DOCSLIB.ORG

Matrix Computations (DAA, M.Tech + Ph.D.)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Matrix Computations (DAA, M.Tech + Ph.D.) Matrix Computations (DAA, M.Tech + Ph.D.) By: Sunil Kumar Singh, PhD Assistant Professor, Department of Computer Science and Information Technology School of Computational Sciences, Information and Communication Technology, Mahatma Gandhi Central University, Motihari 27-04-2020 Bihar, India-845401 1 Outline • Matrix Computation • Strassen’s Algorithm • Divide an conquer paradigm for multiplication • Inverse of a triangular matrix • Conclusion • References 27-04-2020 2 Matrix Computations Input: 퐴 = 푎푖푗 ,퐵 = 푏푖푗 output: 퐶 = 푐푖푗 = 퐴. 퐵 푤ℎ푒푟푒 푖, 푗 = {1,2,3, … … 푛} 푐11 푐12 … 푐1푛 푎11 푎12 … 푎1푛 푏11 푏12 … 푏1푛 푐 푐 … 푐22 푎 푎 … 푎22 푏 푏 … 푏 21 22 = 21 22 . 21 22 22 ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ⋮ 푐푛1 푐푛2 … 푐푛푛 푎푛1 푎푛2 … 푎푛푛 푏푛1 푏푛2 … 푏푛푛 푛 푐푖푗 = ෍ 푎푖푘. 푏푘푗 푘=1 3 28-04-2020 Standard Algorithm 푁 푁 푁 3 3 푇 푁 = σ푖=1 σ푗=1 σ푘=1 푐=c 푁 = 푂(푁 ) 4 27-04-2020 Divide-and-Conquer • Divide-and conquer is a general algorithm design paradigm: ✓ Divide: divide the input data S in two or more disjoint subsets 푆1, 푆2 … ✓ Solve the subproblems recursively ✓ Conquer: combine the solutions for 푆1, 푆2 … into a solution for 푆 • The base case for the recursion are sub problems of constant size • Analysis can be done using recurrence equations 5 28-04-2020 Matrix multiplication through divide and conquer approach 푛 푛 1. Divide: Partition A and B into 푋 submatrices. Form terms 2 2 to be multiplied using + and – 푛 푛 2. Conquer: Perform 7 multiplications of 푋 submatrices 2 2 recursively. 푛 푛 3. Combine: Form C using + and – on 푋 submatrices 2 2 6 28-04-2020 Matrix multiplication through divide and conquer approach 푛 × 푛 푚푎푡푟푖푥 = 2 × 2 matrix of (푛/2) × (푛/2) submatrices: 푟 ⋮ 푠 푎 ⋮ 푏 푒 ⋮ 푓 ⋯ ⋮ ⋯ = ⋯ ⋮ ⋯ . ⋯ ⋮ ⋯ 푡 ⋮ 푢 푐 ⋮ 푑 푔 ⋮ ℎ 퐶 = 퐴. 퐵 푟 = 푎푒 + 푏푔 푠 = 푎푓 + 푏ℎ n n 8 multiplications and 4 adds of X 푡 = 푐푒 + 푑푔 2 2 푢 = 푐푓 + 푑ℎ 7 28-04-2020 Matrix multiplication through divide and conquer approach 푛 푇 푛 = 8푇 + Θ(푛2) 2 #submatrices work adding submatrices submatrix size Complexity: 푇 푛 = Θ(푛3) 8 28-04-2020 Strassen’s Algorithm • In linear algebra, the Strassen algorithm, named after Volker Strassen, is an algorithm for matrix multiplication. • It is faster than standard matrix multiplication algorithm and is useful in practice for large matrices. • But it would be slower than the fastest known algorithms for extremely large matrices. 9 28-04-2020 Strassen’s Idea • Multiply 2 × 2 matrices with only 7 recursive multiplications. 퐶 퐶 퐴 퐴 퐵 퐵 11 12 = 11 12 . 