Systems of Linear Equations(Optional)Special Matrices

QuestionHow would you solve a system Ax = b if A isWould you use Gauss elimination? Forward or backward substitution? LU Decomposition?

Banded MatrixCertain 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.

Tri-diagonal Matrix// Thomas Algorithm

// Decompositionfor k = 2 to n { ek = ek / fk-1 fk = fk ek * gk-1}

// Forward substitutionfor k = 2 to n rk = rk ek * rk-1

// Back subsititionxn = rn / fnfor k = n-1 downto 1 xk = (rk gk * xk+1) / fkA special case of banded matrix.

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

Cholesky DecompositionIf 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