New Algorithm for Determinant of Matrices Via Combinatorial Approach

Methods for finding determinants for matrices have long been explored and attracted interest of numerous researchers. However, most of the existing methods are tedious and require lengthy computation time particularly as the size of matrices becomes larger. Therefore, this study attempts to develop a new method which can reduce determinant computation time for matrices of any order. The developed method was based on permutations which were generated using starter sets. All starter sets for n elements were obtained by using combinatorial approach which then produced all n! distinct permutations. This starter sets strategy was proven to be more efficient if compared to other existing methods for listing all permutations such as lexicographic method. All permutations obtained were then used to construct a new method for finding determinants of n x n matrices. This study also produced a new theorem for finding determinant of n x n matrices and this theorem was proven to be equivalent to the existing theorem i.e Leibniz theorem. Besides that, several new theoretical works and mathematical properties for generating permutation and determining determinant were also constructed to verify the new developed method. The numerical results revealed that the determinant computation time for the new method was faster if compared to the existing methods. Testing of the new method on several special matrices such as Toeplitz, Hilbert and Hessenberg matrices was also carried out to prove the efficiency of the developed method. The numerical results also indicated that the new method outperformed Gauss and Gauss Jordon methods in term of computation time

ALGORITMA ELIMINASI GAUSS INTERVAL DALAM MENDAPATKAN NILAI DETERMINAN MATRIKS INTERVAL DAN MENCARI SOLUSI SISTEM PERSAMAAN INTERVAL LINEAR

Sistem Persamaan Interval Linear (SPIL) merupakan perluasan dari Sistem Persamaan Linear (SPL) dengan koefisien-koefisiennya berupa interval. Bentuk umum dari SPIL dapat ditulis sebagai . Untuk memperoleh solusi dari SPIL dapat digunakan matriks sebagaimana pada SPL. Dalam hal ini, matriks yang digunakan adalah matriks interval dengan entri-entri berupa interval. Selain untuk menyelesaikan SPIL, teori-teori tentang matriks interval juga sangat diperlukan yang salah satunya adalah untuk mendapatkan nilai determinan  matriks interval. Salah satu metode yang digunakan adalah dengan menggunakan Algoritma Eliminasi Gauss Interval. Algoritma ini dimulai dengan mereduksi matriks interval dan matriks interval augmanted dari SPIL dengan  menerapkan  aritmatika  interval yang dimodifikasi untuk mendapatkan matriks interval segitiga atas dan matriks interval augmanted yang lebih sederhana. Selanjutnya, dengan metode substitusi balik pada sistem yang bersesuaian dari matriks interval augmanted yang lebih sederhana sehingga diperoleh solusi dari SPIL dan mengalikan entri-entri diagonal utama dari matriks interval segitiga atas untuk mendapatkan nilai determinan matriks interval segitiga atas. Solusi yang diperoleh adalah solusi yang memenuhi sistem dan vektor interval yang diperoleh dari sistem ekuivalen dengan vektor interval dari sistem yang dapat dilihat dari masing-masing midpoint pada vektor interval . Nilai determinan matriks interval segitiga atas ekuivalen dengan nilai determinan matriks interval. Kata kunci : Aritmatika Interval, Algoritma Eliminasi Gauss Interva

