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Parallel Full Matrix Algorithms

Given by Geoffrey C. Fox, Nancy J. McCracken at CPS615 Basic Simulation Track for Computational Science on Fall Semester 95. Foils prepared 6 December 96

This CPS615 module covers basic full matrix parallel algorithms with a discussion of matrix multiplication, LU decomposition with latter covered for banded as well as true full case
Matrix multiplication covers the approach given in "Solving Problems on Concurrent Processors" as well as Cannon's algorithm.
We review those applications -- especially Computational electromagnetics and Chemistry -- where full matrices are commonly used
Of course sparse matrices are far more important than full matrices!

Table of Contents for Parallel Full Matrix Algorithms

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1 Full Matrices
CPS615 Computational Science for Simulation Applications
December 4, 1995
2 Abstract of Full Matrix CPS615 Module
3 Review of Matrices seen in PDE's
4 Examples of Full Matrices in Chemistry
5 Operations used with Hamiltonian operator
6 Examples of Full Matrices in Electromagnetics
7 Computational Electromagnetics Formalism I
8 Computational Electromagnetics Formalism II
9 Comments on Computational Electromagnetics
10 Summary of Use of Full Matrices in Chemistry
11 Notes on the use of full matrices
12 Full Matrix Multiplication
13 Sub-block definition of Matrix Multiply
14 Some References
15 The First Algorithm
(Broadcast, Multiply, and Roll)
16 The first stage -- index n=0 in sub-block sum -- of the algorithm on N=16 example
17 The second stage -- n=1 in sum over subblock indices -- of the algorithm on N=16 example
18 Second stage, continued
19 Look at the whole algorithm on one element
20 Cartesian Topology in MPI -- General
21 Cartesian Topology in MPI -- Matrix Multiplication
22 Cartesian Topology in MPI -- MPI_CART_SUB
23 Cartesian Topology in MPI -- MPI_CART_shift
24 Cartesian Topology in MPI -- MPI_SENDRECV_REPLACE
25 Cartesian Topology in MPI -- MPI_Cart_Create
26 Matrix Multiplication MPI Style Pseudocode
27 Matrix Multiplication Pseudocode, continued
28 Broadcast in the Full Matrix Case
29 Implementation of Naive and Log Broadcast
30 The Pipe Broadcast Operation
31 Schematic of Pipe Broadcast Operation
32 Performance Analysis of Matrix Multiplication
33 Cannon's Algorithm for Matrix Multiplication
34 Cannon's Algorithm
35 The Set-up Stage of the Algorithm
36 The first iteration of the algorithm
37 Performance Analysis of Cannon's Algorithm
38 Full LU Decomposition
39 Some References
40 Sequential LU Algorithm
41 Sequential LU Algorithm, continued
42 Sequential Pseudocode
43 Parallel Decomposition
44 Better Parallel Decomposition
45 Parallel Pseudocode
46 Performance Analysis of the Parallel LU Decomposition
47 Banded LU Decomposition
48 Some References
49 Banded Matrix Decomposition
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© on Tue Oct 7 1997