G. H. Golub. Some modified matrix eigenvalue problems. SIAM Review, 15:318--334, 1973. 149  Home/Search   Document Not in Database   Summary   Related Articles   Check

....one may ask if it is possible to detect such points in advance. Suppose D was chosen as a rank one modification of A and A(t) is given in the form A(t) Delta taezz (see Sec. 3.1. 1) The eigenvalues of A(t) are the zeros of the secular function f( t) 1 tae (see Golub [43]) Figure 3.3 depicts a typical secular function f( t) for a fixed t. Note that the first and second zero are close. In addition, we define f k ( t) f( t) Gamma tae Now assume that k Gamma1 (t) and k (t) have a pseudo crossing point at b t 2 [0; 1] It is easily seen that f k (ffi ....

G. H. Golub. Some modified matrix eigenvalue problems. SIAM Rev., 15:318-- 334, 1973.

....AMS subject classifications. 65F15, 68C25 1 Introduction The divide and conquer algorithm is an important recent development for solving the tridiagonal symmetric eigenvalue problem. The algorithm was first developed by Cuppen [8] based on previous ideas of Golub [17] and Bunch, Nielson and Sorensen [5] for the solution of the secular equation and made popular as a practical parallel method by Dongarra and Sorensen [14] This simple and attractive algorithm was considered unstable for a while because of a lack of orthogonality in the computed eigenvectors. It ....

....application of the strategy described above on the two tridiagonal matrices in (2.1) leads to the divide and conquer method for the symmetric tridiagonal eigenvalue problem. 2. 1 Computing the Spectral Decomposition of a RankOne Perturbed Matrix An updating technique as described in [5] 8] [17] can be used to compute the spectral decomposition of a rank one perturbed matrix (2.5) where D = diag(d 1 ; d 2 ; d n ) z = z 1 ; z 2 ; z n ) and ae is a nonzero scalar. By setting the characteristic polynomial of D aezz equal to zero, we find that the eigenvalues f i ....

Gene H. Golub. Some modified matrix eigenvalue problems. SIAM Review, 15(2):318--334, 1973.

....U e U ; U = diag(Q 1 ; Q 2 ) e Q: The formation of U is a matrix multiplication and dominates the operation count. This algorithm was originally suggested by Cuppen [30] and how to solve the secular equation efficiently was shown by Bunch, Nielson and Sorensen [23] building on work of Golub [68]. Until recently, it was thought that extended precision arithmetic was needed in the solution of the secular equation to guarantee that sufficiently orthogonal eigenvectors are produced when there are close eigenvalues. However, Gu and Eisenstat [73] have found a new approach that does not ....

Gene H. Golub. Some modified matrix eigenvalue problems. SIAM Review, 15(2):318--334, 1973.

....eigenvalues of A are Gamma2, Gamma0:5 and 2. a) general case with the slope at also plotted; b) exact hard case; c) near hard case; d) detail of box in (c) where differentiation is with respect to , and x satisfies (A Gamma I)x = Gammag. The function OE appears in many contexts [2, 8, 18, 19] and Figure 1.a shows its typical behavior. It is worth noticing that the values of OE and OE at an eigenvalue of B ff , are readily available and contain valuable information with respect to problem (1) as long as has a corresponding eigenvector with nonzero first component. Finding the ....

G.H. Golub. Some modified matrix eigenvalue problems, SIAM Rev., 15(2):318--334, 1973.

....computations and have computational implications. The final remarks are about the relation of the Procruste problem to the constrained linear least squares problem. The quadratically constrained linear least squares problem min kxk=ff kb Gamma Axk (5. 12) arises in many applications [8] 10] [9], 5] By changing variables this problem can be transformed into a special Procrustes problem. The Procrustes problem is min kqk=1 ka Gamma Sigmaqk 2 (5.13) PROCRUSTES PROBLEM FOR STIEFEL MATRICES 11 where A = U SigmaV T , a = U T b=ff and q = V T x=ff. This Procrustes problem is ....

G.H. Golub, Some modified matrix eigenvalue problems, SIAM Review 15 (1973), 318--334.

