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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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Presentation Transcript

• The problem: solve

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

• How to efficiently compute the solution numerically???

• 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

• 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)

• 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

• GMRES for nonsymmetric mehtod

• Rewrite

• The linear system

• Equation form

• Matrix form

• An example

• The method

• Initial guess

• The results

• Idea: Used the new values when they are available

• Equation form

• Matrix form

• An example

• The method

• Initial guess

• The results

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

• Equation form

• Matrix form

• General form of basic iterative methods

• Exact solution

• Define the error at the m-th iteration

• Error equations

• Convergence

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

• Definition: Spectral radius of R

• Lemma: For any square matrix R,

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

• Thm: For the iterative method suppose then

• The iterative method converges

• Linear convergence rate with q<1

• Error bound

• Fact

• Error bound

• Another error bound

• Error bound

• 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

• Definition: A is strictly row diagonally dominant if

• Examples

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

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

• Proof

• 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.

• Thm: For SOR method, we have

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

• Proof:

• 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: