1 / 12

Efficient Parallelization of Backsolve Algorithms in Matrix Computations

This project focuses on parallelizing the backsolve algorithm for upper triangular matrices, solving equations efficiently. It covers the challenges, solution techniques, and implementation using parallel processing with MPI for improved performance.

clark-madden
Télécharger la présentation

Efficient Parallelization of Backsolve Algorithms in Matrix Computations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. PARALLELIZATION OF MULTIPLE BACKSOLVES Project #2 James Stanley April 25, 2002

  2. PARALLELIZATION OF MULTIPLE BACKSOLVESProject #2 • Introduction (Backsolve) • Challenge • Example (m = 5) • Problem Description • Solution Technique • Parallel Implementation • Results

  3. Introduction (Backsolve) If R is an upper triangular matrix, a backsolve is a solution of the equation, Rx=b Where b is a vector of length m. The formulas for the solution are

  4. bm ___ mth equation: rmmxm = bm => xm= rmm Introduction (Backsolve) cont. from mth equation m-1th equation: rm-1,m-1xm-1 + rm-1,mxm = bm-1 bm-1- rm-1,* xm __________________ => xm-1 = rm-1

  5. Introduction (Backsolve) cont. rii,xi + ri,i+1xi+1 + …+ rim,xi = bi , for 0  i  m-1 ith equation: =>

  6. Challenge Storage: To avoid storing zeros, store n(n+1)/2 nonzero elements of R1in a 1-Dimensional array by rows, and the m(m+1)/2 Nonzero elements of R2 in a 1-Dimensional array by rows.

  7. Example Suppose m = 5, then or,

  8. Example cont. Solving for the xi’s provides in memory 

  9. Problem Description Given RHS matrix H to solve for the nxmunknown matrix Y : R1YR2T = H Where R1 is a square upper triangular matrix of order nxn and R2is a square upper triangular matrix of order mxm.

  10. (2) Let Z =R1Y • Then R1YR2T = ZR2T • ZR2T = H Solution Technique  R2ZT = HT Parallel Solution of (1) and (2): (1) R2T=HT = (h1,h2,…,hn) (2) R1Y =Z = (z1,z2,…,zm) 1rst solve (1) for the mxn matrix ZT. 2nd take the Transpose of ZT to get Z. 3rd solve (2) for the nxm solution matrix Y.

  11. Parallel Implementation • Generate HT and R2, R1 on Process 0, using a Random Number Generator. • Move all of R2 to all processes with the MPI_BCAST. (If there are p processors, then make sure n/p and m/p are integers. • Ship the n/p of the rows of H to each process with MPI_Scatter. • On each process solve n/p equations for local Z. • Ship all of the columns of Z using MPI_Gather to process 0. • Perform the transpose of Z on process 0. • Ship m/p of the rows of ZT to each process with MPI_Scatter. • On each process solve m/p equations for local Y. • Ship all of the m/p columns of local Y to process 0 and print the solution on process 0. Denotes communication time Denotes computation time

  12. Results

More Related