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Extended Rank Reduction Formula and its Application to Real and Integer Matrix Factorizations

Golpar Raboky, Efat | 2011

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 42196 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahdavi Amiri, Nezameddin
  7. Abstract:
  8. The Wedderburn rank reduction formula and the ABS algorithms are powerful methods for developing matrix factorizations and many fundamental numerical linear algebra processes such as Gramm- Schmidt, conjugate direction and Lanczos methods. Esmaeili, Mahdavi-Amiri and Spedicato introduced a class of integer ABS algorithms for solving linear systems of Diophantine equations. In a recent work, Khorramizadeh and Mahdavi-Amiri have also presented a new class of extended integer ABS algorithms for solving linear Diophantine systems by computing an integer basis for the null space while controlling the growth of intermediate results. Here, we propose new approches to develop a new class of extended integer ABS algorithms generating an integer basis for the integer null space of a matrix A ∈ Zmn. In this dissertation, we present an extended rank reduction formula for transforming the row, column or row and column simultaneously of A, extending the Wedderburn rank reduction formula. By repeatedly applying the formula to reduce the rank of A, a general extended rank reducing process (GERRP) is derived which presents a new class of algorithms. The biconjuation process associated with the Wedderburn rank reduction process and the scaled extended ABS class of algorithms are shown to be in the rank reducing process, while the process is more general to produce several other effective reduction algorithms to compute various structured factorizations such as banded and Hessenberg. The process provides a general finite iterative approach for constructing the factorizations of A and AT under a common framework of a general decomposition V TAP = Ω. We also show that the biconjuation process associated with the Wedderburn rank reduction process can be derived from the scaled ABS class of algorithms applied to A or AT . Smith proved that any integer matrix can be transformed by elementary row and column operations into a diagonal matrix. We presente an application of extended rank reduction formula for integer matrices. Using this formula, we propose a general extended integer rank reducing process (GEIRRP). The process generates a new class of algorithms contaning the SEIABS class of algorithms applied to A and AT . We describe how to choose the parameters of GEIRRP and SEIABS algorithm so that a Smith normal form of an integer matrix is generated. Then, we develope the integer Wedderburn rank reduction formula and its integer biconjugation process and show that the integer biconjugate process computes the Smith normal form. We also show that the integer biconjugate process can be drived from GEIRRP and SIABS clss of algorithms. GEIRRP need to solve the quadratic Diophantine equation yTAx = b. We present two algorithms for solving such equations. The first algorithm, QEDS, makes use of a divisibility sequence basis for the row space of the matrix A, and the second one, QEIABS, with the intention of controlling the growth of intermediate results and making use of our given conjecture, is based on the integer ABS algorithm. The numerical results on randomly generated test problems showing a better performance of the second algorithm in controlling the size of the solution. We also report the results obtained by our proposed algorithm on the Smith normal form and compare them with the ones obtained from the procedure ismith from Maple, observing a more balanced distribution of the intermediate components obtained by our algorithm.
  9. Keywords:
  10. Diophantine Linear System ; Class of Integer ABS Algorithms ; Matrix Factorization ; Wedderburn Rank Reduction Formula ; Extended Rank Reduction Formula ; Quadratic Diophantine Equation ; Biconjugation Process

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