Coursehero >> California >> San Jose State >> CS 159

Course Hero has millions of student submitted documents similar to the one below including study guides, homework solutions, papers, exam answer keys and textbook solutions.

Computations Matrix Outline: CS 159: Parallel Processing Spring 2007 Matrix review Matrix"vector multiplication Matrix"Matrix multiplicaton Iterative methods for solving linear equations Matrix Review Matrices and transformations Matrices and linear equations Solution of linear equations from the matrix viewpoint Matrix"vector multiplication Ab = c n De!nition: " aij b j = ci j=1 i = 1,2,K,n ! ! More than one way to implement this: row"column method linear combination of columns method ! row"column method Matrix"vector multiplication Parallel implementation of the row"column method each row of A in a slave processor master has the b vector master broadcasts all components of b to slaves slave i computes ci and sends to master Matrix"vector multiplication #cont.$ Parallel implementation of the linear combination of columns method each column of A in a slave processor master has the b vector master scatters the b vector to slaves slave i computes bi ai #scalar"vector product$ slaves do a tree"sum and report to the master ! !! Matrix"Matrix multiplication AB = C A bi = c i #this allows us to treat this as multiple matrix"vector products$ Let A, B, and C be divided into blocks that ! ! conformable. Then the block product works are in the same way as regular matrix multiply Iterative Methods for Solving Linear Equations Problem: Solve x A = b Jacobi method xik = 1 [bi " $ aij x k"1 ] j aii j#i Gauss"Seidel method % #update as you go$ ! Block Iterative Methods Block Jacobi method Where Matrices Arise Example of Laplace&s Equation Inside the rectangular region solve for the function u#x,y$: u(x, yt ) = h t (x) "2 u "2 u + = f (x, y) "x 2 "y 2 r r r Aii xik = bi " $ Aij x k"1 j j#i where a block of equations are assigned to each processor. ! y ! Block Gauss"Seidel method % """update the x values from the previous % blocks as soon as available u(x l , y) = vl (y) u(x, yb ! hb (x) )= x ! ! u(x r , y) = vr (y) ! Where Matrices Arise From the continuous to the discrete approximate the second derivative of g at xi g(x) Where Matrices Arise Example of Laplace&s Equation Inside the rectangular region solve for the function u#x,y$: "2 u "2 u + = f (x, y) "x 2 "y 2 ! xi"1 h ! boundary case y ! xi h xi+1 interior case ! d 2 g(xi ) gi#1 # 2gi + gi+1 " 2 h2 ! dx ! x boundary case ! The resulting matrix equation 1 Au = f h2 Time"dependent problems The heat equation: ! What the matrix looks like... % numbering the unknowns % very sparse matrix What do the block iterative % methods look like... % What&s a block? "u " 2 u " 2 u = + "t "x 2 "y 2 Suppose we have the same rectangular region ! Approximate the time derivative The resulting equations u t+"t # u t = "t Au t h2 there are variations on the right side !

Find millions of documents here - Study Guides, Homework Solutions, Papers, Exam Answer Keys and more. Course Hero has millions of course related materials that will enable you to learn better, faster and get an A in all your courses.
Below is a small sample set of documents: