CSC 445/545 Notes 4.1: Computing LU-factorizations

I used Matlab to do the computations required for making these notes.

We want to LU-factorize a matrix A0.

A0 =


     2    -1     2


     2     3     1


    -1     1    -2

1. The first step is to multiply the first row by 1/2. [ M₁(1/2)].
This is equivalent to premultiplying by B1.

B1 =


         0.5       0         0


         0         1         0


         0         0         1


A1 = B1 * A0 =


    1   -0.5  1


    2    3    1


   -1    1   -2

2. Next, -2 times row 1 is added to row 2 [A_1,2(-2)]
and 1 times row 1 is added to row 3. [A_1,3(1)].
This is equivalent to premultiplying by B2.

B2 =

     1     0     0
    -2     1     0
     1     0     1

A2 = B3 * A1 =

    1.0000   -0.5000    1.0000
         0    4.0000   -1.0000
         0    0.5000   -1.0000

3. Now multiply row 2 by 1/4 [M₂(1/4)].
This is equivalent to premultiplying by B2.

B3 =

    1.0000         0         0
         0    0.2500         0
         0         0    1.0000

A3 = B3 * A2 =

    1.0000   -0.5000    1.0000
         0    1.0000   -0.2500
         0    0.5000   -1.0000

4. Next, -1/2 times row 2 is added to row 3 [A_2,3(-1/2)].
This is equivalent to premultiplying by B4.

B4 =

    1.0000         0         0
         0    1.0000         0
         0   -0.5000    1.0000

A4 = B4 * A3 =

    1.0000   -0.5000    1.0000
         0    1.0000   -0.2500
         0         0   -0.8750

5. Finally, premultiply row 3 by -8/7 [M₃(-1.1429)].
This is equivalent to premultiplying by B5.

B5 =

    1.0000         0         0
         0    1.0000         0
         0         0   -1.1429

A5 = B5 * A4

    1.0000   -0.5000    1.0000
         0    1.0000   -0.2500
         0         0    1.0000

L1inv = B2 * B1 0.5000 0 0 -1.0000 1.0000 0 0.5000 0 1.0000

 
L2inv = B4 * B3

    1.0000         0         0
         0    0.2500         0
         0   -0.1250    1.0000

 
L3inv = B5

    1.0000         0         0
         0    1.0000         0
         0         0   -1.1429

 
L1 =  inv(L1inv)

     2     0     0
     2     1     0
    -1     0     1

 
L2 =  inv(L2inv)

    1.0000         0         0
         0    4.0000         0
         0    0.5000    1.0000

 
L3 =  inv(L3inv)

    1.0000         0         0
         0    1.0000         0
         0         0   -0.8750

 
L = L1 * L2 * L3

    2.0000         0         0
    2.0000    4.0000         0
   -1.0000    0.5000   -0.8750

 
U =  A5

    1.0000   -0.5000    1.0000
         0    1.0000   -0.2500
         0         0    1.0000

 
TESTA = L * U

     2    -1     2
     2     3     1
    -1     1    -2