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09_tridiagonal

# 09_tridiagonal - Band Systems Tridiagonal Systems Cyclic...

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Band Systems Tridiagonal Systems Cyclic Reduction Parallel Numerical Algorithms Chapter 9 – Band and Tridiagonal Systems Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign CSE 512 / CS 554 Michael T. Heath Parallel Numerical Algorithms 1 / 24

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Band Systems Tridiagonal Systems Cyclic Reduction Outline 1 Band Systems 2 Tridiagonal Systems 3 Cyclic Reduction Michael T. Heath Parallel Numerical Algorithms 2 / 24
Band Systems Tridiagonal Systems Cyclic Reduction Banded Linear Systems Bandwidth (or semibandwidth ) of n × n matrix A is smallest value β such that a ij = 0 for all | i - j | > β Matrix is banded if β n If β p , then minor modifications of parallel algorithms for dense LU or Cholesky factorization are reasonably efficient for solving banded linear system Ax = b If β / p , then standard parallel algorithms for LU or Cholesky factorization utilize few processors and are very inefficient Michael T. Heath Parallel Numerical Algorithms 3 / 24

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Band Systems Tridiagonal Systems Cyclic Reduction Narrow Banded Linear Systems More efficient parallel algorithms for narrow banded linear systems are based on divide-and-conquer approach in which band is partitioned into multiple pieces that are processed simultaneously Reordering matrix by nested dissection is one example of this approach Because of fill, such methods generally require more total work than best serial algorithm for system with dense band We will illustrate for tridiagonal linear systems, for which β = 1 , and will assume pivoting is not needed for stability (e.g., matrix is diagonally dominant or symmetric positive definite) Michael T. Heath Parallel Numerical Algorithms 4 / 24
Band Systems Tridiagonal Systems Cyclic Reduction Tridiagonal Linear System Tridiagonal linear system has form b 1 c 1 a 2 b 2 c 2 . . . . . . . . . a n - 1 b n - 1 c n - 1 a n b n x 1 x 2 . . . x n - 1 x n = y 1 y 2 . . . y n - 1 y n For tridiagonal system of order n , LU or Cholesky factorization incurs no fill, but yields serial thread of length Θ( n ) through task graph, and hence no parallelism Neither cdivs nor cmods can be done simultaneously Michael T. Heath Parallel Numerical Algorithms 5 / 24

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Band Systems Tridiagonal Systems Cyclic Reduction Tridiagonal System, Natural Order G ( A ) 10 9 8 7 6 5 4 3 2 1 11 12 13 14 15 A × × × × × × ×× × × × × × × × × × × × × × × × × × ×× × × × × × × × × × × × × × × × × L × × × × × × × × × × × × × × × × × × × × × × × × × × × × × T ( A ) 10 9 8 7 6 5 4 3 2 1 11 12 13 14 15 Michael T. Heath Parallel Numerical Algorithms 6 / 24
Band Systems Tridiagonal Systems Cyclic Reduction Two-Way Elimination Other orderings may enable some degree of parallelism, however For example, elimination from both ends (sometimes called twisted

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09_tridiagonal - Band Systems Tridiagonal Systems Cyclic...

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