DOCSLIB.ORG

Efficient Parallel Algorithm for Dense Matrix LU Decomposition with Pivoting on Hypercubes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Efficient Parallel Algorithm for Dense Matrix LU Decomposition with Pivoting on Hypercubes Computers Math. Applic. Vol. 33, No. 8, pp. 39-50, 1997 Pergamon CopyrightO1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0898-1221/97 $17.00 + 0.00 PII: S0898-1221(97)000$2-7 Efficient Parallel Algorithm for Dense Matrix LU Decomposition with Pivoting on Hypercubes ZHIYONG LIU Institute of Computing Technology, Academia Sinica, Beijing 100080, P.R. China D. W. CHEUNG Department of Computer Science, The University of Hong Kong, Pokfulam, Hong Kong (Received November 1996; accepted December 1996) Abstract--LU decomposition is intensively used in various scientific and engineering compu- tations. A parallel algorithm for dense matrix LU decomposition with pivoting on hypercubes is presented. Using n processors, the presented algorithm can finish LU decomposition of an n x n matrix in 0(n2/3 + O(n~/'~log2 n)) steps, including computations as well as communications, and its efficiency is 1 asymptotically when n becomes large. The algorithm employs row- column- as well as block-parallelisms interchangeably so that the n processors are used efficiently in the whole computation process. Using the rich connectivity, all the data alignment requirements can be realized in O(log2 n) steps. The algorithm proposed here not only is suitable for systems with small numbers of processors, but also is suitable for systems with large numbers of processors. Keywords--Linear systems of equations, LU decomposition, Partial pivoting, Parallel processing, Efficiency of parallel algorithms, Hypercubes. 1. INTRODUCTION Linear systems of equations are used intensively in various scientific and engineering compu- tations. As parallel processing systems advance, various parallel processing strategies for LU decompositions have been developed. Partitioned algorithms are given in [1,2], which are effi- cient for VLSI implementation. Optimal schedule strategies for LU decompositions on MIMD systems are given in [3-5]. The effect of the order of parallelization of the indices "i," "j," "k" in connection with different storage schemes (row or column interleaved storage of the matrix) is analyzed in [6,7] for both vector and parallel computer systems. The effects that data stor- age schemes and pivoting (row or column) strategies have the etticiency of LU decomposition on distributed memory systems are analyzed in [8]. Gallivan eL al. give a review, and provide an overall perspective of parallel algorithms for various linear algebra computations on both shared and distributed memory systems in [9]. Some techniques are given to reduce communication overheads for LU decomposition on message passing multicomputers in [10]. Sequential LU decomposition for a dense n x n matrix needs ns/3 + O(n 2) time steps, where a time steps is the time needed to execute an addition and a multiplication operation [3,11]. The This research is supported partly by a grant from The University of Hong Kong and The China National "863" Project of High Technology. Typeset by .4j~S-T~ 39 40 Z. Lxu AND D. W. CHEUNO algorithm for multiprocessors given in [3], using n/2 processors, can be completed in n 2 - 1 steps with the efficiency of 2/3. When n processors are used, the algorithms given in [8], known as RSRP, CSRP, can be completed in n2/2 steps, which have the same efficiency as the algorithm in [3]. The above algorithms for distributed memory systems use as many processors as possible to process each row (or column) in the updating process. However, as the decomposition pro- ceeds, there are not enough matrix elements to be updated in a row (or column). Thus, some computation resources are lost. Note that the BLAS-3 LU decomposition algorithms, as de- scribed in [9] for shared memory systems have the potential to reach the optimal linear speed-up for a system with n processors. It depends how the n processors are scheduled. It is pointed out in [10] that with the row or column wrap scheme, the sequential BLAS-3 computations in some stages contributes some O(n 2) time, and this can be a bottleneck for the whole computation. Computation and communication time complexities on multiprocessor ring connected by local link and broadcast bus for various task mapping strategies are analyzed in [12]. Blocked LU decomposition algorithms are given in [13,14] for mesh connected multiprocessors. The presented algorithms can balance the workload on the available processors. In the algorithm given in [14], not only data distribution is in a blocked fashion so that workload can be balanced, but the tasks are also scheduled in a blocked fashion and systolic algorithm is used for the submatrices multiplications, thus data reuse ratio is high. For hypercube connected multiprocessors, different algorithms are given in [15-17]. Various strategies for