High performance simplex solver

The dual simplex method is frequently the most efficient technique for solving linear programming (LP) problems. This thesis describes an efficient implementation of the sequential dual simplex method and the design and development of two parallel dual simplex solvers. In serial, many advanced techniques for the (dual) simplex method are implemented, including sparse LU factorization, hyper-sparse linear system solution technique, efficient approaches to updating LU factors and sophisticated dual simplex pivoting rules. These techniques, some of which are novel, lead to serial performance which is comparable with the best public domain dual simplex solver, providing a solid foundation for the simplex parallelization. During the implementation of the sequential dual simplex solver, the study of classic LU factor update techniques leads to the development of three novel update variants. One of them is comparable to the most efficient established approach but is much simpler in terms of implementation, and the other two are specially useful for one of the parallel simplex solvers. In addition, the study of the dual simplex pivoting rules identifies and motivates further investigation of how hyper-sparsity maybe promoted. In parallel, two high performance simplex solvers are designed and developed. One approach, based on a less-known dual pivoting rule called suboptimization, exploits parallelism across multiple iterations (PAMI). The other, based on the regular dual pivoting rule, exploits purely single iteration parallelism (SIP). The performance of PAMI is comparable to a world-leading commercial simplex solver. SIP is frequently complementary to PAMI in achieving speedup when PAMI results in slowdown

Row generation techniques for approximate solution of linear programming problems

Ankara : The Department of Industrial Engineering and the Institute of Engineering and Science of Bilkent University, 2010.Thesis (Master's) -- Bilkent University, 2010.Includes bibliographical references leaves 69-77.In this study, row generation techniques are applied on general linear programming problems with a very large number of constraints with respect to the problem dimension. A lower bound is obtained for the change in the objective value caused by the generation of a specific row. To achieve row selection that results in a large shift in the feasible region and the objective value at each row generation iteration, the lower bound is used in the comparison of row generation candidates. For a warm-start to the solution procedure, an effective selection of the subset of constraints that constitutes the initial LP is considered. Several strategies are discussed to form such a small subset of constraints so as to obtain an initial solution close to the feasible region of the original LP. Approximation schemes are designed and compared to make possible the termination of row generation at a solution in the proximity of an optimal solution of the input LP. The row generation algorithm presented in this study, which is enhanced with a warm-start strategy and an approximation scheme is implemented and tested for computation time and the number of rows generated. Two efficient primal simplex method variants are used for benchmarking computation times, and the row generation algorithm appears to perform better than at least one of them especially when number of constraints is large.Paç, A BurakM.S

Decision procedures for linear arithmetic

In this thesis, we present new decision procedures for linear arithmetic in the context of SMT solvers and theorem provers: 1) CutSat++, a calculus for linear integer arithmetic that combines techniques from SAT solving and quantifier elimination in order to be sound, terminating, and complete. 2) The largest cube test and the unit cube test, two sound (although incomplete) tests that find integer and mixed solutions in polynomial time. The tests are especially efficient on absolutely unbounded constraint systems, which are difficult to handle for many other decision procedures. 3) Techniques for the investigation of equalities implied by a constraint system. Moreover, we present several applications for these techniques. 4) The Double-Bounded reduction and the Mixed-Echelon-Hermite transformation, two transformations that reduce any constraint system in polynomial time to an equisatisfiable constraint system that is bounded. The transformations are beneficial because they turn branch-and-bound into a complete and efficient decision procedure for unbounded constraint systems. We have implemented the above decision procedures (except for Cut- Sat++) as part of our linear arithmetic theory solver SPASS-IQ and as part of our CDCL(LA) solver SPASS-SATT. We also present various benchmark evaluations that confirm the practical efficiency of our new decision procedures.In dieser Arbeit präsentieren wir neue Entscheidungsprozeduren für lineare Arithmetik im Kontext von SMT-Solvern und Theorembeweisern: 1) CutSat++, ein korrekter und vollständiger Kalkül für ganzzahlige lineare Arithmetik, der Techniken zur Entscheidung von Aussagenlogik mit Techniken aus der Quantorenelimination vereint. 2) Der Größte-Würfeltest und der Einheitswürfeltest, zwei korrekte (wenn auch unvollständige) Tests, die in polynomieller Zeit (gemischt-)ganzzahlige Lösungen finden. Die Tests sind besonders effizient auf vollständig unbegrenzten Systemen, welche für viele andere Entscheidungsprozeduren schwer sind. 3) Techniken zur Ermittlung von Gleichungen, die von einem linearen Ungleichungssystem impliziert werden. Des Weiteren präsentieren wir mehrere Anwendungsmöglichkeiten für diese Techniken. 4) Die Beidseitig-Begrenzte-Reduktion und die Gemischte-Echelon-Hermitesche- Transformation, die ein Ungleichungssystem in polynomieller Zeit auf ein erfüllbarkeitsäquivalentes System reduzieren, das begrenzt ist. Vereint verwandeln die Transformationen Branch-and-Bound in eine vollständige und effiziente Entscheidungsprozedur für unbeschränkte Ungleichungssysteme. Wir haben diese Techniken (ausgenommen CutSat++) in SPASS-IQ (unserem theory solver für lineare Arithmetik) und in SPASS-SATT (unserem CDCL(LA) solver) implementiert. Basierend darauf präsentieren wir Benchmark-Evaluationen, die die Effizienz unserer Entscheidungsprozeduren bestätigen