11 12 퐶21 퐶22 퐴21 퐴22 퐵21 퐵22 푷ퟏ = (푨ퟏퟏ + 푨ퟐퟐ)(푩ퟏퟏ + 푩ퟐퟐ) 푪ퟏퟏ = 푷ퟏ + 푷ퟒ − 푷ퟓ + 푷ퟕ 푷ퟐ = 푨ퟐퟏ + 푨ퟐퟐ ∗ (푩ퟏퟏ) 푪ퟏퟐ = 푷ퟑ + 푷ퟓ 푷ퟑ = 푨ퟏퟏ ∗ (푩ퟏퟐ − 푩ퟐퟐ) 푪ퟐퟏ = 푷ퟐ + 푷ퟒ 푷ퟒ = (푨ퟐퟐ)(푩ퟏퟏ + 푩ퟐퟐ) 푪ퟏퟏ = 푷ퟏ + 푷ퟑ − 푷ퟐ + 푷ퟔ 푷ퟓ = 푨ퟏퟏ + 푨ퟐퟐ ∗ (푩ퟐퟐ) 푷ퟔ = 푨ퟐퟏ − 푨ퟏퟏ ∗ (푩ퟏퟏ + 푩ퟏퟐ) 푷ퟕ = 푨ퟏퟐ − 푨ퟐퟐ ∗ (푩ퟐퟏ + 푩ퟐퟐ) 10 28-04-2020 Analysis 푏 푛 ≤ 2 푇 푛 = ቐ 푛 7푇 + Θ(푛2) 푛 > 2 2 • When we apply the masters theorem: 푇 푛 = 푂(푛log2 7) 푇 푛 = 푂(푛2.81) • The number 2.81 may not seem much smaller than 3, but because the difference is in the exponent, the impact on running time is significant. In fact, Strassen’s algorithm beats the ordinary algorithm on today’s machines for 푛 ≥ 32 or so. 11 28-04-2020 Inverse of a Matrix • Inverse of an 푛 × 푛 matrix A to be the 푛 × 푛 matrix, denoted by 퐴−1(if it exists), such that −1 −1 퐴퐴 = 퐼푛 = 퐴 퐴 • for example 1 1 −1 0 1 = 1 0 1 −1 • Many nonzero 푛 × 푛 matrices do not have inverse. A matrix without an inverse is called noninvertible, or singular. • An example of a non-zero singular matrix is 1 0 1 0 • If a matrix has an inverse, it is called invertible, or nonsingular. 12 28-04-2020 Inverse of a Matrix • If A and B are nonsingular 푛 × 푛 matrices, then 퐵퐴 −1 = 퐴−1퐵−1 • Inverse operation commutes with the transpose operation: 퐴−1 푇 = 퐴푇 −1 • Prove that an 푛 × 푛 matrix is singular if and only if det(A)=0. 13 28-04-2020 References 1. Cormen, Thomas H., Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to algorithms. MIT press, 2009. 2. Cormen, Thomas H., Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. "Introduction to algorithms second edition." The Knuth-Morris-Pratt Algorithm, year (2001). 3. Seaver, Nick. "Knowing algorithms." (2014): 1441587647177. 14 28-04-2020 Thank You 27-04-2020 15.
Load more

Recommended publications
  • Fast Matrix Multiplication
    Fast matrix multiplication Yuval Filmus May 3, 2017 1 Problem statement How fast can we multiply two square matrices? To make this problem precise, we need to fix a computational model. The model that we use is the unit cost arithmetic model. In this model, a program is a sequence of instructions, each of which is of one of the following forms: 1. Load input: tn xi, where xi is one of the inputs. 2. Load constant: tn c, where c 2 R. 3. Arithmetic operation: tn ti ◦ tj, where i; j < n, and ◦ 2 f+; −; ·; =g. In all cases, n is the number of the instruction. We will say that an arithmetic program computes the product of two n × n matrices if: 1. Its inputs are xij; yij for 1 ≤ i; j ≤ n. 2. For each 1 ≤ i; k ≤ n there is a variable tm whose value is always equal to n X xijyjk: j=1 3. The program never divides by zero. The cost of multiplying two n × n matrices is the minimum length of an arithmetic program that computes the product of two n × n matrices. Here is an example. The following program computes the product of two 1 × 1 matrices: t1 x11 t2 y11 t3 t1 · t2 The variable t3 contains x11y11, and so satisfies the second requirement above for (i; k) = (1; 1). This shows that the cost of multiplying two 1 × 1 matrices is at most 3. Usually we will not spell out all steps of an arithmetic program, and use simple shortcuts, such as the ones indicated by the following program for multiplying two 2 × 2 matrices: z11 x11y11 + x12y21 z12 x11y12 + x12y22 z21 x21y11 + x22y21 z22 x21y12 + x22y22 1 When spelled out, this program uses 20 instructions: 8 load instructions, 8 product instructions, and 4 addition instructions.