load balancing, communication and computation overlapping, including row or column wrapped, row or column blocked data distribution with corresponding pivoting strategies, and various parallelization forms (fik, jki, and kji), are discussed there thoroughly. A task partition scheme different from the one pre~ented in [3] is presented in [5], which can reduce the length of the critical task path in the scheme given in [3], but it then cannot use the task schedule strategy in [3] to complete the computation using n/2 processors, it's efficiency is 2/3 when n processors are used. The algorithm is implemented on hypercube. When systems with large numbers of processors are used, especially when the number of pro- cessors is comparable to the number of the equations of the linear system (e.g., is of O(n)), row or column oriented data distribution and task schedule is not efficient because the workload be- comes imbalanced as the computation progresses. This is because, once the size of the submatrix becomes much smaller than n, then the number of processors that will be involved in each step of the computation will be much smaller than n. Therefore, algorithms designed for hypercubes, not only efficient for systems with small number of processors, but also efficient for systems with large number of processors, are needed. Our goal here is to develop an algorithm whose efficiency can reach 1 asymptotically even when the number of processors used is comparable with the number of equations. Notice that it is easy to keep high efficiency when the number of processors is much smaller than the matrix size; however, it may not be easy to keep high efficiency when the number of processors is as large as the size of the matrix. The proposed algorithm LU_RCB strives to achieve this goal. We will propose a parallel LU decomposition algorithm with pivoting, named LU_RCB, for hypercube systems, and we will mainly describe it for an MIMD distributed memory hypercube system in this paper; a modification is straightforward for an implementation on SIMD hyper- cubes, but we will omit its description for SIMD hypercube systems. The algorithm exploits parallelisms inherent in rows, columns, and blocks interchangeably. The techniques in some par- allel storage schemes, e.g., the one that we have developed in [18] are applied here for both data distribution and task scheduling, so that the matrix elements and the computational tasks are distributed among all the processors evenly. An optimal algorithm for matrix multiplication on hypercube from [19,20] are adopted here as the basic building block for submatrix modification. When a hypercube with n nodes is used for an n x n matrix decomposition, its efficiency can be 1 asymptotically as n becomes large. Dense Matrix LU Decomposition 41 In short, the special features of LU_RCB are as follows. (1) The data and workload are distributed evenly among all the processors in the whole decomposition process so that it does not have the efficiency problem resulted from load imbalance. (2) It adapts an optimal matrix multiplication algorithm proposed for hypercubes in [19], and its data reuse ratio is high (O(v~) for one memory access). (3) Data movements can be realized efficiently using existing routing algorithms on both SIMD and MIMD hypercube systems. This paper is organized as follows. Section 2 will give some preliminary knowledge. A parallel algorithm for LU decomposition with partial pivoting will be described in Section 3. In Section 4, we will analyze time complexity of the algorithm briefly, and discuss some implementation issue, especially the implementation method of data alignment requirements. Section 5 will give some conclusions. 2. PRELIMINARIES A linear system of equations Ax = b, where A is a n x n matrix, and x and b are vectors of size n with x being the unknown, can be solved in two stages. First, decomposing A into a lower triangular matrix L and an upper triangular matrix U (such that A = L x U). Second, solving the two triangular systems by substitutions, i.e., solving Ly = b by substitutions and solution for the vector y is obtained, then solving Ux = y by substitutions, and the solution for x is obtained. In the whole computation process, the decomposition stage is the dominant part, which needs n3/3 + O(n 2) time steps when executed sequentially. Sequential LU decomposition algorithms based on Ganssian elimination can be found in literature, e.g., in [10]. Various parallel algorithms have been developed to speed up the decomposition process [1-5,7,9,11,13-16,21-23]. The reader is referred to [9] for a review. Let the n rows (and n columns) of A be numbered from 0 to n - 1, from top to bottom (from left to right). Let A (k) denote the submatrix which includes only the elements ai,j with k < i, j < n - 1. In iteration k of the LU decomposition algorithm, only the submatrix A (~) is still active (need to be further updated), the other part of the matrix has reached its final form.
Load more

Recommended publications
  • Recursive Approach in Sparse Matrix LU Factorization
    51 Recursive approach in sparse matrix LU factorization Jack Dongarra, Victor Eijkhout and the resulting matrix is often guaranteed to be positive Piotr Łuszczek∗ definite or close to it. However, when the linear sys- University of Tennessee, Department of Computer tem matrix is strongly unsymmetric or indefinite, as Science, Knoxville, TN 37996-3450, USA is the case with matrices originating from systems of Tel.: +865 974 8295; Fax: +865 974 8296 ordinary differential equations or the indefinite matri- ces arising from shift-invert techniques in eigenvalue methods, one has to revert to direct methods which are This paper describes a recursive method for the LU factoriza- the focus of this paper. tion of sparse matrices. The recursive formulation of com- In direct methods, Gaussian elimination with partial mon linear algebra codes has been proven very successful in pivoting is performed to find a solution of Eq. (1). Most dense matrix computations. An extension of the recursive commonly, the factored form of A is given by means technique for sparse matrices is presented. Performance re- L U P Q sults given here show that the recursive approach may per- of matrices , , and such that: form comparable to leading software packages for sparse ma- LU = PAQ, (2) trix factorization in terms of execution time, memory usage, and error estimates of the solution. where: – L is a lower triangular matrix with unitary diago- nal, 1. Introduction – U is an upper triangular matrix with arbitrary di- agonal, Typically, a system of linear equations has the form: – P and Q are row and column permutation matri- Ax = b, (1) ces, respectively (each row and column of these matrices contains single a non-zero entry which is A n n A ∈ n×n x where is by real matrix ( R ), and 1, and the following holds: PPT = QQT = I, b n b, x ∈ n and are -dimensional real vectors ( R ).
    [Show full text]
  • Lecture 5 - Triangular Factorizations & Operation Counts
    Lecture 5 - Triangular Factorizations & Operation Counts LU Factorization We have seen that the process of GE essentially factors a matrix A into LU. Now we want to see how this factorization allows us to solve linear systems and why in many cases it is the preferred algorithm compared with GE. Remember on paper, these methods are the same but computationally they can be different. First, suppose we want to solve A~x = ~b and we are given the factorization A = LU. It turns out that the system LU~x = ~b is \easy" to solve because we do a forward solve L~y = ~b and then back solve U~x = ~y. We have seen that we can easily implement the equations for the back solve and for homework you will write out the equations for the forward solve. Example If 0 2 −1 2 1 0 1 0 0 1 0 2 −1 2 1 A = @ 4 1 9 A = LU = @ 2 1 0 A @ 0 3 5 A 8 5 24 4 3 1 0 0 1 solve the linear system A~x = ~b where ~b = (0; −5; −16)T . ~ We first solve L~y = b to get y1 = 0, 2y1 +y2 = −5 implies y2 = −5 and 4y1 +3y2 +y3 = −16 T implies y3 = −1. Now we solve U~x = ~y = (0; −5; −1) . Back solving yields x3 = −1, 3x2 + 5x3 = −5 implies x2 = 0 and finally 2x1 − x2 + 2x3 = 0 implies x1 = 1 giving the solution (1; 0; −1)T . If GE and LU factorization are equivalent on paper, why would one be computationally advantageous in some settings? Recall that when we solve A~x = ~b by GE we must also multiply the right hand side by the Gauss transformation matrices.
    [Show full text]
  • LU Factorization LU Decomposition LU Decomposition: Motivation LU Decomposition
    LU Factorization LU Decomposition To further improve the efficiency of solving linear systems Factorizations of matrix A : LU and QR A matrix A can be decomposed into a lower triangular matrix L and upper triangular matrix U so that LU Factorization Methods: Using basic Gaussian Elimination (GE) A = LU ◦ Factorization of Tridiagonal Matrix ◦ LU decomposition is performed once; can be used to solve multiple Using Gaussian Elimination with pivoting ◦ right hand sides. Direct LU Factorization ◦ Similar to Gaussian elimination, care must be taken to avoid roundoff Factorizing Symmetrix Matrices (Cholesky Decomposition) ◦ errors (partial or full pivoting) Applications Special Cases: Banded matrices, Symmetric matrices. Analysis ITCS 4133/5133: Intro. to Numerical Methods 1 LU/QR Factorization ITCS 4133/5133: Intro. to Numerical Methods 2 LU/QR Factorization LU Decomposition: Motivation LU Decomposition Forward pass of Gaussian elimination results in the upper triangular ma- trix 1 u12 u13 u1n 0 1 u ··· u 23 ··· 2n U = 0 0 1 u3n . ···. 0 0 0 1 ··· We can determine a matrix L such that LU = A , and l11 0 0 0 l l 0 ··· 0 21 22 ··· L = l31 l32 l33 0 . ···. . l l l 1 n1 n2 n3 ··· ITCS 4133/5133: Intro. to Numerical Methods 3 LU/QR Factorization ITCS 4133/5133: Intro. to Numerical Methods 4 LU/QR Factorization LU Decomposition via Basic Gaussian Elimination LU Decomposition via Basic Gaussian Elimination: Algorithm ITCS 4133/5133: Intro. to Numerical Methods 5 LU/QR Factorization ITCS 4133/5133: Intro. to Numerical Methods 6 LU/QR Factorization LU Decomposition of Tridiagonal Matrix Banded Matrices Matrices that have non-zero elements close to the main diagonal Example: matrix with a bandwidth of 3 (or half band of 1) a11 a12 0 0 0 a21 a22 a23 0 0 0 a32 a33 a34 0 0 0 a43 a44 a45 0 0 0 a a 54 55 Efficiency: Reduced pivoting needed, as elements below bands are zero.
    [Show full text]
  • LU-Factorization 1 Introduction 2 Upper and Lower Triangular Matrices
    MAT067 University of California, Davis Winter 2007 LU-Factorization Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (March 12, 2007) 1 Introduction Given a system of linear equations, a complete reduction of the coefficient matrix to Reduced Row Echelon (RRE) form is far from the most efficient algorithm if one is only interested in finding a solution to the system. However, the Elementary Row Operations (EROs) that constitute such a reduction are themselves at the heart of many frequently used numerical (i.e., computer-calculated) applications of Linear Algebra. In the Sections that follow, we will see how EROs can be used to produce a so-called LU-factorization of a matrix into a product of two significantly simpler matrices. Unlike Diagonalization and the Polar Decomposition for Matrices that we’ve already encountered in this course, these LU Decompositions can be computed reasonably quickly for many matrices. LU-factorizations are also an important tool for solving linear systems of equations. You should note that the factorization of complicated objects into simpler components is an extremely common problem solving technique in mathematics. E.g., we will often factor a polynomial into several polynomials of lower degree, and one can similarly use the prime factorization for an integer in order to simply certain numerical computations. 2 Upper and Lower Triangular Matrices We begin by recalling the terminology for several special forms of matrices. n×n A square matrix A =(aij) ∈ F is called upper triangular if aij = 0 for each pair of integers i, j ∈{1,...,n} such that i>j. In other words, A has the form a11 a12 a13 ··· a1n 0 a22 a23 ··· a2n A = 00a33 ··· a3n .
