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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)
- 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

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)

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

Basic iterative methods

- Rewrite

Jacobi iterative method

- The linear system
- Equation form
- Matrix form

Jacobi iterative method

- An example
- The method
- Initial guess

Jacobi iterative method

- The results

Gauss-Seidel method

- Idea: Used the new values when they are available
- Equation form
- Matrix form

Gauss-Seidel method

- An example
- The method
- Initial guess

Gauss-Seidel method

- The results

SOR method

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

Convergence analysis

- General form of basic iterative methods
- Exact solution
- Define the error at the m-th iteration
- Error equations

Convergence analysis

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

Convergence analysis

- Definition: Spectral radius of R
- Lemma: For any square matrix R,
- Theorem: The iterative method converges to the exact solution of iff

Convergence rate

- Thm: For the iterative method suppose then
- The iterative method converges
- Linear convergence rate with q<1
- Error bound

Proof for convergence rate

- Fact
- Error bound
- Another error bound
- Error bound

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

Convergence results

- Definition: A is strictly row diagonally dominant if
- Examples
- Thm: If A is strictly row diagonally dominant, it is invertible!

Convergence results

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

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.

Convergence results

- Thm: For SOR method, we have
Thus the relaxation parameter be in (0,2) is necessary for SOR converge

- Proof:

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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