CmpE 360 Homework 6

 

 

  1. Suppose that you are computing the QR factorization of the matrix:

 

 

by Householder transformations.

 

    1. How many Householder transformations are required?
    2. What does the first column of A become as a result of applying the first Householder transformation?
    3. What does the first column of A become as a result of applying the second Householder transformation?
    4. How many Givens rotations would be required to compute the QR factorization of A?

 

 

  1. Given the data:

 

xi                     4.0           4.2           4.5           4.7           5.1           5.5           5.9           6.3           6.8           7.1

yi                     102.56     113.18     130.11     142.05     167.53     195.14     224.87     256.73     299.50     326.72

 

a.       Construct the least squares polynomial of degree 1, and compute the error

b.      Construct the least squares polynomial of degree 2, and compute the error

c.       Construct the least squares polynomial of degree 3, and compute the error

 

  1. Hotelling Deflation

Assume that the largest eigenvalue λ1 in magnitude and an associated eigenvector v(1) have been obtained for the n x n symmetric matrix A. Show that the matrix:

                       

                                   

 

has the same eigenvalues λ2, … , λn as A, except that B has eigenvalue 0 with eigenvector v(1) instead of eigenvector λ1. Use this deflation method to find the first two eigenvalues of the matrix:

                       

                       

 

Theoretically, this method can be continued to find more eigenvalues, but roundoff error soon makes the effort worthless.