1 / 6

Systems of Linear Equations

Systems of Linear Equations. (Optional) Special Matrices. Question. How would you solve a system Ax = b if A is. Would you use Gauss elimination? Forward or backward substitution? LU Decomposition?. Banded Matrix.

floyd
Download Presentation

Systems of Linear Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Systems of Linear Equations (Optional) Special Matrices

  2. Question How would you solve a system Ax = b if A is Would you use Gauss elimination? Forward or backward substitution? LU Decomposition?

  3. Banded Matrix Certain matrices have particular structures that can be exploited to develop efficient solution schemes. A banded matrix is a square matrix that has all elements equal to zero, with the exception of a band centered on the main diagonal. The dimensions of a banded system can be quantified by two parameters: the band width BW and half-bandwidth HBW. These two values are related by BW=2HBW+1.

  4. Tri-diagonal Matrix A special case of banded matrix. // Thomas Algorithm // Decomposition for k = 2 to n { ek = ek / fk-1 fk = fk – ek * gk-1 } // Forward substitution for k = 2 to n rk = rk – ek * rk-1 // Back subsitition xn = rn / fn for k = n-1 downto 1 xk = (rk – gk * xk+1) / fk

  5. Block Diagonal Matrix Another special case of banded matrix where each of B1, B2, …, Bm are square matrices of various dimension. We can solve Biyi = ci, independently

  6. Cholesky Decomposition If A is symmetry (A = AT), and A is positive definite (i.e., xTAx > 0 for any x≠0), then we can decompose A in to LLT as

More Related