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### Fall2011-HW6-soln

Course: MATH 371, Fall 2011
School: Michigan
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Word Count: 321

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6 Homework Solution Sketches Math 371, Fall 2011 Assigned: Thursday, October 20, 2011 Due: Thursday, October 26, 2011 Clearly label all plots using title, xlabel, ylabel, legend Use the subplot command to compare multiple plots Include printouts of all Matlab code, labeled with your name, date, section, etc. (1) (Norms) (a) Do problems #3(a), (b) from page 180 (b) Consider the matrix 20 0 A = 0 3 4 . 0 2...

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6 Homework Solution Sketches Math 371, Fall 2011 Assigned: Thursday, October 20, 2011 Due: Thursday, October 26, 2011 Clearly label all plots using title, xlabel, ylabel, legend Use the subplot command to compare multiple plots Include printouts of all Matlab code, labeled with your name, date, section, etc. (1) (Norms) (a) Do problems #3(a), (b) from page 180 (b) Consider the matrix 20 0 A = 0 3 4 . 0 2 1 Compute the spectrum of AT A. What is the 2 matrix norm of A? (c) Compute the matrix norm of A. Find a vector x such that A = Ax (d) Verify your calculations in (b) and (c) using Matlab. x . We prove that || ||1 is a vector norm as follows. It is clear that x 1 0. If x 1 = 0, then |xi | = 0 for each i. So x = 0. x = || |xi | = || |xi |. x + y 1 |xi | + |yi | = x 1 + y 1 . / #3(b). The following 1 norms are easily determined. (3, 5, 2) 1 = 9.4142. (2, 1, 3, 4) 1 = 10. (4, 8, 1) 1 = 13. (e, , 1) 1 = 6.8599. The eigenvalues of AT A are 4, 4.8020, and 25.1980. The 2 matrix norm A of is 5.0198. (2) (Condition numbers) Let A be the n n matrix of the form 2 1 1 2 1 1 2 .. . .. .. . . 1 . 1 2 Read the help of the Matlab command diag() and use it create such a matrix for any given n. (a) For n = 50, 100, 500, 1000, 2000, 4000, nd the condition number of the matrix using cond(A) and make a table. (b) Plot the condition numbers against n using loglog(). Include your plot. 1 2 (c) How do do the condition numbers depend upon n? The following program was used to solve this problem in class: nlist clist for n A= for = [ 5 0 , 100 , 500 , 1000 , 5000 , 10000 , 20000] = []; = nlist sparse ( n , n ) ; i = 1:n A( i , i ) = 2 ; end f o r i = 1 : n1 A( i , i +1) = 1; A( i +1, i ) = 1; end c l i s t =[ c l i s t , cond (A ) ] ; end loglog ( n l i s t , c l i s t ) ; hold on ; grid ; loglog ( n l i s t , c l i s t , ko , M a r k e r S i z e , 6 , MarkerFaceColor , k ) ; Alternatively, you can use A=2*diag(ones(n,1))-diag(ones(n-1,1),-1)-diag(ones(n-1,1),1).
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