Crout Factorization 45. Crout LU factorization requires the diagonal entries of U be 1's, while L is a general lower triangular matrix. Perform direct calculation of the entries of L and U for the case of a 3x3 matrix, similar to that in Example 4.6. Based on the find- ings, write a user-defined function with function call [L, U] Crout Factor (A) that returns the desired lower and upper triangular matrices for any nxn matrix. Apply Crout Factor to 4 1 7 A = 2 3 -6 -5 -1 -5 Thomasethod (A, B) 1.0000 -3.0000 1.0000 4.1.4.2 MATLAB Built-In Function"\" The built-in function in MATLAB for solving a linear system Ax-b is the backslash (\), and the solution vector is obtained via x Ab It is important to note that x Ab computes the solution vector by Gauss elimination and not by x- A b For the linear system in Example 4.4, this yields 1.0000 -3.0000 1.0000 Numerical Solution of Systems of Equations 107 4.4 LU Factorization Methods In the last section, we learned that solving a large nxn system Ax-busing Gauss elimina tion requires approximately operations. There are other direct methods that require fewer operations than Gauss elimination. These methods make use of the LU factorization of the coefficient matrix A LU factorization for decomposition) of a matrix A... means expressing the matrix as A-LU, where L is a lower triangular matrix and U., is upper triangular. Subsequently, the original system Ax-bis rewritten as [LU]xbLUxb Lux-b Letting Ux-y the above can be solved in two steps ty-b Ux-y Note that each of the two systems is triangular, hence easy to solve. Because Ly-b is a lower triangular system, it can be solved using forward substitution. The system Ux-y is upper triangular and is solved via back substitution There are different ways to accomplish the factorization of matrix A, depending on the specific restrictions imposed on L or U. For example, Crout factorization (see Problem Set) requires the diagonal entries of U be l's, while L is a general lower triangular matrix. Another technique, known as Doolittle factorization, uses the results from different steps of Gauss elimination. These two approaches have similar performances, but we will present Doolittle factorization here." 4.4.1 Doolittle Factorization Doolittle factorization of A is A-LU, where L is lower triangular consisting of I's along the diagonal, and U is upper triangular. That the diagonal elements of Lare chosen as 1's can be explained using a generic 3x3 matrix A [A] If Land U are selected in their most general forms, then A-LU p d 333 0 0 0 0 Ma 0 0 ཎྞ ༤ རྦ 132 of 494ements in the first column of are found 110 The entries in the second row of are calculated via --- The element in the second column of is found a Finally, the entry in the third row of U is given by --- Using the entries of matrix A and solving the nine equations just listed, we find 100 L-2 1 0 U- 2 13 -11 • This clearly agrees with the outcome of Example 4.5 The direct calculation of the entries of L and U in Doolittle factorization can be performed systematically for an ex matrix A using the steps outlined in Example 4.6 The user-defined function Doolittlefactor performs all the Numerical Methods for Engineers and Scientists Using MATLAB, Second Edition operations in the order suggested in Example 46 and returns the appropriate Land U function [D Doolittlefactor returns the Doolittle factorization of matris A. U is an upper triangular tri for 1 for The findings of the last example can readily be confirmed by executing this function. 1 0 D 。 •