    [Show full text]
  • Efficient Matrix Multiplication on Simd Computers* P
    SIAM J. MATRIX ANAL. APPL. () 1992 Society for Industrial and Applied Mathematics Vol. 13, No. 1, pp. 386-401, January 1992 025 EFFICIENT MATRIX MULTIPLICATION ON SIMD COMPUTERS* P. BJORSTADt, F. MANNEr, T. SOI:tEVIK, AND M. VAJTERIC?$ Dedicated to Gene H. Golub on the occasion of his 60th birthday Abstract. Efficient algorithms are described for matrix multiplication on SIMD computers. SIMD implementations of Winograd's algorithm are considered in the case where additions are faster than multiplications. Classical kernels and the use of Strassen's algorithm are also considered. Actual performance figures using the MasPar family of SIMD computers are presented and discussed. Key words, matrix multiplication, Winograd's algorithm, Strassen's algorithm, SIMD com- puter, parallel computing AMS(MOS) subject classifications. 15-04, 65-04, 65F05, 65F30, 65F35, 68-04 1. Introduction. One of the basic computational kernels in many linear algebra codes is the multiplication of two matrices. It has been realized that most problems in computational linear algebra can be expressed in block algorithms and that matrix multiplication is the most important kernel in such a framework. This approach is essential in order to achieve good performance on computer systems having a hier- archical memory organization. Currently, computer technology is strongly directed towards this design, due to the imbalance between the rapid advancement in processor speed relative to the much slower progress towards large, inexpensive, fast memories. This algorithmic development is highlighted by Golub and Van Loan [13], where the formulation of block algorithms is an integrated part of the text. The rapid devel- opment within the field of parallel computing and the increased need for cache and multilevel memory systems are indeed well reflected in the changes from the first to the second edition of this excellent text and reference book.
    [Show full text]
  • Matrix Multiplication: a Study Kristi Stefa Introduction
    Matrix Multiplication: A Study Kristi Stefa Introduction Task is to handle matrix multiplication in an efficient way, namely for large sizes Want to measure how parallelism can be used to improve performance while keeping accuracy Also need for improvement on “normal” multiplication algorithm O(n3) for conventional square matrices of n x n size Strassen Algorithm Introduction of a different algorithm, one where work is done on submatrices Requirement of matrix being square and size n = 2k Uses A and B matrices to construct 7 factors to solve C matrix Complexity from 8 multiplications to 7 + additions: O(nlog27) or n2.80 Room for parallelism TBB:parallel_for is the perfect candidate for both the conventional algorithm and the Strassen algorithm Utilization in ranges of 1D and 3D to sweep a vector or a matrix TBB: parallel_reduce would be possible but would increase the complexity of usage Would be used in tandem with a 2D range for the task of multiplication Test Setup and Use Cases The input images were converted to binary by Matlab and the test filter (“B” matrix) was generated in Matlab and on the board Cases were 128x128, 256x256, 512x512, 1024x1024 Matlab filter was designed for element-wise multiplication and board code generated for proper matrix-wise filter Results were verified by Matlab and an import of the .bof file Test setup (contd) Pictured to the right are the results for the small (128x128) case and a printout for the 256x256 case Filter degrades the image horizontally and successful implementation produces
    [Show full text]
  • The Strassen Algorithm and Its Computational Complexity
    Utrecht University BSc Mathematics & Applications The Strassen Algorithm and its Computational Complexity Rein Bressers Supervised by: Dr. T. van Leeuwen June 5, 2020 Abstract Due to the many computational applications of matrix multiplication, research into efficient algorithms for multiplying matrices can lead to widespread improvements of performance. In this thesis, we will first make the reader familiar with a universal measure of the efficiency of an algorithm, its computational complexity. We will then examine the Strassen algorithm, an algorithm that improves on the computational complexity of the conventional method for matrix multiplication. To illustrate the impact of this difference in complexity, we implement and test both algorithms, and compare their runtimes. Our results show that while Strassen’s method improves on the theoretical complexity of matrix multiplication, there are a number of practical considerations that need to be addressed for this to actually result in improvements on runtime. 2 Contents 1 Introduction 4 1.1 Computational complexity .................................................................................. 4 1.2 Structure ........................................................................................................... 4 2 Computational Complexity ................................................................ 5 2.1 Time complexity ................................................................................................ 5 2.2 Determining an algorithm’s complexity ..............................................................
