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# QR Algorithm Presentation

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### QR Algorithm Presentation

1. 1. Eigen Problem Power and Inverse Power Method QR Decomposition QR Algorithm Techniques used to Accelerate Convergence References Understanding the QR Algorithm Kenneth Mwangi Northern Arizona University December 3, 2008 Kenneth Mwangi Northern Arizona University Understanding the QR Algorithm
2. 2. Eigen Problem Power and Inverse Power Method QR Decomposition QR Algorithm Techniques used to Accelerate Convergence References Table of contents 1 Eigen Problem 2 Power and Inverse Power Method Power Method Inverse Power Method 3 QR Decomposition QR Decomposition 4 QR Algorithm QR Algorithm 5 Techniques used to Accelerate Convergence 6 References Kenneth Mwangi Northern Arizona University Understanding the QR Algorithm
3. 3. Eigen Problem Power and Inverse Power Method QR Decomposition QR Algorithm Techniques used to Accelerate Convergence References Introduction to the Eigen Value Problem The classical mathematical eigenvalue problem is deﬁned as the solution of the following equation: Avn = λvn ; n = 1, 2, . . . , N Eigen Problem reveals the following facts; 1 A is a n matrix either real or complex. 2 n is small (1000,2000,. . . ) 3 Calculate all the Eigenvalue. Eigenvectors? The generalised eigenvalue problem is; Avn = Bλvn Kenneth Mwangi Northern Arizona University Understanding the QR Algorithm
4. 4. Eigen Problem Power and Inverse Power Method QR Decomposition Power Method QR Algorithm Inverse Power Method Techniques used to Accelerate Convergence References Power Method The power iteration is a very simple algorithm. It does not decompose the matrix and thus it can be used when A is a large matrix. Nevertheless, it will ﬁnd only one eigenvalue (one with the greatest absolute value) and it may converge only slowly. Kenneth Mwangi Northern Arizona University Understanding the QR Algorithm
5. 5. Eigen Problem Power and Inverse Power Method QR Decomposition Power Method QR Algorithm Inverse Power Method Techniques used to Accelerate Convergence References The power method algorithm is highlighted below; u (0) = x0 k=1 Repeat w = Au (k−1) λ = entry of u with the largest magnitude. u (k) = wλ k ←k +1 until u (k) − u (k−1) < Kenneth Mwangi Northern Arizona University Understanding the QR Algorithm
6. 6. Eigen Problem Power and Inverse Power Method QR Decomposition Power Method QR Algorithm Inverse Power Method Techniques used to Accelerate Convergence References Inverse Power Method Inverse iteration is also an iterative eigenvalue algorithm.From the power method, the method seeks to improve the performance While the power method always converges to the largest eigenvalue, inverse iteration also enables the choice of eigenvalue to converge to. The inverse iteration algorithm converges fairly quickly.However, we can achieve even faster convergence by using a better eigenvalue approximation each time (Rayleigh quotient iteration). The inverse iteration algorithm requires solving a linear system at each step. This would appear to require O(n3 ) operations. However, this can be reduced to O(n2 ) if we ﬁrst reduce A to Hessenberg form. Kenneth Mwangi Northern Arizona University Understanding the QR Algorithm
7. 7. Eigen Problem Power and Inverse Power Method QR Decomposition Power Method QR Algorithm Inverse Power Method Techniques used to Accelerate Convergence References To get the eigenvalue of A closet to q Eigenvalue of (A − qI )−1 1 1 1 = λ1 −q , λ2 −q , . . . , λn −q Suppose λk is the eigenvalue of A closet to q. Thus the modiﬁcation on the power algorithm is; w (k) = (A − qI )−1 u (k−1) Kenneth Mwangi Northern Arizona University Understanding the QR Algorithm
8. 8. Eigen Problem Power and Inverse Power Method QR Decomposition QR Decomposition QR Algorithm Techniques used to Accelerate Convergence References QR Decomposition QR Decomposition is the factorization of a matrix into an orthogonal and a right triangular matrix; A = QR where Q is an orthogonal matrix (meaning thatQ T Q = I ) and R is an upper triangular matrix (also called right triangular matrix). Although the QR factorization is more computationally rigorous than the LU factorization the method oﬀers superior stability properties. Kenneth Mwangi Northern Arizona University Understanding the QR Algorithm
