MATH 5620 NUMERICAL ANALYSIS IIOPTIONAL HOMEWORK 6, DUE APRIL 27 2010Note References are to the B&F 8th edition.Problem 1 B&F 9.2.2 a,b and 9.2.8 a,b (power method)Problem 2 B&F 9.2.6 a,c and 9.2.12 a,c (symmetric power method)Problem 3 B&F 9.2.6 a,c and 9.2.12 a,c, but use Rayleigh Quotient It-eration (page 63 in the class notes math5620s11 09.pdf) instead of thesymmetric power method.Problem 4 B&F 9.4.2 a,c and 9.4.4 a,c (QR Algorithm). Please do notimplement Algorithm 9.6 in B&F. A simpler algorithm is outlined below.Here are some implementation guidelines:• Please display at least 7 digits of precision (or use format long)• In Problems 1–4 I expect to see a few iterates of the method in questionon the specified matrix and then the approximation to the eigenvalues (andeigenvectors in Problems 1–3). You can always check that your eigenvaluesare correct using Matlab’s command eig (try [V,D]=eig(A);). Of coursethe eigenvectors you get maybe different from those given by eig becauseof normalization or signs.• To check convergence in Problems 1–3 simply determine if two consecu-tive iterates (eigenvalues or eigenvectors) are within the desired toleranceusing the Euclidean norm (this is not an industrial strength convergencecriterion!)• In Problem 4: Please use directly Matlab’s qr function. The algorithmin p73 (notes: math5620 s11.pdf) needs some kind of deflation techniqueto work well. Deflation simply means that when the desired accuracy ofan eigenvalue is met the procedure continues with the next eigenvalue. Todetect convergence of the entry T (m, m) to an eigenvalue of T its enoughto require that |T (m, m − 1)| < T OL. If T (m, m) is within the desiredtolerance, we can set m = m − 1 and continue with T (1 : m, 1 : m). Seenext page for a suggested implementation.12 MATH 5620 NUMERICAL ANALYSIS II OPTIONAL HOMEWORK 6, DUE APRIL 27 2010Given an n × n tridiagonal matrix T and a prescribed tolerance  hereis a practical shifted QR algorithm:m = nfor k = 0 , 1 , . . .Determine s h i f t µ (∗ Wilki n son s h i f t recommended ∗)QR = T (1 : m, 1 : m) − µI (∗ QR f a c t o r i z a t i o n ∗)T (1 : m, 1 : m) = RQ + µI(∗ ch e ck c o nv er gen ce o f T(m ,m) t o an e i g e nv al ue o f T ∗)i f |A(m, m − 1)| <  theni f m > 2 then(∗ go t o n e x t e ig en va lu e ∗)m = m − 1e l s e(∗ a l l e i g e n v a l u e s have co n ve rg e d t o w i th i n p r e s c r i b e dt o le r a n ce ∗)print ’ done i n ’ k ’ i t e r a t i o n s ’stopend i fend i fend f o r• For determining the shift µ please use the Wilkinson shift (the one givenin the notes can have problems in pathological cases):d =12(A(m − 1, m − 1) − A(m, m))i f d > 0µ = A(m, m) + d −pd2+ A(m, m − 1)2e l s eµ = A(m, m) + d +pd2+ A(m, m − 1)2end i