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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

- 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 BW2HBW1.

Tri-diagonal 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 xk1) / fk

A special case of banded matrix.

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

Cholesky Decomposition

- If A is symmetry (A AT), and A is positive

definite (i.e., xTAx gt 0 for any x ? 0), then we

can decompose A in to LLT as

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