9. 9. Eigen Problem Power and Inverse Power Method QR Decomposition QR Decomposition QR Algorithm Techniques used to Accelerate Convergence References The QR decomposition is usually calculated using the following ways; 1 Gram-Schmidt process; 2 Householder reﬂections; and 3 Givens rotations. Mostly used process is the Householder reﬂection because it has greater numerical stability than the Gram-Schmidt method. Kenneth Mwangi Northern Arizona University Understanding the QR Algorithm
10. 10. Eigen Problem Power and Inverse Power Method QR Decomposition QR Decomposition QR Algorithm Techniques used to Accelerate Convergence References Householder Transformation If X and Y are vectors with the same norm, there exists an orthogonal symmetric matrix P such that; Y = PX where P = I − 2WW T and X −Y W = X −Y . Since P is both orthogonal and symmetric, it follows that P −1 = P. * * * * * * * * * * * * * * * * * * * Conclusion; X = * * * * * and Y = * * * * * * * * * * * * * * * * Kenneth Mwangi Northern Arizona University Understanding the QR Algorithm
11. 11. Eigen Problem Power and Inverse Power Method QR Decomposition QR Algorithm QR Algorithm Techniques used to Accelerate Convergence References QR Algorithm The overall stability theory for the QR algorithm was given by Wilkinson’s in 1965 in his work the algebraic eigenvalue problem. We now investigate a well known and eﬃcient method for ﬁnding all the eigenvalues of a general n × n real matrix; but working with a general matrix takes many iterations and becomes expensive and time consuming. Therefore the QR Method works much faster on special matrices, hence the reason to decompose the general matrices to a Hessenberg form. Hessenberg matrix is one that is ”almost” triangular. To be exact, an upper Hessenberg matrix has zero entries below the ﬁrst subdiagonal. Kenneth Mwangi Northern Arizona University Understanding the QR Algorithm
12. 12. Eigen Problem Power and Inverse Power Method QR Decomposition QR Algorithm QR Algorithm Techniques used to Accelerate Convergence References QR Algorithm Implementation To ﬁnd eigenvalues of a real matrix A using the QR factorization, we will generate a sequence of matrices A(m) that are orthogonal and similar to A, the QR proceeds by forming a sequences of matrices A = A(1) , A(2) , . . . , as follows. 1 A(1) = A is factored as a product A(1) = Q (1) R (1) , where Q (1) is orthogonal and R (1) is upper triangular. 2 A(2) is deﬁned as A(2) = R (1) Q (1) . Thus if the eigenvalues satisfy |λ1 | > |λ2 | > . . . > |λ2 |, the iterates converge to T , an upper triangular matrix with the eigenvalues on the diagonal. Kenneth Mwangi Northern Arizona University Understanding the QR Algorithm
13. 13. Eigen Problem Power and Inverse Power Method QR Decomposition QR Algorithm QR Algorithm Techniques used to Accelerate Convergence References QR Algorithm Given A ∈ R n×n Deﬁne A1 = A For k = 1, 2, . . . , do Calculate the QR decomposition Ak = Qk Rk , Deﬁne Ak+1 = Ak . Kenneth Mwangi Northern Arizona University Understanding the QR Algorithm
14. 14. Eigen Problem Power and Inverse Power Method QR Decomposition QR Algorithm Techniques used to Accelerate Convergence References Techniques used to Accelerate Convergence Deﬂation Aggressive deﬂation (due to Francis 1961, Watkins 1995) Ad hoc shifts Kenneth Mwangi Northern Arizona University Understanding the QR Algorithm
15. 15. Eigen Problem Power and Inverse Power Method QR Decomposition QR Algorithm Techniques used to Accelerate Convergence References References 1 Burden and Faires , Numerical Analysis , 8th Edition 2 Laurene V. Fausett, Applied Numerical Analysis Using MATLAB, 2nd Edition 3 Marco Latini ,The QR Algorithm, Past and Future 4 Wikipedia 5 Dr. Shaﬁu Jibrin, Numerical Analysis notes. Kenneth Mwangi Northern Arizona University Understanding the QR Algorithm