Tools and Algorithms for the Construction and Analysis of Systems

This open access book constitutes the proceedings of the 28th International Conference on Tools and Algorithms for the Construction and Analysis of Systems, TACAS 2022, which was held during April 2-7, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 46 full papers and 4 short papers presented in this volume were carefully reviewed and selected from 159 submissions. The proceedings also contain 16 tool papers of the affiliated competition SV-Comp and 1 paper consisting of the competition report. TACAS is a forum for researchers, developers, and users interested in rigorously based tools and algorithms for the construction and analysis of systems. The conference aims to bridge the gaps between different communities with this common interest and to support them in their quest to improve the utility, reliability, exibility, and efficiency of tools and algorithms for building computer-controlled systems

Coarse-grained modelling of protein structure and internal dynamics: comparative methods and applications

The first chapter is devoted to a brief summary of the basic techniques commonly used to characterise protein's internal dynamics, and to perform those primary analyses which are the basis for our further developments. To this purpose we recall the basics of Principal Component Analysis of the covariance matrix of molecular dynamics (MD) trajectories. The overview is aimed at motivating and justifying a posteriori the introduction of coarse-grained models of proteins. In the second chapter we shall discuss dynamical features shared by different conformers of a protein. We'll review previously obtained results, concerning the universality of the vibrational spectrum of globular proteins and the self-similar free energy landscape of specific molecules, namely the G-protein and Adk. Finally, a novel technique will be discussed, based on the theory of Random Matrices, to extract the robust collective coordinates in a set of protein conformers by comparison with a stochastic reference model. The third chapter reports on an extensive investigation of protein internal dynamics modelled in terms of the relative displacement of quasi-rigid groups of amino acids. Making use of the results obtained in the previous chapters, we shall discuss the development of a strategy to optimally partition a protein in units, or domains, whose internal strain is negligible compared to their relative uctuation. These partitions will be used in turn to characterise the dynamical properties of proteins in the framework of a simplified, coarse-grained, description of their motion. In the fourth chapter we shall report on the possibility to use the collective uctuations of proteins as a guide to recognise relationships between them that may not be captured as significant when sequence or structural alignment methods are used. We shall review a method to perform the superposition of two proteins optimising the similarity of the structures as well as the dynamical consistency of the aligned regions; then, we shall next discuss a generalisation of this scheme to accelerate the dynamics-based alignment, in the perspective of dataset-wide applications. Finally, the fifth chapter focuses on a different topic, namely the occurrence of topologically-entangled states (knots) in proteins. Specifically, we shall investigate the sequence and structural properties of knotted proteins, reporting on an exhaustive dataset-wide comparison with unknotted ones. The correspondence, or the lack thereof, between knotted and unknotted proteins allowed us to identify, in knotted chains, small segments of the backbone whose `virtual' excision results in an unknotted structure. These `knot-promoting' loops are thus hypothesised to be involved in the formation of the protein knot, which in turn is likely to cover some role in the biological function of the knotted proteins