Matrix Multiplication General Case M. A. Hameed ECSE Department - PowerPoint PPT Presentation

Jan 12, 2023 •380 likes •430 views

Matrix Multiplication General Case M. A. Hameed ECSE Department Rensselaer Polytechnic Institute Intro to ECSE Multiply matrices A and B In order for matrix multiplication to be possible: number of columns in matrix A = number of rows in

Matrix Multiplication General Case M. A. Hameed ECSE Department Rensselaer Polytechnic Institute Intro to ECSE
Multiply matrices A and B In order for matrix multiplication to be possible: number of columns in matrix A = number of rows in matrix B # of rows m Let the size (or order or dimension) of matrix A = N x M ↳ # of columns ① → # of columns Let the size (or order or dimension) of matrix B = M x K ↳ # of rows O Size of the resulting matrix Z = A.B will be N x K # columns L ro w s M. Hameed 5 October 2019
" ¥ ::÷÷÷÷÷:÷¥ . ::÷i:÷:÷÷÷÷÷ . . ) . . = TETA ?÷ Ali Biz § ⇐ ¥Bi ¥ , :B , K , Alibi , . - - - ¥2 Az :B , . 772¥13 AB = - - - - , . " " ' - i En ' E. Ani Bik viii. a. Bi . - . * . M. Hameed 5 October 2019

Recommend

Matrix Multiplication Matrix multiplication is an operation with properties quite different from

Matrix Multiplication Matrix multiplication is an operation with properties quite different from its scalar counterpart. To begin with, order matters in matrix multiplication. That is, the matrix product AB need not be the same as the matrix

695 views • 58 slides

Shared Memory with Cilk++ Matrix-matrix multiplication Matrix-vector multiplication

CS 140 : Numerical Examples on Shared Memory with Cilk++ Matrix-matrix multiplication Matrix-vector multiplication Hyperobjects Thank nks to Charles E. L Leiserson on for some of t these slides 1 Work and Span (Recap) T P =

449 views • 29 slides

Complexity of matrix multiplication (For Hierarchical matrix) For Usual matrix The

Complexity of matrix multiplication (For Hierarchical matrix) For Usual matrix The naive multiplication algorithm for nxn matrix needs n^3 multiplications (and n^3 additions) Is it Optimal ? No! [Strassen] do better (n^log2

579 views • 26 slides

CS 140 : Matrix multiplication Warmup: Matrix times vector: communication volume Matrix

CS 140 : Matrix multiplication Warmup: Matrix times vector: communication volume Matrix multiplication I: parallel issues Matrix multiplication II: cache issues Thanks to Jim Demmel and Kathy Yelick (UCB) for some of these slides

473 views • 45 slides

Matrix Calculations: Kernels & Images, Matrix Multiplication A. Kissinger (and H. Geuvers)

Matrix multiplication Matrix inverse Radboud University Nijmegen Kernel and image Matrix Calculations: Kernels & Images, Matrix Multiplication A. Kissinger (and H. Geuvers) Institute for Computing and Information Sciences Intelligent

618 views • 40 slides

Chapter VI All Pair Shortest Paths and Matrix Multiplication VI.1 APSPs and Matrix

Chapter VI All Pair Shortest Paths and Matrix Multiplication VI.1 APSPs and Matrix Multiplication There is a close similarity between the inner loop in the APSP algorithm and matrix multiplication. Recall that for n n matrices A = ( a ij )

93 views • 7 slides

Efficient multiplication 2 Matrix multiplication If you have square matrices A and B, then C =

1 Efficient multiplication 2 Matrix multiplication If you have square matrices A and B, then C = A*B is defined as: Takes O(n 3 ) time 3 Matrix multiplication Can we do better? What is the theoretical lowest running time possible? 4

622 views • 18 slides

Proposing a Fast and Scalable Systolic Array for Matrix Multiplication Bahar Asgari , , Ra

Proposing a Fast and Scalable Systolic Array for Matrix Multiplication Bahar Asgari , , Ra Ramyad Ha Hadidi, , Hy Hyesoon esoon Ki Kim Click to edit Master subtitle style Matrix Multiplication 2 Matrix multiplication is the key operation

1.1k views • 44 slides

Matrix Multiplication Matrix Multiplication via Matrix-Vector Mult Defn. If matrix A is m n

Matrix Multiplication Matrix Multiplication via Matrix-Vector Mult Defn. If matrix A is m n and matrix B is r s , then for the product AB to be valid it must be that n = r . If valid, the product AB has size m s . The columns of the

400 views • 12 slides

Dual-Entangled Polynomial Code: Three-Dimensional Coding for Distributed Matrix Multiplication

Dual-Entangled Polynomial Code: Three-Dimensional Coding for Distributed Matrix Multiplication Pedro Soto, Jun Li, Xiaodi Fan Florida International University Matrix Multiplication Matrix multiplication is a fundamental building block in

670 views • 23 slides

Parallel Scientific Computing Matrix-vector multiplication. Matrix-matrix multiplication.

CS140 IV-1 Parallel Scientific Computing Matrix-vector multiplication. Matrix-matrix multiplication. Direct method for solving a linear equation. Gaussian Elimination. Iterative method for solving a linear equation.

