1.Chapter Objectives

Transcription

1 LU Factorization

2 INDEX 1.Chapter objectives 2.Overview of LU factorization 2.1GAUSS ELIMINATION AS LU FACTORIZATION 2.2LU Factorization with Pivoting 2.3 MATLAB Function: lu 3. CHOLESKY FACTORIZATION 3.1 CHOLESKY FACTORIZATION 3.2 MATLAB Function: chol 4. MATLAB LEFT DIVISION

3 1.Chapter Objectives Understanding that LU factorization involves decomposing the coefficient matrix into two triangular matrices that can then be used to efficiently evaluate different right-handside vectors. Knowing how to express Gauss elimination as an LU factorization Given an LU factorization, knowing how to evaluate multiple right-hand-side vectors. Recognizing that Cholesky s method provides an efficient way to decompose a symmetric matrix and that the resulting triangular matrix and its transpose can be used to evaluate right-hand-side vectors efficiently. Understanding in general terms what happens when MATLAB.s backslash operator is used to solve linear systems.

4 2. OVERVIEW OF LU FACTORIZATION

5 OVERVIEW OF LU FACTORIZATION

6 OVERVIEW OF LU FACTORIZATION

7 2.1 GAUSS ELIMINATION AS LU FACTORIZATION The first step in Gauss elimination is to multiply row 1 by the factor and subtract the result from the second row to eliminate a 21. Similarly, row 1 is multiplied by and the result subtracted from the third row to eliminate a 31. The final step is to multiply the modified second row by and subtract the result from the third row to eliminate a 32.

8 GAUSS ELIMINATION AS LU FACTORIZATION The result of forward elimination is the [U] matrix. And the factors are used for elements of the [L] matrix.

9 GAUSS ELIMINATION AS LU FACTORIZATION Example 10.1 GAUSS ELIMINATION AS LU FACTORIZATION Derive an LU factorization based on the Gauss elimination performed previously in Example 9.3. After forward elimination, the following upper triangular matrix was obtained: The factors employed to obtain the upper triangular matrix can be assembled into a lower triangular matrix. The elements a 21 and a 31, a 32 were eliminated by using the factors

10 GAUSS ELIMINATION AS LU FACTORIZATION Thus, the lower triangular matrix is Consequently, the LU factorization is This result can be verified by performing the multiplication of [L][U] to give where the minor discrepancies are due to roundoff.

11 GAUSS ELIMINATION AS LU FACTORIZATION Example 10.2 The Substitution Steps Complete the problem initiated in Example 10.1 by generating the final solution with forward and back substitution. and that the forward-elimination phase of Gauss elimination resulted in The forward-substitution phase is implemented by applying Eq. (10.8):

12 GAUSS ELIMINATION AS LU FACTORIZATION or multiplying out the left-hand side: We can solve the first equation for d 1 = 7.85, which can be substituted into the second equation to solve for Both d1 and d2 can be substituted into the third equation to give Thus,

13 GAUSS ELIMINATION AS LU FACTORIZATION This result can then be substituted into Eq. (10.3), [U]{x} = {d}: which can be solved by back substitution for the final solution:

14 2.2LU Factorization with Pivoting AS for Gauss elimination, partial pivoting is necessary to obtain reliable solutions with LU factorization. One way to do this involves using a permutation matrix. The approach consists of the following steps: 1. Elimination : The LU factorization with pivoting of a matrix [A] can be represented in matrix form as 2. Forward substitution : The matrices [L] and [P] are used to perform the elimination step with pivoting on {b} in order to generate the intermediate right-hand-side vector, {d}. 3. Back substitution : The final solution is generated in the same fashion as done previously for Gauss elimination.

15 LU Factorization with Pivoting Example 10.3 LU Factorization with Pivoting Compute the LU factorization and find the solution for the same system analyzed in Example 9.4 Before elimination, we set up the initial permutation matrix: prior to elimination, we switch the rows:

16 LU Factorization with Pivoting At the same time, we keep track of the pivot by switching the rows of the permutation matrix: eliminate a 21 by subtracting l 21 = a 21 /a 11 =0.0003/1 = from the second row of A. In so doing, we compute that the new value of a 22 = (1) =

17 LU Factorization with Pivoting Before implementing forward substitution, the permutation matrix is used to reorder the right-hand-side vector to reflect the pivots as in Forward substitution Backward substitution

18 2.3 MATLAB Function: lu [L, U] = lu(x) Example 10.4 LU Factorization with MATLAB Use MATLAB to compute the LU factorization and find the solution for the same linear system analyzed.

19 MATLAB Function: lu To generate the solution, we compute These results conform to those obtained by hand in Example 10.2.

20 3. CHOLESKY FACTORIZATION a symmetric matrix is one where a ij = a ji for all i and j. In other words, [A] = [A] T. Special solution techniques are available for such systems. One of the most popular approaches involves Cholesky factorization. This algorithm is based on the fact that a symmetric matrix can be decomposed, as in The factorization can be generated efficiently by recurrence relations. For the ith row:

21 3. CHOLESKY FACTORIZATION After obtaining the factorization, it can be used to determine a solution for a righthand-side vector {b} in a manner similar to LU factorization. First, an intermediate vector {d} is created by solving Then, the final solution can be obtained by solving

22 CHOLESKY FACTORIZATION Example 10.5 Cholesky factorization Compute the Cholesky factorization for the symmetric matrix. For the first row (i = 1), Eq. (10.15) is employed to compute Then, Eq. (10.16) can be used to determine

23 CHOLESKY FACTORIZATION For the second row (i = 2): For the third row (i = 3): Thus, the Cholesky factorization yields

24 MATLAB Function: chol U = chol(x) Example 10.6 Cholesky Factorization with MATLAB Use MATLAB to compute the Cholesky factorization for the same matrix we analyzed in Example Also obtain a solution for a right-hand-side vector that is the sum of the rows of [A]. Note that for this case, the answer will be a vector of ones.

25 MATLAB Function: chol

26 10.4 MATLAB LEFT DIVISION Left division with the backslash operator Examining the structure of the coefficient matrix Sparse and banded Triangular Symmetric Square matrix Banded solvers Back and forward substitution Cholesky factorization General triangular factorization