The linear system
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The linear system. The problem: solve Suppose A is invertible, then there exists a unique solution How to efficiently compute the solution numerically???. Review of direct methods. Gaussian elimination with pivoting Memory cost: O(n^2) Computational cost: O(n^3)

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The linear system l.jpg
The linear system

  • The problem: solve

    • Suppose A is invertible, then there exists a unique solution

    • How to efficiently compute the solution numerically???


Review of direct methods l.jpg
Review of direct methods

  • Gaussian elimination with pivoting

    • Memory cost: O(n^2)

    • Computational cost: O(n^3)

    • Can only be used for small n, e.g. n<=1000

  • LU decomposition

    • Memory cost: O(n^2)

    • Computational cost: O(n^2)

    • Can only be used for small n, e.g. n<=1000

    • Good for problem to solve the linear system with different right hand


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Review of direct methods

  • For tri-diagonal matrix

    • Thomas algorithm based on Crout factorization

    • Memory cost: O(n) & Computational cost: O(n)

    • Can be extended to band-limited matrix

  • For linear system from discretization of Poisson equation by FDM

    • Direct Poisson solver based on FFT

    • Memory cost: O(n) & Computational cost: O(n ln n)

  • For linear system from discretization of elliptic equation by FEM

    • Multigrid method (MG) or Algebraic Multigrid method (AMG)

    • Memory cost: O(n) & Computational cost: O(n)

  • For linear system from discretization of Poisson equation by integral formulation

    • Fast Multipole method

    • Memory cost: O(n) & Computational cost: O(n)


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Iterative methods

  • Aim: to solve large sparse linear system

  • Basic iterative methods

    • Jacobi method

    • Gauss-Seidel method

    • Successive overrelaxation method (SOR)

  • Krylov subspace (modern iterative) methods

    • Steepest decent method

    • Conjugate gradient (CG) method

    • GMRES for nonsymmetric mehtod



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Jacobi iterative method

  • The linear system

  • Equation form

  • Matrix form


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Jacobi iterative method

  • An example

  • The method

  • Initial guess



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Gauss-Seidel method

  • Idea: Used the new values when they are available

  • Equation form

  • Matrix form


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Gauss-Seidel method

  • An example

  • The method

  • Initial guess


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Gauss-Seidel method

  • The results


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SOR method

  • Idea: To improve the Gauss-Seidel method by a linear combinationof the old value and new

  • Equation form

  • Matrix form


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Convergence analysis

  • General form of basic iterative methods

  • Exact solution

  • Define the error at the m-th iteration

  • Error equations


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Convergence analysis

  • Convergence

  • Lemma: For any square matrix R, there exists a nonsingular matrix T such that – Jordan canonical form


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Convergence analysis

  • Definition: Spectral radius of R

  • Lemma: For any square matrix R,

  • Theorem: The iterative method converges to the exact solution of iff


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Convergence rate

  • Thm: For the iterative method suppose then

    • The iterative method converges

    • Linear convergence rate with q<1

    • Error bound


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Proof for convergence rate

  • Fact

  • Error bound

  • Another error bound

  • Error bound


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Convergence results

  • If A is strictly row diagonally dominant, then both Jacobi and Gauss-Seidel methods converge.

  • Gauss-Seidel method converges if A is symmetric positive definite

  • The relaxation parameter be in (0,2) is the necessary condition for the convergence of SOR method. In addition, if A is symmetric positive definite, then the condition is also sufficient for the convergence of SOR method


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Convergence results

  • Definition: A is strictly row diagonally dominant if

  • Examples

  • Thm: If A is strictly row diagonally dominant, it is invertible!


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Convergence results

  • Thm: If A is strictly row diagonally dominant, then both Jacobi and Gauss-Seidel methods converge. In fact,

  • Proof


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Convergence results

  • Thm: Let A be symmetric positive definite matrix, then the Gauss-Seidel method converges for any initial guess.

  • Proof: See details in class

  • Remark: There are linear system, for which the Jacobi method converges, but the Gauss-Seidel method diverges, e.g.


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Convergence results

  • Thm: For SOR method, we have

    Thus the relaxation parameter be in (0,2) is necessary for SOR converge

  • Proof:


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Convergence results

  • Thm: If A is symmetric positive definite, then for . That is, SOR converges for all

  • Proof: See details in class

  • Remark:

    • Over relaxation:

    • Under relaxation:

    • Optimal relaxation parameter:


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