The Cholesky factorization
one
3. Consider the linear system Ax = b, where 6.25 -1 0.5 2.12 3.6 and [ 7.51 b= -8.68 [ -0.24 Write a MATLAB program for LU-factorization with a unit lower triangular L (meaning that the diagonal entries should be equal to one). Then write a program for the Cholesky factorization. WARNING: avoid using MATLAB shortcuts. The programming should be done "from scratch"
3. (a) Compute the Cholesky decomposition of A given by 16 16 -4 -24 16 32-12 -24 -4 -12 301 -24-24 46 by hand (b) Solve the linear system Ax- b, where 16 -63 20 using your Cholesky factorization from Part (a)
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Cholesky factorization versus QR factorization. In this problem we compare the accuracy of the two methods for solving a least-squares problem minimize Ar - b We take b10- 1-10-k ke A10 0 10k for k 6, k- 7 and k 8. (a) Write the normal equations, and solve them analytically (i.e., on paper, without using MATLAB). (b) Solve the least-squares problem in MATLAB, for k = 6. K-7 and k = 8, using the recommended...
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Find the Cholesky factorization, A = CCT for the following matrix. Be sure to clearly identify the matrix C. A = 14 2 21 2 2 -1 2 -16
3 (The UL factorization.) Show how to compute the factorization A = UL where U is upper triangular with ls along the diagonal and L is lower triangular. Show how this relates to a way of solving Ax = b by transforming the system into an equivalent system with a lower triangular matrix. (In other words, show that what we did for the LU factorization also works for a UL factorization.) Note: For the purposes of this exercise you may...
3 What is the QR factorization of the matrix 2 8 13 A-4 77? 4 -2 -13 You can use MAILAB to check your answer, but you must provide the details of all intermediate steps on paper.
1. (5 pts) Use partial pivoting to compute (by hand) the PA -LU factorization of the matrix A-12-1 3
1. (5 pts) Use partial pivoting to compute (by hand) the PA -LU factorization of the matrix A-12-1 3
(4) Q4a) Find the QR factorization of the matrix 13 3 -1 1 7 -4 2 1 -1 b) Test using the spectral method or suitable matrix norms, the guaranteed convergence of Gauss Jacobi method for the following system (2) 1x + 4z = 8 4y + 2z = 9 4.0 + 2y - 2z = 10
(4 points) 3 3 -2 Find the LU factorization of the 3 x 3 matrix A3 5 6 -12 20 22 and use it to solve the system 13 #2 | = 1-23 -88 T1 3 5 6 12 20 22 C3
2 3 3. Let A = 2 4 and b = 3 . Find QR factorization of A. (1 1