    [Show full text]
  • Using Machine Learning to Improve Dense and Sparse Matrix Multiplication Kernels
    Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2019 Using machine learning to improve dense and sparse matrix multiplication kernels Brandon Groth Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Applied Mathematics Commons, and the Computer Sciences Commons Recommended Citation Groth, Brandon, "Using machine learning to improve dense and sparse matrix multiplication kernels" (2019). Graduate Theses and Dissertations. 17688. https://lib.dr.iastate.edu/etd/17688 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Using machine learning to improve dense and sparse matrix multiplication kernels by Brandon Micheal Groth A dissertation submitted to the graduate faculty in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Major: Applied Mathematics Program of Study Committee: Glenn R. Luecke, Major Professor James Rossmanith Zhijun Wu Jin Tian Kris De Brabanter The student author, whose presentation of the scholarship herein was approved by the program of study committee, is solely responsible for the content of this dissertation. The Graduate College will ensure this dissertation is globally accessible and will not permit alterations after a degree is conferred. Iowa State University Ames, Iowa 2019 Copyright c Brandon Micheal Groth, 2019. All rights reserved. ii DEDICATION I would like to dedicate this thesis to my wife Maria and our puppy Tiger.
    [Show full text]
  • Divide-And-Conquer Algorithms
    Chapter 2 Divide-and-conquer algorithms The divide-and-conquer strategy solves a problem by: 1. Breaking it into subproblems that are themselves smaller instances of the same type of problem 2. Recursively solving these subproblems 3. Appropriately combining their answers The real work is done piecemeal, in three different places: in the partitioning of problems into subproblems; at the very tail end of the recursion, when the subproblems are so small that they are solved outright; and in the gluing together of partial answers. These are held together and coordinated by the algorithm's core recursive structure. As an introductory example, we'll see how this technique yields a new algorithm for multi- plying numbers, one that is much more efficient than the method we all learned in elementary school! 2.1 Multiplication The mathematician Carl Friedrich Gauss (1777–1855) once noticed that although the product of two complex numbers (a + bi)(c + di) = ac bd + (bc + ad)i − seems to involve four real-number multiplications, it can in fact be done with just three: ac, bd, and (a + b)(c + d), since bc + ad = (a + b)(c + d) ac bd: − − In our big-O way of thinking, reducing the number of multiplications from four to three seems wasted ingenuity. But this modest improvement becomes very significant when applied recur- sively. 55 56 Algorithms Let's move away from complex numbers and see how this helps with regular multiplica- tion. Suppose x and y are two n-bit integers, and assume for convenience that n is a power of 2 (the more general case is hardly any different).
    [Show full text]
  • Algebraic Elimination of Epsilon-Transitions Gérard Duchamp, Hatem Hadj Kacem, Éric Laugerotte
    Algebraic Elimination of epsilon-transitions Gérard Duchamp, Hatem Hadj Kacem, Éric Laugerotte To cite this version: Gérard Duchamp, Hatem Hadj Kacem, Éric Laugerotte. Algebraic Elimination of epsilon-transitions. 2004. hal-00001038v1 HAL Id: hal-00001038 https://hal.archives-ouvertes.fr/hal-00001038v1 Preprint submitted on 15 Jan 2004 (v1), last revised 13 Dec 2004 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Algebraic elimination of ε-transitions G´erard Duchamp, Hatem Hadj Kacem, Eric´ Laugerotte∗ LIFAR, Facult´edes Sciences et des Techniques, 76821 Mont-Saint-Aignan Cedex, France. Abstract We present here algebraic formulas associating a k-automaton to a k-ε-automaton. The existence depends on the definition of the star of matrices and of elements in the semiring k. For this reason, we present the theorem which allows the transformation of k-ε-automata into k-automata. The two automata have the same behaviour. Keywords : Automata with multiplicities, ε-transitions, behaviour, star of matri- ces. 1 Introduction Automata with multiplicities (or weighted automata) are a versatile class of transition systems which can modelize as well classical (boolean), stochas- tic, transducer automata and be applied to various purposes such as image compression, speech recognition, formal linguistic (and automatic treatment of natural languages too) and probabilistic modelling.