367 views • 25 slides

I/O Lower Bounds and Algorithms for Matrix-Matrix Multiplication Tyler M. Smith July 5, 2017 1

I/O Lower Bounds and Algorithms for Matrix-Matrix Multiplication Tyler M. Smith July 5, 2017 1 / 69 Introduction Dense matrix-matrix multiplication (MMM) Goal: Reduce I/O cost for machines with hierarchical memory Novel

926 views • 69 slides

High-performance and Memory-saving Sparse General Matrix-Matrix Multiplication for Pascal GPU

High-performance and Memory-saving Sparse General Matrix-Matrix Multiplication for Pascal GPU Yusuke Nagasaka, Akira Nukada, Satoshi Matsuoka Tokyo Institute of Technology Sparse General Matrix-Matrix Multiplication (SpGEMM) Numerical

943 views • 34 slides

Communication Lower Bounds for Matrix-Matrix Multiplication Dagstuhl Seminar #15281 July 6-9,

Communication Lower Bounds for Matrix-Matrix Multiplication Dagstuhl Seminar #15281 July 6-9, 2015 Julien Langou M OTIVATIONS 2 Motivations Communication Lower Bound for Sequential Matrix-Matrix multiplication Application to Parallel

1.07k views • 95 slides

Column-Based Matrix Partitioning for Parallel Matrix Multiplication on Heterogeneous Processors

Motivation Parallel Matrix Multiplication Routine Matrix Partitioning Algorithms Experimental Results Conclusions Column-Based Matrix Partitioning for Parallel Matrix Multiplication on Heterogeneous Processors Based on Functional Performance

514 views • 33 slides

Matrix-chain multiplication Carola Wenk 1 CMPS 6610 Algorithms Matrix-chain multiplication

CMPS 6610 Fall 2018 Matrix-chain multiplication Carola Wenk 1 CMPS 6610 Algorithms Matrix-chain multiplication Given: A sequence/chain of n matrices A 1 , A 2 ,, A n , where A i is a p i-1 p i matrix Task: Compute their product A 1

627 views • 14 slides

MATH 105: Finite Mathematics 2-5: Matrix Multiplication Prof. Jonathan Duncan Walla Walla

The Matrix Form of a System of Equations Matrix Multiplication The Identity Matrix Conclusion MATH 105: Finite Mathematics 2-5: Matrix Multiplication Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 The Matrix Form of a System

724 views • 38 slides

Error Correction in Fast Matrix Multiplication and Inverse Daniel S. Roche Computer Science

Introduction Matrix Multiplication Matrix Inverse Perspective Error Correction in Fast Matrix Multiplication and Inverse Daniel S. Roche Computer Science Department United States Naval Academy Annapolis, Maryland, U.S.A. ISSAC 2018 19

566 views • 55 slides

Matrix-Chain Multiplication Given : chain of matrices ( A 1 , A 2 , . . . A n ) , with A i

Matrix-Chain Multiplication Given : chain of matrices ( A 1 , A 2 , . . . A n ) , with A i having dimension ( p i 1 p i ) . Goal: compute product A 1 A 2 A n as quickly as possible Dynamic Programming 1 Multiplication of (

339 views • 21 slides

CS 401 Integer Multiplication / Matrix Multiplication Xiaorui Sun 1 Integer Multiplication

CS 401 Integer Multiplication / Matrix Multiplication Xiaorui Sun 1 Integer Multiplication Integer Arithmetic 1 1 1 1 1 1 0 1 Add: Given two ! -bit integers 1 1 0 1 0 1 0 1 " and # , compute " + # . Add + 0 1 1 1

468 views • 19 slides

Matrix Multiply in Hadoop Botong Huang and You Wu (Will) Content Dense Matrix Multiplication

Matrix Multiply in Hadoop Botong Huang and You Wu (Will) Content Dense Matrix Multiplication Previous Work Our Approach and Strategy Analysis Experiment Sparse Matrix Multiplication Our Approach Experiment Previous

365 views • 23 slides

Module 4.3 - Memory Model and Locality Tiled Matrix Multiplication Objective To understand

GPU Teaching Kit Accelerated Computing Module 4.3 - Memory Model and Locality Tiled Matrix Multiplication Objective To understand the design of a tiled parallel algorithm for matrix multiplication Loading a tile Phased execution

452 views • 15 slides

Matrix Multiplication and Graph Algorithms Uri Zwick Tel Aviv University NoNA Summer School on

Matrix Multiplication and Graph Algorithms Uri Zwick Tel Aviv University NoNA Summer School on Complexity Theory Saint Petersburg August 15-16, 2009 Outline 1. Algebraic matrix multiplication a. Strassens algorithm b. Rectangular

1.34k views • 131 slides

Quiz Give the definition of matrix-matrix multiplication ( A times B ) in which one of the matrices

Quiz Give the definition of matrix-matrix multiplication ( A times B ) in which one of the matrices (which?) is broken into rows and columns (which?). In your definition, specify how the output matrix is obtained from the input matrices A and B .

690 views • 28 slides