1 / 21

CS 584

CS 584. Assignment. Systems of Linear Equations. A linear equation in n variables has the form A set of linear equations is called a system. A solution exists for a system iff the solution satisfies all equations in the system. Many scientific and engineering problems take this form.

basil-brennan
Download Presentation

CS 584

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. CS 584 • Assignment

  2. Systems of Linear Equations • A linear equation in n variables has the form • A set of linear equations is called a system. • A solution exists for a system iff the solution satisfies all equations in the system. • Many scientific and engineering problems take this form. a0x0 + a1x1 + … + an-1xn-1 = b

  3. Solving Systems of Equations • Many such systems are large. • Thousands of equations and unknowns a0,0x0 + a0,1x1 + … + a0,n-1xn-1 = b0 a1,0x0 + a1,1x1 + … + a1,n-1xn-1 = b1 an-1,0x0 + an-1,1x1 + … + an-1,n-1xn-1 = bn-1

  4. Solving Systems of Equations • A linear system of equations can be represented in matrix form a0,0 a0,1 … a0,n-1 x0 b0 a1,0 a1,1 … a1,n-1 x1 b1 an-1,0 an-1,1 … an-1,n-1 xn-1 bn-1 = Ax = b

  5. Solving Systems of Equations • Solving a system of linear equations is done in two steps: • Reduce the system to upper-triangular • Use back-substitution to find solution • These steps are performed on the system in matrix form. • Gaussian Elimination, etc.

  6. a0,0 a0,1 … a0,n-1 x0 b0 0 a1,1 … a1,n-1 x1 b1 0 0 … an-1,n-1 xn-1 bn-1 = Solving Systems of Equations • Reduce the system to upper-triangular form • Use back-substitution

  7. Reducing the System • Gaussian elimination systematically eliminates variable x[k] from equations k+1 to n-1. • Reduces the coefficients to zero • This is done by subtracting a appropriate multiple of the kth equation from each of the equations k+1 to n-1

  8. Procedure GaussianElimination(A, b, y) for k = 0 to n-1 /* Division Step */ for j = k + 1 to n - 1 A[k,j] = A[k,j] / A[k,k] y[k] = b[k] / A[k,k] A[k,k] = 1 /* Elimination Step */ for i = k + 1 to n - 1 for j = k + 1 to n - 1 A[i,j] = A[i,j] - A[i,k] * A[k,j] b[i] = b[i] - A[i,k] * y[k] A[i,k] = 0 endfor endfor end

  9. Parallelizing Gaussian Elim. • Use domain decomposition • Rowwise striping • Division step requires no communication • Elimination step requires a one-to-all broadcast for each equation. • No agglomeration • Initially map one to to each processor

  10. Communication Analysis • Consider the algorithm step by step • Division step requires no communication • Elimination step requires one-to-all bcast • only bcast to other active processors • only bcast active elements • Final computation requires no communication.

  11. Communication Analysis • One-to-all broadcast • log2q communications • q = n - k - 1 active processors • Message size • q active processors • q elements required T = (ts + twq)log2q

  12. Computation Analysis • Division step • q divisions • Elimination step • q multiplications and subtractions • Assuming equal time --> 3q operations

  13. Computation Analysis • In each step, the active processor set is reduced by one resulting in: å - 1 n = - - CompTime n k 1 = 0 k = - CompTime 3 n ( n 1 ) / 2

  14. Can we do better? • Previous version is synchronous and parallelism is reduced at each step. • Pipeline the algorithm • Run the resulting algorithm on a linear array of processors. • Communication is nearest-neighbor • Results in O(n) steps of O(n) operations

  15. Pipelined Gaussian Elim. • Basic assumption: A processor does not need to wait until all processors have received a value to proceed. • Algorithm • If processor p has data for other processors, send the data to processor p+1 • If processor p can do some computation using the data it has, do it. • Otherwise, wait to receive data from processor p-1

  16. Conclusion • Using a striped partitioning method, it is natural to pipeline the Gaussian elimination algorithm to achieve best performance. • Pipelined algorithms work best on a linear array of processors. • Or something that can be linearly mapped • Would it be better to block partition? • How would it affect the algorithm?

More Related