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J−HOUSEHOLDER MATRICES AND CONDENSED FORMS | ||
| Journal of New Researches in Mathematics | ||
| مقاله 9، دوره 8، شماره 35، مهر و آبان 2022، صفحه 109-116 اصل مقاله (267.34 K) | ||
| نوع مقاله: research paper | ||
| نویسنده | ||
Mojtaba Ghasemi 
| ||
| Department of Mathematics, Lorestan University, Khorramabad, Iran | ||
| چکیده | ||
| Abstract. The main concept in this paper is the notion of the J-Householder matrix and its main applications. From these cases are the achievement to QR-decomposition, where Q is a J-Orthogonal matrix and R is an upper triangular matrix and reduction to the Hessenberg form and the tridiagonal form, for J-symmetric matrices. The reduction problem to condensed forms of triangular, Hessenberg and tridiagonal is one of the important problem in the numerical linear algebra. It is the structures of these condensed forms that are exploited in the solution of the reduced problem. For example, as we have seen in [2], [3],[7], [8], [6], [9] and [10], the solution of the linear system Ax = b is usually obtained by first triangularizing the matrix A and then solving an equivalent triangular system. In [8], for reduction to a condensed form, the concept of J−unitary similarity is used, while in the rest is used in the ordinary sense. In eigenvalue computations, the matrix A is transformed to a Hessenberg form befor applying the QR iterations. In [1], for reduction to a condensed form, the concept of J−unitary similarity is used. These condensed forms are Householder transformations and mybe J−Householder transformations. | ||
| کلیدواژهها | ||
| Indefinite inner product؛ J-Householder matrix؛ J-Orthonormal matrix؛ J-symmetric matrices؛ QR-Factorization | ||
| مراجع | ||
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]1[ حسین موسائی، سعید کتابچی، محمد تقی فولادی، روش لاگرانژ بهبود یافته برای حل دستگاه معادلات قدرمطلق و کاربرد آن در مسایل مقدار مرزی دو نقطهای، پژوهش های نوین در ریاضی، 5 (20)( 1398)
]2[ محمود پریپو، اسمعیل بابلیان، لیلا اسدبیگی، یک مدل جدید ABS سهگامی برای حل دستگاههای معادلات خطی تمام رتبه سطری، پژوهشهای نوین در ریاضی، 6 (26)( 1399)
[3] M. Dana, A. G. Zykov, Kh.D. Ikramov, A minimal residual method for a special class of the linear systems with normal coefficient matrices.-Comput. Math. Math. Phys., 45 (2005) 1854-1863.
[4] M. Ghasemi Kamalvand, Kh. D. Ikramov, A Method of Congruent Type for Linear Systems with Conjugate-Normal Coefficient Matrices.- Computational Mathematics and Mathematical Physics, 2009, Vol. 49, No. 2, pp.203-216.
[5] K. Niazi Asil, M. Ghasemi Kamalvand, On reduction of K-almost normal and K-almost conjugate normal matrices to a block tridiagonal form. - J. Korean Soc. Ind. Appl. Math. Vol.23, No.3, 267282, 2019.
[6] K. Niazi asil, M. Ghasemi Kamalvand , Some Hyperbolic Iterative Method for Linear Systems , Journal of Applied Mathematics Volume 2020, Article ID 9874162, 8 pages.
[7] R.A. Horn and C.R. Johnson, Matrix Analysis. Cambridge University Press, Cambridge, 1985.
[8] Biswa Nath Datta, Numerical linear algebra and applications. Delhi: PHI Learning Private Limited, 1985.
[9] I.Gohberg, P.Lancaster, L.Rodman, Indefinite linear algebra and applications. Birkh- auser, 2005.
[10] N.J. Higham, J-orthogonal matrices: properties and generations. SIAM, Rev. | ||
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