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diophantine-equation
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ABS solution of a class of linear integer inequalities and integer LP problems
, Article Optimization Methods and Software ; Volume 16, Issue 1-4 , 2001 , Pages 179-192 ; 10556788 (ISSN) ; Mahdavi Amiri, N ; Spedicato, E ; Sharif University of Technology
2001
Abstract
Using the recently developed ABS algorithm for solving linear Diophantine equations we give a representation of the solutions of a system of m linear integer inequalities in n variables, m ≤ n, with full rank coefficient matrix. We apply this result to solve linear integer programming problems with m ≤ n inequalities. © 2001 OPA (Overseas Publishers Association) N.V. Published by license under the Gordon and Breach Science Publishers imprint, a member of the Taylor & Francis Group
Real and Integer Extended Rank Reduction Formulas and Matrix Decompositions: A Review
, Article Springer Proceedings in Mathematics and Statistics, 5 January 2014 through 9 January 2014 ; Volume 134 , January , 2015 , Pages 237-259 ; 21941009 (ISSN) ; 9783319176888 (ISBN) ; Golpar Raboky, E ; Al Baali M ; Grandinetti L ; Purnama A ; AMPL, USA; et al; German University of Technology (GUtech), Oman; SQU; The International Center for Theoretical Physics, ICTP, Italy; The Research Council of Oman ; Sharif University of Technology
Springer New York LLC
2015
Abstract
We have recently developed an extended rank reducing process for rank reduction of a matrix leading to various matrix decompositions containing the Abaffy-Broyden-Spedicato (ABS) and Wedderburn processes. Notably, the extended process contains both the Wedderburn biconjugation process and the scaled extended ABS class of algorithms. The process provides a general finite iterative approach for constructing factorizations of a matrix and its transpose under a common framework of a general decomposition having various useful structures such as triangular, orthogonal, diagonal, banded and Hessenberg and many others. One main new result is the derivation of an extended rank reducing process for...
Study of Invariants of Diophantine Forms and Equations
, M.Sc. Thesis Sharif University of Technology ; Jafari, Amir (Supervisor)
Abstract
The aim of this thesis is to study invariants of Diophantine forms and equations. These invariants are the transformations that do not change the original equations and forms. This thesis has a different view to the concept of invariants from Hlibertian view. Based on this new definition, by having one of the solutions of the equation, new solutions can be found with invariant method. The number of these solutions can be finite or infinite. In case of infinite solutions this method is more important. The focus of this thesis is on forms and linear invariants. The most important result in this text is about two main theorems that can classify the invariants of all completely decomposable...
Diophantine Quadratic Equation and Smith Normal Form Using Scaled Extended Integer Abaffy-Broyden-Spedicato Algorithms
, Article Journal of Optimization Theory and Applications ; Volume 152, Issue 1 , January , 2012 , Pages 75-96 ; 00223239 (ISSN) ; Mahdavi Amiri, N ; Sharif University of Technology
2012
Abstract
Classes of integer Abaffy-Broyden-Spedicato (ABS) methods have recently been introduced for solving linear systems of Diophantine equations. Each method provides the general integer solution of the system by computing an integer solution and an integer matrix, named Abaffian, with rows generating the integer null space of the coefficient matrix. The Smith normal form of a general rectangular integer matrix is a diagonal matrix, obtained by elementary nonsingular (unimodular) operations. Here, we present a class of algorithms for computing the Smith normal form of an integer matrix. In doing this, we propose new ideas to develop a new class of extended integer ABS algorithms generating an...
Finding Invariants and Parametric Solutions for Some Systems of Diophantine Equations with Arbitrary Coefficients and Variables Over Q
, Ph.D. Dissertation Sharif University of Technology ; Jafari, Amir (Supervisor)
Abstract
The main topic of this dissertation is to find methods for obtaining parametric solutions and linear/nonlinear invariants of Diophantine equations and consists of 4 chapters. The first chapter consists of some introductory discussions. The second chapter begins with a review of linear invariant, U-invariants, covariants and the concept of semi-invariants. The generators of maximum degree 3 for producing all linear invariants are introduced in Chapter 2 as well. Moreover, the relations between 3rd and 4th degree Hilbert invariants in terms of Procesi bases are included in this chapter.In chapter 3, a general conjecture is given to check whether there are finitely many solutions to a...
