Math 5610 - Computational Linear Algebra

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The purpose of this page is to document three different preconditioning strategies for solving linear systems of equations. A preconditioner of a matrix A is a matrix P such that (P^-1)A has a smaller condition number than A alone. Multiplying the original system of equations by P^-1, (P^-1)Ax=(P^-1)b, means that the system is more suitable for iterative methods. Three preconditioners are listed below.

  1. Jacobi Preconditioner
    • The Jacobi Preconditioner is the simplest preconditioner. It is chosen to be the diagonal of A, P = diag(A). This preconditioner has very small storage requirements and is easy to implement. However, it does not yield particularly large improvements when compared to more sophisticated preconditioners. Source
  2. SSOR Preconditioner
    • The SSOR preconditioner is similar to the Jacobi Preconditioner in that it can be derived from the coefficient matrix without any work. If a matrix is decomposed into A = D + L + U where D, L, and U are the diagonal, lower, and upper triangular parts respectively, then the preconditioner is P = (D + L)(D^-1)(D + L)^T. Source
  3. Incomplete Cholesky Preconditioner
    • This preconditioner is created by performing Cholesky decomposition on the coefficient matrix while ignoring the fill elements. The corresponding preconditioner is then P = (G^T)G. Source