Fall2011-HW5

# Fall2011-HW5 - r that encodes the permutation without...

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Homework 5 Math 371, Fall 2011 Assigned: Sunday, October 9, 201 1 Due: Thursday, October 20, 2011 Clearly label all plots using title , xlabel , ylabel , legend Use the subplot command to compare multiple plots Include printouts of all Matlab code and output, labeled with your name, date, section, etc. (1) (Pivoting) (a) Prove that the matrix ± 0 1 1 1 ² does not have an LU factorization. Hint : Try computing elements l ij and u ij directly. (b) Does the system ± 0 1 1 1 ²± x y ² = ± a b ² have a unique solution for all a,b R ? Why? (c) How can you modify the matrix in part (a) so that it has an LU factorization? (2) (Gaussian Elimination with Partial Pivoting) Do page 169, # 7(a) by hand. (3) (Implementing Gaussian Elimination) Write a Matlab program to perform Gaussian Elimination with partial pivoting and back substitution to solve a given linear system Ax = b . It is ok to implement explicit switching of rows. Alternatively, you can try to follow the steps in the textbook and create a pivot vector

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Unformatted text preview: r that encodes the permutation, without explicitly switching the rows. Re-do problem #2 as a test. (4) Consider the matrix A = 2 7 5 6 20 10 4 3 . Using your Matlab program, solve linear systems with the matrix A and (a) b = 14 36 7 . (b) b = -4-16-7 . (c) b = -3-12 6 . (5) Write a Matlab function that implements the secant method. Your implementation can use three function evaluations for the ﬁrst iteration. But the later iterations must use just one function evaluation. Use your secant method to solve the following problems: (a) Finding √ 2 to 14 or more digits of accuracy. 1 2 (b) Finding 3 √ 3 to 14 or more digits of accuracy. (c) Improve the approximation given as x ≈ 4 . 965 on p.103/104 of the textbook to 14 or more digits of accuracy....
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Fall2011-HW5 - r that encodes the permutation without...

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