1. Lecture 11 - LU Decomposition
CVEN 302
June 26, 2002

2. Lecture’s Goals LU Decomposition Pivoting of matrices
Doolittle’s technique
Cholesky’s technique
Pivoting of matrices
Tridiagonal Method

3. LU Decomposition (Doolittle’s method)
Matrix decomposition

4. Doolitte’s method
The method alternates from solving from the upper triangular to the lower triangular

5. General formulation of Doolittle’s
The problem is reverse of the Crout’s reduction, starting with the upper triangular matrix and going to the lower triangular matrix.

6. LU Programs There are two programs
LU_factor - the program does a Doolittle decomposition of a matrix and returns the L and U matrices
LU_solver uses an L and U matrix combination to solve the system of equations.

7. Example
The matrix is broken into a lower and upper triangular matrices.

8. Cholesky’s method
The Cholesky decomposition is used on symmetric positive definite matrix:
where, lii = uii

9. Cholesky’s Method
The method does not alternate but does it from the diagonal out.

10. Cholesky’s Method
The second row in.

11. Cholesky’s Method
General Method

12. Cholesky’s Method
General Method

13. Example
The matrices can contain imaginary values.

14. LU Programs Test program
LU_cholesky_factor - the program does a Cholesky’s decomposition of a matrix and returns the L and U matrices
LU_Solve uses an L and U matrix combination to solve the system of equations.

15. Tridiagonal Matrix
For a banded matrix using Doolittle’s method, i.e. a tridiagonal matrix.

16. Tridiagonal LU Decomposition
The tridiagonal solver first step:
The second step:

17. Tridiagonal LU Decomposition
The tridiagonal solver for LU decomposition breaks down into form:

18. Tridiagonal Example

19. LU Programs for Tridiagonal Matrices
Test program
[dd,bb]=LU_tridiag(a,d,b) - the program does a decomposition of a tridiagonal matrix and returns the lower diagonal of L and the diagonal of U matrices
x = LU_tridiag_solve(a,dd,b,r) uses the lower diagonal of the L matrix and the diagonal and a vectors of the U matrix combination to solve the system of equations r is the right hand side.

20. Pivoting of the LU Decomposition
Still need pivoting in LU decomposition
Messes up order of [L]
What to do?
Need to pivot both [L] and a permutation matrix [P]
Initialize [P] as identity matrix and pivot when [A] is pivoted  Also pivot [L]

21. Pivoting of the LU Decomposition
Permutation matrix [ P ]
- permutation of identity matrix [ I ]
Permutation matrix performs “bookkeeping” associated with the row exchanges
Permuted matrix [ P ] [ A ]
LU factorization of the permuted matrix
[ P ] [ A ] = [ L ] [ U ]

22. Permutation Matrix Bookkeeping for row exchanges
Example: [ P1] interchanges row 1 and 3
Multiple permutations [ P ]

23. LU Decomposition with Pivoting
Start with
No need to consider {b} in decomposition

24. Forward Elimination Gaussian elimination of first column
Interchange rows 1 & 4
Gaussian elimination of first column
Save -mi1 in the first column of [L]

25. Forward Elimination Gaussian elimination of second column
No interchange required
Gaussian elimination of second column
Save -mi2 in the second column of [L]

26. Forward Elimination Partial pivoting for [L] and [P]
Interchange rows 3 &4
Partial pivoting for [L] and [P]

27. Forward Elimination Gaussian elimination of third column
Save -mi3 in third column of [L]

28. Doolittle LU with Pivoting
Gauss elimination with partial pivoting

29. Doolittle LU with Pivoting
Forward substitution
Back substitution

30. LU Pivoting Program Test program
[L U P]=LU_pivot(A) - the program does a decomposition of a matrix and returns the L and U matrices and P matrix which represents the pivoting problem.

31. Summary Tridiagonal matrices compression techniques
Setup of the LU decomposition techniques.
Doolittle’s method
Cholesky’s method is for symmetric positive definite matrices
Tridiagonal matrices compression techniques
Pivoting of matrices

32. Homework
Check the Homework webpage