    [Show full text]
  • LU, QR and Cholesky Factorizations Using Vector Capabilities of Gpus LAPACK Working Note 202 Vasily Volkov James W
    LU, QR and Cholesky Factorizations using Vector Capabilities of GPUs LAPACK Working Note 202 Vasily Volkov James W. Demmel Computer Science Division Computer Science Division and Department of Mathematics University of California at Berkeley University of California at Berkeley Abstract GeForce Quadro GeForce GeForce GPU name We present performance results for dense linear algebra using 8800GTX FX5600 8800GTS 8600GTS the 8-series NVIDIA GPUs. Our matrix-matrix multiply routine # of SIMD cores 16 16 12 4 (GEMM) runs 60% faster than the vendor implementation in CUBLAS 1.1 and approaches the peak of hardware capabilities. core clock, GHz 1.35 1.35 1.188 1.458 Our LU, QR and Cholesky factorizations achieve up to 80–90% peak Gflop/s 346 346 228 93.3 of the peak GEMM rate. Our parallel LU running on two GPUs peak Gflop/s/core 21.6 21.6 19.0 23.3 achieves up to ~300 Gflop/s. These results are accomplished by memory bus, MHz 900 800 800 1000 challenging the accepted view of the GPU architecture and memory bus, pins 384 384 320 128 programming guidelines. We argue that modern GPUs should be viewed as multithreaded multicore vector units. We exploit bandwidth, GB/s 86 77 64 32 blocking similarly to vector computers and heterogeneity of the memory size, MB 768 1535 640 256 system by computing both on GPU and CPU. This study flops:word 16 18 14 12 includes detailed benchmarking of the GPU memory system that Table 1: The list of the GPUs used in this study. Flops:word is the reveals sizes and latencies of caches and TLB.
    [Show full text]
  • LU and Cholesky Decompositions. J. Demmel, Chapter 2.7
    LU and Cholesky decompositions. J. Demmel, Chapter 2.7 1/14 Gaussian Elimination The Algorithm — Overview Solving Ax = b using Gaussian elimination. 1 Factorize A into A = PLU Permutation Unit lower triangular Non-singular upper triangular 2/14 Gaussian Elimination The Algorithm — Overview Solving Ax = b using Gaussian elimination. 1 Factorize A into A = PLU Permutation Unit lower triangular Non-singular upper triangular 2 Solve PLUx = b (for LUx) : − LUx = P 1b 3/14 Gaussian Elimination The Algorithm — Overview Solving Ax = b using Gaussian elimination. 1 Factorize A into A = PLU Permutation Unit lower triangular Non-singular upper triangular 2 Solve PLUx = b (for LUx) : − LUx = P 1b − 3 Solve LUx = P 1b (for Ux) by forward substitution: − − Ux = L 1(P 1b). 4/14 Gaussian Elimination The Algorithm — Overview Solving Ax = b using Gaussian elimination. 1 Factorize A into A = PLU Permutation Unit lower triangular Non-singular upper triangular 2 Solve PLUx = b (for LUx) : − LUx = P 1b − 3 Solve LUx = P 1b (for Ux) by forward substitution: − − Ux = L 1(P 1b). − − 4 Solve Ux = L 1(P 1b) by backward substitution: − − − x = U 1(L 1P 1b). 5/14 Example of LU factorization We factorize the following 2-by-2 matrix: 4 3 l 0 u u = 11 11 12 . (1) 6 3 l21 l22 0 u22 One way to find the LU decomposition of this simple matrix would be to simply solve the linear equations by inspection. Expanding the matrix multiplication gives l11 · u11 + 0 · 0 = 4, l11 · u12 + 0 · u22 = 3, (2) l21 · u11 + l22 · 0 = 6, l21 · u12 + l22 · u22 = 3.
    [Show full text]
  • Math 2270 - Lecture 27 : Calculating Determinants
    Math 2270 - Lecture 27 : Calculating Determinants Dylan Zwick Fall 2012 This lecture covers section 5.2 from the textbook. In the last lecture we stated and discovered a number of properties about determinants. However, we didn’t talk much about how to calculate them. In fact, the only general formula was the nasty formula mentioned at the beginning, namely n det(A)= sgn(σ) a . X Y iσ(i) σ∈Sn i=1 We also learned a formula for calculating the determinant in a very special case. Namely, if we have a triangular matrix, the determinant is just the product of the diagonals. Today, we’re going to discuss how that special triangular case can be used to calculate determinants in a very efficient manner, and we’ll derive the nasty formula. We’ll also go over one other way of calculating a deter- minnat, the “cofactor expansion”, that, to be honest, is not all that useful computationally, but can be very useful when you need to prove things. The assigned problems for this section are: Section 5.2 - 1, 3, 11, 15, 16 1 1 Calculating the Determinant from the Pivots In practice, the easiest way to calculate the determinant of a general matrix is to use elimination to get an upper-triangular matrix with the same de- terminant, and then just calculate the determinant of the upper-triangular matrix by taking the product of the diagonal terms, a.k.a. the pivots. If there are no row exchanges required for the LU decomposition of A, then A = LU, and det(A)= det(L)det(U).
