Numerical Methods Contents - SAM

❶: elimination step, ❷: backsubstitution step
2.2 LU-Decomposition/LU-Factorization
⎛
⎛
Assumption: (sub-)matrices regular, if required.
Remark 2.1.5 (Gaussian elimination for non-square matrices).
“fat matrix”: A ∈ K n,m , m>n:
⎛
⎞
⎞
⎜
⎟
⎝
⎠ −→ ⎜
⎟
⎝ 0 ⎠ −→ ⎜
⎝
elimination step
back substitution
1
1 0
0
1
⎞
⎟
⎠
△
The gist of Gaussian elimination:
⎛
⎞ ⎛
⎞ ⎛
⎜
⎟
⎝
⎠ −→ ⎜
⎟
⎝ 0 ⎠ −→ ⎜
⎝
1
1 0
row transformations
row transformations
row transformation = adding a multiple of a matrix row to another row, or
multiplying a row with a non-zero scalar (number)
(ger.: Zeilenumformung)
Note: row transformations preserve regularity of a matrix (why ?)
0
1
⎞
⎟
⎠
Simultaneous solving of
LSE with multiple right hand sides
Given regular A ∈ K n,n , B ∈ K n,m ,
seek X ∈ K n,m
MATLAB:
AX = B ⇔
X = A\B;
X = A −1 B
asymptotic complexity: O(n 2 m)
Code 2.1.6: Gaussian elimination with multiple r.h.s.
1 function X = gausselim (A, B)
2 % Gauss elimination without pivoting, X = A\B
3 n = size (A, 1 ) ; m = n + size (B, 2 ) ; A = [ A,B ] ;
4 for i =1:n−1, p i v o t = A( i , i ) ;
5 for k= i +1:n , fac = A( k , i ) / p i v o t ;
6 A( k , i +1:m) = A( k , i +1:m) −
fac∗A( i , i +1:m) ;
7 end
8 end
9 A( n , n+1:m) = A( n , n+1:m) / A( n , n ) ;
10 for i =n−1:−1:1
11 for l = i +1:n
12 A( i , n+1:m) = A( i , n+1:m) −
A( l , n+1:m) ∗A( i , l ) ;
13 end
14 A( i , n+1:m) = A( i , n+1:m) /A( i , i ) ;
15 end
16 X = A ( : , n+1:m) ;
△
Ôº½ ¾º½
A matrix factorization (ger. Matrixzerlegung) writes a general matrix A as product of two special
(factor) matrices. Requirements for these special matrices define the matrix factorization.
Mathematical issue:
existence & uniqueness
Ôº¾ ¾º¾
Numerical issue: algorithm for computing factor matrices
Ôº¿
¾º¾
Example 2.2.1 (Gaussian elimination and LU-factorization).
LSE from Ex. 2.1.1: consider (forward) Gaussian elimination:
⎛
⎝ 1 1 0
⎞ ⎛
2 1 −1⎠
⎝ x ⎞ ⎛
1
x 2 ⎠ = ⎝ 4 ⎞
x 1 + x 2 = 4
1 ⎠ ←→ 2x 1 + x 2 − x 3 = 1 .
3 −1 −1 x 3 −3 3x 1 − x 2 − x 3 = −3
⎛
⎝ 1 ⎞ ⎛
1 ⎠ ⎝ 1 1 0
⎞ ⎛
2 1 −1 ⎠ ⎝ 4 ⎞ ⎛
1 ⎠ ➤ ⎝ 1 ⎞ ⎛
2 1 ⎠ ⎝ 1 1 0
⎞ ⎛
0 −1 −1 ⎠ ⎝ 4
⎞
−7 ⎠ ➤
1 3 −1 −1 −3 0 1 3 −1 −1 −3
⎛
⎞
⎝ 1 2 1 ⎠
3 0 1
⎛
⎝ 1 1 0
⎞
0 −1 −1 ⎠
0 −4 −1
⎛
⎝ 4
−7
−15
⎞
⎠ ➤
⎛
⎞
⎝ 1 2 1 ⎠
3 4 1
} {{ }
=L
= pivot row, pivot element bold, negative multipliers red
⎛ ⎞
⎝ 1 1 0
0 −1 −1 ⎠
}
0 0
{{
3
}
=U
Perspective: link Gaussian elimination to matrix factorization → Ex. 2.2.1
⎛
⎝ 4
−7
13
⎞
⎠
✸
Ôº ¾º¾