LU Factorization

Shahid Hussain

Department of Computer Science & EngineeringBahria University, Karachi Campus

March 3, 2009

Outline

1 LU Factorization

Elimination = Factroization

Let A =[

2 16 8

]. Subtract 3 times row 1 from 2. That step is

E21 in the forward direction i.e.,

Forward: E21A =[

1 0−3 1

] [2 16 8

]=[

2 10 5

]= U

Backward: E−121 A =

[1 03 1

] [2 10 5

]=[

2 16 8

]= A

LU Decomposition

A = LU

Let A be a square matrix. An LU decomposition is adecomposition of the form

A = LU

where L and U are lower and upper triangular matrices (of thesame size), respectively. This means that L has only zeros abovethe diagonal and U has only zeros below the diagonal. For a 3× 3matrix, this becomes:a11 a12 a13

a21 a22 a23

a31 a32 a33

=

l11 0 0l21 l22 0l31 l32 l33

u11 u12 u13

0 u22 u23

0 0 u33

.

Example

8 2 94 9 46 7 79

=

l11 0 0l21 l22 0l31 l32 l33

u11 u12 u13

0 u22 u23

0 0 u33

.

Example cont.

Firs row of U , first column of L:8 2 94 9 46 7 9

=

1 0 01/2 1 03/4 l32 1

8 2 90 u22 u23

0 0 u33

.

Second row of U , second column of L:[9 47 9

]−[1/23/4

] [2 9

]=

[1 0l32 1

] [u22 u23

0 u33

][

8 −1/211/2 9/4

]=

[1 0

11/16 1

] [8 −1/20 u33

]Third row of U : u33 = 9/4 + 11/32 = 83/32.

Result

8 2 94 9 46 7 9

=

1 0 01/2 1 03/4 11/16 1

8 2 90 8 −1/20 0 83/32

.

Do it yourself

Factorize the following matrix A into its LU factors.3 2 12 1 46 2 5

The result. (How?)

3 2 12 1 46 2 5

=

1 0 02/3 1 02 6 1

3 2 10 −1/3 10/30 0 −17

.

A = LDU Factorization

A = LDU

The triangular factorization can be written as

A = LU or A = LDU

Whenever you see LDU it is understood that U has 1’s on thediagonal. For example:[

1 03 1

] [2 80 5

]=[1 03 1

] [2

5

] [1 40 1

].