    [Show full text]
  • LU Decomposition S = LU, Then We Can  1 0 0   2 4 −2   2 4 −2  Use It to Solve Sx = F
    LU-Decomposition Example 1 Compare with Example 1 in gaussian elimination.pdf. Consider I The Gaussian elimination for the solution of the linear system Sx = f transforms the augmented matrix (Sjf) into (Ujc), where U 0 2 4 −21 is upper triangular. The transformations are determined by the S = 4 9 −3 : (1) matrix S only. @ A If we have to solve a system Sx = f~ with a different right hand side −2 −3 7 f~, then we start over. Most of the computations that lead us from We express Gaussian Elimination using Matrix-Matrix-multiplications (Sjf~) into (Ujc~), depend only on S and are identical to the steps that we executed when we applied Gaussian elimination to Sx = f. 0 1 0 0 1 0 2 4 −2 1 0 2 4 −2 1 We now express Gaussian elimination as a sequence of matrix-matrix I @ −2 1 0 A @ 4 9 −3 A = @ 0 1 1 A multiplications. This representation leads to the decomposition of S 1 0 1 −2 −3 7 0 1 5 into a product of a lower triangular matrix L and an upper triangular | {z } | {z } | {z } matrix U, S = LU. This is known as the LU-Decomposition of S. =E1 =S =E1S I If we have computed the LU decomposition S = LU, then we can 0 1 0 0 1 0 2 4 −2 1 0 2 4 −2 1 use it to solve Sx = f. @ 0 1 0 A @ 0 1 1 A = @ 0 1 1 A I We replace S by LU, 0 −1 1 0 1 5 0 0 4 LUx = f; | {z } | {z } | {z } and introduce y = Ux.
    [Show full text]
  • Scalable Sparse Symbolic LU Factorization on Gpus
    JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1 GSOFA: Scalable Sparse Symbolic LU Factorization on GPUs Anil Gaihre, Xiaoye Sherry Li and Hang Liu Abstract—Decomposing a matrix A into a lower matrix L and an upper matrix U, which is also known as LU decomposition, is an essential operation in numerical linear algebra. For a sparse matrix, LU decomposition often introduces more nonzero entries in the L and U factors than in the original matrix. A symbolic factorization step is needed to identify the nonzero structures of L and U matrices. Attracted by the enormous potentials of the Graphics Processing Units (GPUs), an array of efforts have surged to deploy various LU factorization steps except for the symbolic factorization, to the best of our knowledge, on GPUs. This paper introduces GSOFA, the first GPU-based symbolic factorization design with the following three optimizations to enable scalable LU symbolic factorization for nonsymmetric pattern sparse matrices on GPUs. First, we introduce a novel fine-grained parallel symbolic factorization algorithm that is well suited for the Single Instruction Multiple Thread (SIMT) architecture of GPUs. Second, we tailor supernode detection into a SIMT friendly process and strive to balance the workload, minimize the communication and saturate the GPU computing resources during supernode detection. Third, we introduce a three-pronged optimization to reduce the excessive space consumption problem faced by multi-source concurrent symbolic factorization. Taken together, GSOFA achieves up to 31× speedup from 1 to 44 Summit nodes (6 to 264 GPUs) and outperforms the state-of-the-art CPU project, on average, by 5×.