Robust controller design for discrete unstable non-minimum-phase delayed stochastic processes
, Article International Journal of Control, Automation and Systems ; Volume 11, Issue 5 , 2013 , Pages 893-902 ; 15986446 (ISSN) ; Shahrokhi, M ; Sharif University of Technology
2013
Abstract
Control of unstable non-minimum-phase delayed stochastic processes is a challenging problem. In this work based on the Diophantine equation and using pole-placement technique, a discrete control scheme for such processes has been proposed. Robust stability of the suggested control structure has been shown. Advantages of the proposed scheme over the existing algorithms have been shown through computer simulations. It has been shown that performance of the proposed scheme for handling model mismatch and colored noise is superior to the previous work proposed in the literature
Smith normal form using scaled extended integer ABS algorithms
, Article Advances in Intelligent and Soft Computing ; Volume 145 AISC, Issue VOL. 2 , 2012 , Pages 367-372 ; 18675662 (ISSN) ; Mahdavi Amiri, N ; Sharif University of Technology
2012
Abstract
Classes of integer ABS methods have recently been introduced for solving linear systems of Diophantine equations. The Smith normal form of a general integermatrix is a diagonal integer matrix, obtained by elementary nonsingular (unimodular) operations. Such a form may conveniently be used in solving integer systems of equations and integer linear programming problems. Here, we present a class of algorithms for computing the Smith normal form of an integer matrix. In doing this, we propose new ideas to develop a new class of extended integer ABS algorithms generating an integer basis for the integer null space of the matrix. Finally, we test our algorithms and report the obtained numerical...
Real and integer Wedderburn rank reduction formulas for matrix decompositions
, Article Optimization Methods and Software ; Volume 30, Issue 4 , 2015 , Pages 864-879 ; 10556788 (ISSN) ; Golpar Raboky, E ; Sharif University of Technology
Taylor and Francis Ltd
2015
Abstract
The Wedderburn rank reduction formula is a powerful method for developing matrix factorizations and many fundamental numerical linear algebra processes. We present a new interpretation of the Wedderburn rank reduction formula and its associated biconjugation process and see a more extensive result of the formula. In doing this, we propose a new formulation based on the null space transformations on rows and columns of A simultaneously, and show several matrix factorizations that can be derived from the Wedderburn rank reduction formula. We also present a generalization of the biconjugation process and compute banded and Hessenberg factorizations. Using the new formulation, we compute the WZ...
How to Explicitly Solve a Thue-Mahler Equation
, M.Sc. Thesis Sharif University of Technology ; Ebrahimi Borojeni, Javad (Supervisor) ; Ghadermarzi, Amir (Supervisor)
Abstract
David Hilbert posed a list of mathematical problems in 1900. Hilbert's tenth problem was "Is there an algorithm to determine whether a given Diophantine equation, has a solution with all unknowns taking integer values." Although Matiyasevich Showed that the answer to this problem is negative in the general case; the answer of this problem for a specific diophantine equation $f(x,y) $ that has rational coefficients, is unknown. Thue-Mahler equation is a diophantine equation of the form: $$F(x,y)=c\cdot p_1^{z_1}\cdots p_v^{z_v}$$ where $F$ is homogeneous with integer coefficients, degree $n\geq3$ and $p_1,\ldots p_v$ are distinct rational primes ($v\geq1$). All the unknowns are...
Extended Rank Reduction Formula and its Application to Real and Integer Matrix Factorizations
, Ph.D. Dissertation Sharif University of Technology ; Mahdavi Amiri, Nezameddin (Supervisor)
Abstract
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...