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 BW2HBW1.
4
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.
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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
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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