Arrowheaded enhanced multivariance products representations for matrices in denumerably infinite matrices

In this work, Enhanced Multivariance Products Representation (EMPR) approach which is a Demiralp–and–his–group extension to the Sobol's High Dimensional Model Representation (HDMR) has been used as the basic tool. Even though HDMR was first proposed by Sobol its first version was constructed on the unit hypercube geometry whose edges lie on the positive halves of the cooerdinate axes while its one corner was located on the origin. It was also assuming the unit constant weight function(s). H. A. Rabitz and has changed the orthogonal geometry from Sobol's case to a general hyperprism which can stand anywhere, as an extension. In addition, semi infinite or completely infinite geometries have been allowed by Rabitz who has also added the nonunit constant weight functions as an extension even though these functions should have been the product of univariate weight functions each of which depends on a different independent variable. EMPR which is an extended form of HDMR involves univariate support functions each of which depends on a different independent variable such that all possible univariances in the support functions appear at the end. EMPR like HDMR is not developed for only continuous entities like functions. Their discrete form have also been developed and used in practice by Demiralp and his group in addition to some other authors for the decomposition of the arrays like vectors, matrices, or multiway arrays. This work specifically focuses on the decomposition of infinite matrices involving denumerable infinitely many rows and columns. To this end the target matrix is first decomposed to the sum of certain outer products and then each outer product is treated by Tridiagonal Matrix Enhanced Multivariance Products Representation (TMEMPR) which has been developed by Demiralp and his group. The result is a three–matrix–factor–product whose kernel (the middle factor) is an arrowheaded matrix while the pre and post factors are invertable matrices decomposed of the support vectors of TMEMPR. This new method is called as Arrowheaded Enhanced Multivariance Products Representation for Matrices. The general purpose is approximation of denumerably infinite matrices with the new method. EMPR is a method which is based on a divide-and-conquer philosophy and is used for representing a given multivariate function in terms of less variate functions with the support functions. In the expansion, one unknown constant factor containing term, N number of unknown univariate factor involving terms, N(N − 1)/2 number of unknown bivariate factor including terms and so on. 2 N additive terms each of which is a product which may contain at most N number of factors appear in the EMPR expansion. The main goal is to obtain the general structure of these constant, univariate and the higher variate terms of the expansion. In this work we do not focus on continuous target functions but denumerable infinitely many rows and colums involving matrices. Hence, not functions but denumerably infinite vectors and matrices are considered. Similarly not support functions but infinite support vectors are under consideration. It is also proven that the additive terms of TMEMPR are mutually orthogonal in the denumerably infinite (separable) Cartesian space. Using this orthogonality it is possible to analyse the truncation approximation quality of TMEMPR. To this end, so–called Quality Measuerers which are the cumulative sum of ratios of the norm squares of TMEMPR summands to the norm square of TMEMPR focus function can be effectively used. We have not mentioned specifically the weight matrix utilization in the inner products of TMEMPR even though they may be employed to get better efficiency. This works basically uses unit matrix weight. Hence, the target matrix and the support vectors must have some bounded norms to proceed through the scheme presented here. There are various versions of EMPR method for the specific cases. Tridiagonal Matrix Enhanced Multivariance Products Representation is one of theme, we have told something above. There are some other works in Demiralp's group under intense study. Tridiagonal Kernel EMPR is one of these extension. The purpose therein is to decompose of the kernel of an integral operator acting on univariate functions. The resulting decomposition, in denumerable infinite matrix notation, is composed of three factors the middle one of which is a denumerably infinite tridiagonal matrix. Another work in this framework has been quite recently launched and aims at the three folmat (folded matrix) factorization of a given folmat which is considered composed of as if folded rows and columns. When the target matrix is consisted of only outer products, Arrowheaded Enhanced Multivariance Products Representation for Matrices (AEMPRM) which uses the basic philosophy of TMEMPR method almost exactly in the same way becomes the target object of the study like in this thesis. In the fourth section, AEMPRM mehod and its special cases are described in detail. Transformation of arrowhead matrix to tridiagonal form by using TMEMPR method and relation between these two methods have also been indicated. In the fifth section, implementation results of these methods, realized via Mathematica, and efficiency of the results are analysed. In conclusion, the new and original findings with an emphasis on somehow revolutionary aspects in AEMPRM are given in this thesis at a state–of–art status as much as we can do within today's findings.