ECE 530 – Analysis Techniques for Large-Scale

Transcript:ECE 530 – Analysis Techniques for Large-Scale:
ECE 530 – Analysis Techniques for Large-Scale Electrical Systems Prof. Hao Zhu Dept. of Electrical and Computer Engineering University of Illinois at Urbana-Champaign haozhu@illinois.edu 9/14/2015 1 Lecture 6: LU Decomposition; Sparse Matrices Announcements Homework 2 posted, due Sep 23(Wed) 2 A triangularization procedure to solving Ax = b Step k = 1,…,n a) eliminate rows 1,…, k-1 from row k: b) normalize row k: Gaussian Elimination 3 Upper Triangular Matrix The triangular decomposition scheme results in an upper triangular matrix U 4 LU Decomposition Theorem 5 One Corollary 6 LU Decomposition Application 7 LU Composite Matrix We can store the L and U triangular factors into a composite n by n matrix denoted by [L\U] Hence 8 LU Composite Matrix, cont’d The diagonal terms of U are not stored since they are known to be 1’s by the steps of the Gaussian elimination Using the composite matrix [L\U], there is no need to include the vector b during the process of Gaussian elimination since it has all the required information to compute A-1 b for any b 9 LDU Decomposition In the previous case we required that the diagonals of U be unity, while there was no such restriction on the diagonals of L An alternative decomposition is where D is the diagonal part of L, and the lower triangular matrix is modified to require unity for the diagonals 10 Symmetric Matrix Factorization The LDU formulation is quite useful for the case of a symmetric matrix Hence only the upper triangular elements and the diagonal elements need to be stored, reducing storage by almost a factor of 2 11 Pivoting An immediate problem that can occur with Gaussian elimination is the issue of zeros on the diagonal; for example This problem can be solved by a process known as “pivoting,” which involves the interchange of either both rows and columns (full pivoting) or just the rows (partial pivoting) Partial pivoting is much easier to implement, and actually can be shown to work quite well 12 Pivoting, cont. In the previous example the (partial) pivot would just be to interchange the two rows For LU decomposition, we need to keep track of the interchanged rows! Partial pivoting can be helpful in improving numerical stability even when the diagonals are not zero When factoring row k interchange rows so the new diagonal is the largest element in column k for rows j