Lecture 11 - LU Decomposition

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

2. Lecture’s Goals • LU Decomposition • 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 Interchange rows 1 & 4 Gaussian elimination of first column Save -mi1in the first column of [L]

25. Forward Elimination No interchange required Gaussian elimination of second column Save -mi2in the second column of [L]

26. Forward Elimination 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 • 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