1. Analysis of Triangular Factorization Lecture #8 EEE 574 Dr. Dan Tylavsky

2. Analysis Triangular Factorization © Copyright 1999 Daniel Tylavsky 4 What do you do if you run into this problem? Eigenvalues (A not pos. def.) 4 Let’s look at some limitation of Triangular Factorization.

3. Analysis Triangular Factorization © Copyright 1999 Daniel Tylavsky –Solution Pivoting: –Defn: Rows and columns of the unreduced portion of the matrix are interchange (by pre- and post-multiplication by a permutation matrix, P) so that a nonzero (typically largest) element is brought into the diagonal position. –Diagonal Pivoting - Pivots are chosen as any element (typically the largest) from the diagonal elements of the unreduced (active) matrix. –Partial Pivoting - Pivots are chosen as any element from a group of rows and columns of the unreduced (active) matrix. (Usually restricted to the first column of the active matrix.) –Complete Pivoting - Pivots are chosen as any element from the unreduced (active) matrix. If the matrix is positive definite, pivoting is not necessary for numerical stability.

4. Analysis Triangular Factorization © Copyright 1999 Daniel Tylavsky –Numerical Stability: EX: –For, –Matrix is sensitive to changes in b. ( It also is sensitive to small changes in A.)

5. Analysis Triangular Factorization © Copyright 1999 Daniel Tylavsky –To determine the sensitivity of a matrix we need some more definitions: Consistent Matrix Norms: Any measure of a matrix that satisfies: –One consistent matrix norm we’ll use is defined by:

6. Analysis Triangular Factorization © Copyright 1999 Daniel Tylavsky –It can be shown that the sensitivity of a solution to a change in b or A values is related to the condition number, k(A), of matrix. –The expensive part of this calculation occurs when we try to calculate A -1. –For our problem:

7. Analysis Triangular Factorization © Copyright 1999 Daniel Tylavsky –Another measure of k(A), is the ratio of the largest to smallest eigenvalues, another expensive calculation. –Large condition numbers indicate that the matrix is sensitive to small changes in matrix or b values.

8. Analysis Triangular Factorization © Copyright 1999 Daniel Tylavsky –Further, the result we calculate may be incorrect due to roundoff error. –For base 10 arithmetic, with t digits of precision we have the following heuristic: Triangular Factorization and forward/backward substitution produce a result with the following number of correct digits: (Assuming Partial Pivoting) –For 32 bit computation (9 + digits of precision) our previous example is correct to: decimal places. –How can we correct this problem?

9. Analysis Triangular Factorization © Copyright 1999 Daniel Tylavsky –Use Iterative Improvement. –Suppose we have a soln. with a few accurate digits, –The following algorithm can be used for iterative improvement: Residual, r, must be calc. in 2t precision. If 1t precision is used the small residual we will have will be largely error and z=A -1 *err=err.

10. Analysis Triangular Factorization © Copyright 1999 Daniel Tylavsky –Number of multiplications and additions for LU factorization: AddsDivsMults –Row 1 –Row 2 –Row 3 –Row n-1 –Row n * No division need be done for the last row; however, we will need the inverses of all diagonal elements for forward/backward substitution. So we’ll calculate it here.

11. Analysis Triangular Factorization © Copyright 1999 Daniel Tylavsky –Teams: Number of multiplications for factorizing LDU if A is not symmetric.

12. The End