Page 218 - MATLAB an introduction with applications

Numerical Methods ——— 203

The process of computing L and U for a given A is known as LU decomposition or LU factorization.
The given equations can be rewritten as LUx = b and using the notation Ux = y, then
Ly = b
which can be solved for y by forward substitution.
Hence, Ux = y which gives x by the back substitution process.

4.5 CHOLESKI’S DECOMPOSITION
T
Choleski’s decomposition A = LL requires that A to be symmetric. The decomposition process involves
taking square roots of certain combinations of the elements of A.
A typical element in the lower triangular portion of LL is of the form,
T
j
T
( ) = L L + L L + + ij jj ∑ L L jk i ≥ j
... L L =
LL
1 i
2 j
2 i
ik
1 j
ij
k= 1
Equating this term to the corresponding element of A gives
j
ij ∑
A = L L jk i = j, j + 1, …, n j = 1, 2, …, n ...(4.5)
ik
k= 1
Taking the term containing L outside the summation in Eq.(4.5), we obtain
ij
j− 1
ij ∑
A = L L + L L jj
ik
jk
ij
k= 1
If i = j, then the solution is
j− 1
ij ∑
L = A − L 2 jk j = 2, 3, …, n
ij
k= 1
or a non-diagonal term, we get
 j− 1 
ij ∑
L =  A − L L jk  L jj j = 2, 3, …, n–1 i = j + 1, j + 2, …, n
ik
ij
 k= 1 
4.6 GAUSS-SEIDEL METHOD
The equations Ax = b can be written in scalar form as
n
∑ Ax = b i i = 1, 2, …, n
j
ij
j= 1
Extracting the term containing x from the summation sign gives
i
n
ii i ∑
Ax + A x = b i i = 1, 2, …, n
j
ij
j= 1
ji ≠
Solving for x , we obtain
i