    [Show full text]
  • Implementing Strassen's Algorithm with BLIS
    Implementing Strassen's Algorithm with BLIS FLAME Working Note # 79 Jianyu Huang∗ Tyler M. Smith∗y Greg M. Henryz Robert A. van de Geijn∗y April 16, 2016 Abstract We dispel with \street wisdom" regarding the practical implementation of Strassen's algorithm for matrix-matrix multiplication (DGEMM). Conventional wisdom: it is only practical for very large ma- trices. Our implementation is practical for small matrices. Conventional wisdom: the matrices being multiplied should be relatively square. Our implementation is practical for rank-k updates, where k is relatively small (a shape of importance for libraries like LAPACK). Conventional wisdom: it inher- ently requires substantial workspace. Our implementation requires no workspace beyond buffers already incorporated into conventional high-performance DGEMM implementations. Conventional wisdom: a Strassen DGEMM interface must pass in workspace. Our implementation requires no such workspace and can be plug-compatible with the standard DGEMM interface. Conventional wisdom: it is hard to demonstrate speedup on multi-core architectures. Our implementation demonstrates speedup over TM 1 conventional DGEMM even on an Intel R Xeon Phi coprocessor utilizing 240 threads. We show how a distributed memory matrix-matrix multiplication also benefits from these advances. 1 Introduction Strassen's algorithm (Strassen) [1] for matrix-matrix multiplication (gemm) has fascinated theoreticians and practitioners alike since it was first published, in 1969. That paper demonstrated that multiplication of n × n matrices can be achieved in less than the O(n3) arithmetic operations required by a conventional formulation. It has led to many variants that improve upon this result [2, 3, 4, 5] as well as practical implementations [6, 7, 8, 9].
    [Show full text]
  • A Fast and Scalable Matrix Inversion Method in Apache Spark ICDCN’18,January4–7,2018,Varanasi,India
    SPIN: A Fast and Scalable Matrix Inversion Method in Apache Spark Chandan Misra Swastik Haldar Indian Institute of Technology Kharagpur Indian Institute of Technology Kharagpur West Bengal, India West Bengal, India [email protected] [email protected] Sourangshu Bhaacharya Soumya K. Ghosh Indian Institute of Technology Kharagpur Indian Institute of Technology Kharagpur West Bengal, India West Bengal, India [email protected] [email protected] ABSTRACT many of the workloads. In the big data era, many of these applica- e growth of big data in domains such as Earth Sciences, Social tions have to work on huge matrices, possibly stored over multiple Networks, Physical Sciences, etc. has lead to an immense need servers, and thus consuming huge amounts of computational re- for efficient and scalable linear algebra operations, e.g. Matrix in- sources for matrix inversion. Hence, designing efficient large scale version. Existing methods for efficient and distributed matrix in- distributed matrix inversion algorithms, is an important challenge. version using big data platforms rely on LU decomposition based Since its release in 2012, Spark [19] has been adopted as a dom- block-recursive algorithms. However, these algorithms are com- inant solution for scalable and fault-tolerant processing of huge plex and require a lot of side calculations, e.g. matrix multiplica- datasets in many applications, e.g., machine learning [11], graph tion, at various levels of recursion. In this paper, we propose a processing [7], climate science [12], social media analytics [2], etc. different scheme based on Strassen’s matrix inversion algorithm Spark has gained its popularity for its in-memory distributed data (mentioned in Strassen’s original paper in 1969), which uses far processing ability, which runs interactive and iterative applications fewer operations at each level of recursion.
