Numerical Analysis I
Numerical Analysis I MA 580
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This 4 page Class Notes was uploaded by Braeden Lind on Thursday October 15, 2015. The Class Notes belongs to MA 580 at North Carolina State University taught by Staff in Fall. Since its upload, it has received 6 views. For similar materials see /class/223723/ma-580-north-carolina-state-university in Mathematics (M) at North Carolina State University.
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Date Created: 10/15/15
MASSOCSCSSO Review Please also see Midterm Review Homework Assignments Short Proofs and Questions asked in class Concepts Iteration matrix Spectral radius A and quot quot matrix Residual Vector the least squares solution Householder matrix curve fitting by polynomi als or by other base functions overdeterminedunderdetermined system Gershgorin circles Eigenvalues and Eigenvectors Similarity transformation Orthogonal matrice The domi nant least dominant eigenvalue Gerschgorin circles Householder transformation Upper Hessenberg matrices Least squares solutions Singular value decomposition SVD The nor mal equations Pseudoinverse Stationary Iterative Algorithms a Tmik ll c k 1 2 Jacobi Iterative Method Tg DquotL U c Dquot 1 1 71 7 my 7 hi 2 aijmgk l ij jzl The Jacobi method converges if A is strictly row diagonally dominant GavssSeidel Iterative Method Tg D LquotU c D L 1b mm i 7 m The Jacobi method converges if A is strictly row diagonally dominant or symmetric positive definite SOB Iterative Method Tw D wL l 1 wD c wD wLquotb 1 71 7 1 wa kil bi Erma 2 lijf gkil i 1 2 rt 77 gt 31 y1l The SOR method converges if A is symmetric positive definite and 0 lt w lt 2 The Power method yk1 yk1 4 vallz 139 MM ma aw The shifted inverse Power method to find a specific eigenvalue M if q a M A II3kl wk yk1 r 1 7 H 9M1 H 2 Mt1 whirAJ Im QR method and shifted QR method At It QkRk Ak1 13955ka It Theories I Error estimates of the direct methods and iterative methods I Show that the iterative method will converge if HRH lt 1 Derive the convergence speed I Show that oA g I Orthogonal matrices 9 1 QT QTQ 1 Wash HQAHZ n2QA n2A etc I The Householder transformation theorem Pa 9 How to choose 9 y ml need to find a pay attention to the sign and w writ 1 to form P I 2 1 71 Supplementary exercises 1 Homework problems problems from class notes etc 2 What is the essential condition for the convergence of the Power method or the symmetric Power method The dominant eigenvalue is a simple eigenvalue 3 What is the convergence speed of the Power method 4 Can we use the Householder similarity transformations to reduce a matrix A to a diagonal matrix 5 What is the relation between the eigenvalues and eigenvectors of A and the eigenvalues and eigenvectors of a Aquot b A q I c A 11quot 6 If A is an eigenvalue of a matrix A 1 gt 9 0 a show that A g and so A g b use Gerschgorin theorem to show that A g 3 0 1 Show that A 0 0 1 has three distinct eigenvalues A1 gt A2 gt A In order to 1 1 5 approximate the eigenvalue A2 apply the first two iterations of the inverse Power method to the shifted matrix A 31 Let 1 0 0 1 0 l l w m 4 W W A um 0 5 0 i i 1 0 6 0 0 Use above to solve the system of equations Ax I where I 0 1 1 Let A E Rmm with mnMA n We can use the QR method or the Cholesky decomposition for the normal equation ATAy 2 AT to solve the least squares problem 1 Compare the two methods and make a recommendation Hint Consider the operation account the storage accuracy etc Given the following linear system of equations 331 2 3 3 272 3 2 y2 2303 2 a With new 1 1 1 find the rst and second iteration of the Jacobi GaussSeidel and 80R 0 15 methods b Write down the Jacobi and GaussSeidel iteration matrices RJ and Ray c Do the Jacobi and GaussSeidel iterative methods converge d Use the QR method and the normal equation to solve the least squares problem 1 2 2 0 0 0 5 0 A 1 A 0 0 0 2 0 1 3 3 0 0 0 where I 3 0 0 6 8 Find the 2norm of the residual vector corresponding to the least squares solution 11 Assume we use the following iterative method to solve Ag 2 I where A is a nonsingular arbitrary matrix M71 mm ck 0amp4 crkpk ck A lb Tic W71 MAW W b AM for given pk Determine wk according to the following Minimize b Minimize c Which approach is computable For the second case also show that rzApk 0 Hint Use the fact that 1130113
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