Systems of Linear Equations

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