×
Log in to StudySoup
Get Full Access to Linear Algebra With Applications - 9 Edition - Chapter 3 - Problem 4
Join StudySoup for FREE
Get Full Access to Linear Algebra With Applications - 9 Edition - Chapter 3 - Problem 4

Already have an account? Login here
×
Reset your password

(Rank-1 Updates of Linear Systems) (a) Set A = round(10 rand(8)) b = round(10 rand(8

Linear Algebra with Applications | 9th Edition | ISBN: 9780321962218 | Authors: Steven J. Leon ISBN: 9780321962218 437

Solution for problem 4 Chapter 3

Linear Algebra with Applications | 9th Edition

  • Textbook Solutions
  • 2901 Step-by-step solutions solved by professors and subject experts
  • Get 24/7 help from StudySoup virtual teaching assistants
Linear Algebra with Applications | 9th Edition | ISBN: 9780321962218 | Authors: Steven J. Leon

Linear Algebra with Applications | 9th Edition

4 5 1 312 Reviews
19
2
Problem 4

(Rank-1 Updates of Linear Systems) (a) Set A = round(10 rand(8)) b = round(10 rand(8, 1)) M = inv(A) Use the matrix M to solve the system Ay = b for y. (b) Consider now a new system Cx = b, where C is constructed as follows: u = round(10 rand(8, 1)) v = round(10 rand(8, 1)) E = u v C = A + E The matrices C and A differ by the rank-1 matrix E. Use MATLAB to verify that the rank of E is 1. Use MATLABs \ operation to solve the system Cx = b and then compute the residual vector r = b Ax. (c) Let us now solve Cx = b by a new method that takes advantage of the fact that A and C differ by a rank-1 matrix. This new procedure is called a rank-1 update method. Set z = M u, c = v y, d = v z, e = c/(1 + d) and then compute the solution x by x = y e z Compute the residual vector b Cx and compare it with the residual vector in part (b). This new method may seem more complicated, but it actually is much more computationally efficient. (d) To see why the rank-1 update method works, use MATLAB to compute and compare Cy and b + cu Prove that if all computations had been carried out in exact arithmetic, these two vectors would be equal. Also, compute Cz and (1 + d)u Prove that if all computations had been carried out in exact arithmetic, these two vectors would be equal. Use these identities to prove that Cx = b. Assuming that A is nonsingular, will the rank-1 update method always work? Under what conditions could it fail? Explain.

Step-by-Step Solution:
Step 1 of 3

- -1- _ Jgk _ A hjje(.,oC_m'-tc

Step 2 of 3

Chapter 3, Problem 4 is Solved
Step 3 of 3

Textbook: Linear Algebra with Applications
Edition: 9
Author: Steven J. Leon
ISBN: 9780321962218

The answer to “(Rank-1 Updates of Linear Systems) (a) Set A = round(10 rand(8)) b = round(10 rand(8, 1)) M = inv(A) Use the matrix M to solve the system Ay = b for y. (b) Consider now a new system Cx = b, where C is constructed as follows: u = round(10 rand(8, 1)) v = round(10 rand(8, 1)) E = u v C = A + E The matrices C and A differ by the rank-1 matrix E. Use MATLAB to verify that the rank of E is 1. Use MATLABs \ operation to solve the system Cx = b and then compute the residual vector r = b Ax. (c) Let us now solve Cx = b by a new method that takes advantage of the fact that A and C differ by a rank-1 matrix. This new procedure is called a rank-1 update method. Set z = M u, c = v y, d = v z, e = c/(1 + d) and then compute the solution x by x = y e z Compute the residual vector b Cx and compare it with the residual vector in part (b). This new method may seem more complicated, but it actually is much more computationally efficient. (d) To see why the rank-1 update method works, use MATLAB to compute and compare Cy and b + cu Prove that if all computations had been carried out in exact arithmetic, these two vectors would be equal. Also, compute Cz and (1 + d)u Prove that if all computations had been carried out in exact arithmetic, these two vectors would be equal. Use these identities to prove that Cx = b. Assuming that A is nonsingular, will the rank-1 update method always work? Under what conditions could it fail? Explain.” is broken down into a number of easy to follow steps, and 297 words. Linear Algebra with Applications was written by and is associated to the ISBN: 9780321962218. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 9. This full solution covers the following key subjects: . This expansive textbook survival guide covers 47 chapters, and 935 solutions. Since the solution to 4 from 3 chapter was answered, more than 523 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 4 from chapter: 3 was answered by , our top Math solution expert on 03/15/18, 05:26PM.

Other solutions

People also purchased

Related chapters

Unlock Textbook Solution

Enter your email below to unlock your verified solution to:

(Rank-1 Updates of Linear Systems) (a) Set A = round(10 rand(8)) b = round(10 rand(8