1. 1 Systems of Linear Equations (Optional) Special Matrices

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

3. 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. 4 Tri-diagonal Matrix A special case of banded matrix. // Thomas Algorithm // Decomposition for k = 2 to n { e k = e k / f k-1 f k = f k – e k * g k-1 } // Forward substitution for k = 2 to n r k = r k – e k * r k-1 // Back subsitition x n = r n / f n for k = n-1 downto 1 x k = (r k – g k * x k+1 ) / f k

5. 5 Block Diagonal Matrix Another special case of banded matrix where each of B 1, B 2, …, B m are square matrices of various dimension. We can solve B i y i = c i, independently

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