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PARALLELIZATION OF MULTIPLE BACKSOLVES

PARALLELIZATION OF MULTIPLE BACKSOLVES. Project #2. James Stanley April 25, 2002. PARALLELIZATION OF MULTIPLE BACKSOLVES Project #2. Introduction (Backsolve) Challenge Example ( m = 5) Problem Description Solution Technique Parallel Implementation Results. Introduction (Backsolve).

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PARALLELIZATION OF MULTIPLE BACKSOLVES

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  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

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