1. LU Decomposition

2. Solve A . x = b (system of linear equations)
Decompose A = L . U *
L : Lower Triangular Matrix U : Upper Triangular Matrix

3. [L][U]=[A]  [L][U]{x}={b}
To solve [A]{x}={b}
[L][U]=[A]  [L][U]{x}={b}
Consider [U]{x}={d}
[L]{d}={b}
Solve [L]{d}={b} using forward substitution to get {d}
Use back substitution to solve [U]{x}={d} to get {x}

4. [ L ]
[ U ]

5. [ U ] Gauss Elimination  
Coefficients used during the elimination step

6. [ L . U ]

7. [ L ] [ U ] Example: A = L . U Gauss Elimination Coefficients

8. Using the Forward Elimination Procedure of Gauss Elimination
Finding the [U] matrix
Using the Forward Elimination Procedure of Gauss Elimination
Step 1:

9. Finding the [U] Matrix Matrix after Step 1: Step 2:

10. Using the multipliers used during the Forward Elimination Procedure
Finding the [L] matrix
Using the multipliers used during the Forward Elimination Procedure
From the first step of forward elimination

11. Finding the [L] Matrix From the second step of forward elimination

12. Does [L][U] = [A]? ?

13. Example Set [L][Z] = [C] Solve for [Z]

14. Example Complete the forward substitution to solve for [Z]

15. Example Math 685/CSI 700 Spring 08
George Mason University, Department of Mathematical Sciences

16. LU decomposition
Gauss Elimination can be used to decompose [A] into [L] and [U]. Therefore, it requires the same total FLOPs as for Gauss elimination:
In the order of (proportional to) N3 where N is the # of unknowns.
lij values (the factors generated during the elimination step) can be stored in the lower part of the matrix to save storage. This can be done because these are converted to zeros anyway and unnecessary for the future operations.
Saves computing time by separating time-consuming elimination step from the manipulations of the right hand side.
Provides efficient means to compute the matrix inverse