    [Show full text]
  • 28 Matrix Operations
    28 Matrix Operations Operations on matrices are at the heart of scientific computing. Efficient algo- rithms for working with matrices are therefore of considerable practical interest. This chapter provides a brief introduction to matrix theory and matrix operations, emphasizing the problems of multiplying matrices and solving sets of simultaneous linear equations. After Section 28.1 introduces basic matrix concepts and notations, Section 28.2 presents Strassen’s surprising algorithm for multiplying two n n matrices in 2(nlg7) O(n2.81) time. Section 28.3 shows how to solve a set of× linear equations using LUP= decompositions. Then, Section 28.4 explores the close relationship be- tween the problem of multiplying matrices and the problem of inverting a matrix. Finally, Section 28.5 discusses the important class of symmetric positive-definite matrices and shows how they can be used to find a least-squares solution to an overdetermined set of linear equations. One important issue that arises in practice is numerical stability. Due to the limited precision of floating-point representations in actual computers, round-off errors in numerical computations may become amplified over the course of a com- putation, leading to incorrect results; such computations are numerically unstable. Although we shall briefly consider numerical stability on occasion, we do not fo- cus on it in this chapter. We refer the reader to the excellent book by Golub and Van Loan [125] for a thorough discussion of stability issues. 28.1 Properties of matrices In this section, we review some basic concepts of matrix theory and some fun- damental properties of matrices, focusing on those that will be needed in later sections.
    [Show full text]
  • Minimizing Communication in All-Pairs Shortest Paths
    2013 IEEE 27th International Symposium on Parallel & Distributed Processing Minimizing communication in all-pairs shortest paths Edgar Solomonik Aydın Buluç James Demmel Univ. of California, Berkeley Lawrence Berkeley Nat. Lab. Univ. of California, Berkeley Department of EECS Computational Research Division Department of EECS [email protected] [email protected] [email protected] Abstract— over the (min,+) semiring. Several theoretical improvements We consider distributed memory algorithms for the all-pairs have been made, resulting in subcubic algorithms for the shortest paths (APSP) problem. Scaling the APSP problem APSD problem. However, in practice, these algorithms are to high concurrencies requires both minimizing inter-processor communication as well as maximizing temporal data locality. typically not competitive with simpler cubic algorithms. The 2.5D APSP algorithm, which is based on the divide-and- Variants of the Floyd-Warshall algorithm are most suitable conquer paradigm, satisfies both of these requirements: it can for dense graphs. Johnson’s algorithm [26], which is based utilize any extra available memory to perform asymptotically less on repeated application of Dijkstra’s single-source shortest communication, and it is rich in semiring matrix multiplications, path algorithm (SSSP), is theoretically faster than the Floyd- which have high temporal locality. We start by introducing a block-cyclic 2D (minimal memory) APSP algorithm. With Warshall variants on sufficiently sparse graphs. However, the a careful choice of block-size, this algorithm achieves known data dependency structure of this algorithm (and Dijkstra’s communication lower-bounds for latency and bandwidth. We algorithm in general) make scalable parallelization difficult. extend this 2D block-cyclic algorithm to a 2.5D algorithm, which SSSP algorithms based on Δ-stepping [32] scale better in can use c extra copies of data to reduce the bandwidth cost by a c1/2 practice but their performance is input dependent and scales factor of , compared to its 2D counterpart.
    [Show full text]
  • High-Performance Matrix Multiplication on Intel and FGPA
    Copyright Warning & Restrictions The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or reproduction is not to be “used for any purpose other than private study, scholarship, or research.” If a, user makes a request for, or later uses, a photocopy or reproduction for purposes in excess of “fair use” that user may be liable for copyright infringement, This institution reserves the right to refuse to accept a copying order if, in its judgment, fulfillment of the order would involve violation of copyright law. Please Note: The author retains the copyright while the New Jersey Institute of Technology reserves the right to distribute this thesis or dissertation Printing note: If you do not wish to print this page, then select “Pages from: first page # to: last page #” on the print dialog screen The Van Houten library has removed some of the personal information and all signatures from the approval page and biographical sketches of theses and dissertations in order to protect the identity of NJIT graduates and faculty. ABSTRACT HIGH-PERFORMANCE MATRIX MULTIPLICATION ON INTEL AND FPGA PLATFORMS by Gang Li Matrix multiplication is at the core of high-performance numerical computation. Software methods of accelerating matrix multiplication fall into two categories. One is based on calculation simplification. The other one is based on increasing the memory access efficiency.
    [Show full text]