    [Show full text]
  • LU, QR and Cholesky Factorizations Using Vector Capabilities of Gpus LAPACK Working Note 202 Vasily Volkov James W
    LU, QR and Cholesky Factorizations using Vector Capabilities of GPUs LAPACK Working Note 202 Vasily Volkov James W. Demmel Computer Science Division Computer Science Division and Department of Mathematics University of California at Berkeley University of California at Berkeley Abstract GeForce Quadro GeForce GeForce GPU name We present performance results for dense linear algebra using 8800GTX FX5600 8800GTS 8600GTS the 8-series NVIDIA GPUs. Our matrix-matrix multiply routine # of SIMD cores 16 16 12 4 (GEMM) runs 60% faster than the vendor implementation in CUBLAS 1.1 and approaches the peak of hardware capabilities. core clock, GHz 1.35 1.35 1.188 1.458 Our LU, QR and Cholesky factorizations achieve up to 80–90% peak Gflop/s 346 346 228 93.3 of the peak GEMM rate. Our parallel LU running on two GPUs peak Gflop/s/core 21.6 21.6 19.0 23.3 achieves up to ~300 Gflop/s. These results are accomplished by memory bus, MHz 900 800 800 1000 challenging the accepted view of the GPU architecture and memory bus, pins 384 384 320 128 programming guidelines. We argue that modern GPUs should be viewed as multithreaded multicore vector units. We exploit bandwidth, GB/s 86 77 64 32 blocking similarly to vector computers and heterogeneity of the memory size, MB 768 1535 640 256 system by computing both on GPU and CPU. This study flops:word 16 18 14 12 includes detailed benchmarking of the GPU memory system that Table 1: The list of the GPUs used in this study. Flops:word is the reveals sizes and latencies of caches and TLB.
    [Show full text]
  • Gaussian Elimination and Lu Decomposition (Supplement for Ma511)
    GAUSSIAN ELIMINATION AND LU DECOMPOSITION (SUPPLEMENT FOR MA511) D. ARAPURA Gaussian elimination is the go to method for all basic linear classes including this one. We go summarize the main ideas. 1. Matrix multiplication The rule for multiplying matrices is, at first glance, a little complicated. If A is m × n and B is n × p then C = AB is defined and of size m × p with entries n X cij = aikbkj k=1 The main reason for defining it this way will be explained when we talk about linear transformations later on. THEOREM 1.1. (1) Matrix multiplication is associative, i.e. A(BC) = (AB)C. (Henceforth, we will drop parentheses.) (2) If A is m × n and Im×m and In×n denote the identity matrices of the indicated sizes, AIn×n = Im×mA = A. (We usually just write I and the size is understood from context.) On the other hand, matrix multiplication is almost never commutative; in other words generally AB 6= BA when both are defined. So we have to be careful not to inadvertently use it in a calculation. Given an n × n matrix A, the inverse, if it exists, is another n × n matrix A−1 such that AA−I = I;A−1A = I A matrix is called invertible or nonsingular if A−1 exists. In practice, it is only necessary to check one of these. THEOREM 1.2. If B is n × n and AB = I or BA = I, then A invertible and B = A−1. Proof. The proof of invertibility of A will need to wait until we talk about deter- minants.
    [Show full text]
  • Effective GPU Strategies for LU Decomposition H
    Effective GPU Strategies for LU Decomposition H. M. D. M. Bandara* & D. N. Ranasinghe University of Colombo School of Computing, Sri Lanka [email protected], [email protected] Abstract— GPUs are becoming an attractive computing approaches, Section 5 shows the results and platform not only for traditional graphics computation performance studies obtained on our test environment but also for general-purpose computation because of the and then finally Section 6 concludes the paper. computational power, programmability and comparatively low cost of modern GPUs. This has lead to II. RELATED WORK a variety of complex GPGPU applications with significant performance improvements. The LU A few research works on LU decomposition with decomposition represents a fundamental step in many the support of highly parallel GPU environments have computationally intensive scientific applications and it is been carried out and had not been able to accelerate it often the costly step in the solution process because of the much. In [4] Ino et al. they have developed and impact of size of the matrix. In this paper we implement evaluated some implementation methods for LU three different variants of the LU decomposition algorithm on a Tesla C1060 and the most significant LU Decomposition in terms of (a) loop processing (b) decomposition that fits the highly parallel architecture of branch processing and (c) vector processing. In [5] modern GPUs is found to be Update through Column Gappalo et al. have reduced matrix decomposition and with shared memory access implementation. row operations to a series of rasterization problems on GPU. They have mapped the problem to GPU Keywords—LU decomposition, CUDA, GPGPU architecture based on the fact that fundamental I.
    [Show full text]