Question

In this exercise, you will work with a QR factorization of an mxn matrix. We will proceed in the way that is chosen by MATLAB, which is different from the textbook presentation. An mxn matrix A can be presented as a product of a unitary (or orthogonal) mxm matrix Q and an upper-triangular m × n matrix R, that is, A = Q * R . Theory: a square mxm matrix Q is called unitary (or orthogona) if -,or equivalently, * ee(m. It can be shown that the matrix AR is known that similar matrices have the same sets of eigenvalues. is similar to A.It For certain square matrices, such as, symmetrical, tridiagonal, Hessenberg matrices, the process of decomposing A into a product *R and interchanging the factors to create a matrix AR , then, repeating the process with the matrix A (instead of A) to create the matrix A,-R *Q, , and so on, produces a sequence of similar matrices A, A1, A2, that converges to a matrix, which is close to an upper triangular matrix, and its diagonal entries approximate the eigenvalues of A. Note: QR is a powerful algorithm for finding the eigenvalues of certain matrices. **Create a function in MATLAB: function [] quer (A) format, format compact [m, n]-size (A) Part I *Generate the first factorization by using the MATLAB in-built function Q,RJ-qr(A) Output Q and R (do not put a semicolon after the command [Q,R]=qr (A) ) . *Then, your function has to verify that you did get the factorization, that is, check if the condition below holds: closetozeroroundoff (A-Q*R)0 If yes, output the message: disp('the product of and R forms a QR decomposition of A) If not, output something like: disp ('No, it cannot be true! The code for closetozeroroundoff is here: function B closetozeroroundofE (A) [m, n]-size (A) for i=1:n for j-1:n if abs (A(i,j))

Answer 1

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