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Get Full Access to Linear Algebra With Applications - 9 Edition - Chapter 3 - Problem 4
Get Full Access to Linear Algebra With Applications - 9 Edition - Chapter 3 - Problem 4

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# (Rank-1 Updates of Linear Systems) (a) Set A = round(10 rand(8)) b = round(10 rand(8 ISBN: 9780321962218 437

## Solution for problem 4 Chapter 3

Linear Algebra with Applications | 9th Edition

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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.

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##### 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.

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