & Vareniuk, N. A. (2019). Expansions of weighted pseudoinverses with mixed weights into matrix power series and power products. Cybern. Syst. Anal., 55, pp. 760-771. https://doi.org/10.1007/s10559-019-00186-9
12. Nikolaevskaya, E. A. & Khimich, A. N. (2009). Error estimation for a weighted minimum-norm least squares solution with positive definite weights. Comput. Math. and Math. Phys., 49, pp. 409-417. https://doi.org/10.1134/S0965542509030038