....# ; where J G n (d) is as in (3.2) e T n = 0; 0; 1] is the nth coordinate vector in R n , and (3:6) ff R n = a Gamma fi n n Gamma1 (a) n (a) with k ( Delta ) k ( Delta ; d) as before. One then has the following theorem analogous to Theorem 3.1. Theorem 3. 2 (Golub [9]) The Gauss Radau nodes R 0 = a, R 1 ; R n in (3.4) are the eigenvalues of J R n 1 , and the Gauss Radau weights R are given by (3:7) R = fi 0 [u R ;1 ] 2 ; 0; 1; 2; n; 6 where u R is the normalized eigenvector of J R n 1 corresponding to the ....

....e n 1 0 q fi L n 1 e T n 1 ff L n 1 3 7 7 5 ; with J G n (d) and e n as before, and ff L n 1 , fi L n 1 the solution of the 2 Theta2 linear system (3:10) n 1 (a) n (a) n 1 (b) n (b) # ff L n 1 fi L n 1 # = a n 1 (a) b n 1 (b) # : We now have Theorem 3. 3 (Golub [9]) The Gauss Lobatto nodes L 0 = a, L 1 ; L n , L n 1 = b in (3.8) are the eigenvalues of J L n 2 , and the Gauss Lobatto weights L are given by (3:11) L = fi 0 [u L ;1 ] 2 ; 0; 1; 2; n; n 1; where u L is the normalized eigenvector of J L n 2 ....

Golub, G.H. 1973. Some modified matrix eigenvalue problems. SIAM Rev. 15, 318--334.

.... Gamma oe j Gamma1;j Gamma1 ) 0, then j 1 j 1 j Gamma1 1 j 2 Delta Delta Delta j j Gamma1 j Gamma1 j Gamma1 j j j j : 3.60) Proof. These results follow immediately from well know results on the eigenvalues of Hermitian tridiagonal matrices (cf. [Go73], GoVL89] We offer the following brief proof for completeness. Consider the eigenvalues of the Hermitian matrices Delta 1=2 j H j Delta Gamma1=2 j and Delta 1=2 j U j L Gamma1 j Delta Gamma1=2 j , which are similar to H j and U j L Gamma1 j respectively. Since Delta 1=2 j ....

G. H. Golub, Some Modified Matrix Eigenvalue Problems, SIAM Review, Vol. 15, No. 2, April, 1973, pp 318-334.

....(4.5) are known as secular equations and functions like OE( OE 0 ( are sometimes called secular functions. The term seems to come from celestial mechanics (see [1] Secular equations appear in many contexts such as in the solution of certain eigenvalue problems, see for example [1] 3] [23], 7] 49] 51] 50] and [52] These works are concerned with computing all the solutions of a secular equation and they usually assume that fl i 6= 0; i = 1; 2; n, therefore excluding the possibility of the hard case. This fact prevents the use of such methods for solving the general ....

....of the hard case. This fact prevents the use of such methods for solving the general TRS. The use of the secular equation (4. 5) in connection with the TRS or related problems such as quadratically equality constrained least squares problems, can be traced back to [18] Other works include [23], 22] 64] 65] 39] 53] 69] and [55] Let us continue now with the study of the TRS, specifically with a special case: the hard case. 4.2.2 The Hard Case The hard case refers to a special situation in which the boundary solution of problem (4.3) is not unique. The hard case can only occur ....

G.H. Golub. Some modified matrix eigenvalue problems. SIAM Review, 15(2):318--334, 1973.

....(S t s x) We apply the Gauss Hermite quadrature rule. The Gauss quadrature rule is given by Z 1 ;1 g(x) 1 (2) 1=2 e ;x 2 dx = m X i=1 w i g(x i ) where the equality holds for all polynomials of degree up to 2m; 1 and the quadrature points x i and the weights are determined (e.g. see [3]) as follows. Let J be the symmetric tri diagonal matrix with zero diagonals and J i#i 1 = p i=2# 1 i m ; 1. Then fx i g are the eigenvalues of J and w i equal to j(v i ) 1 j 2 where (v i ) 1 is thefirstelement of the i th normalized eigenvector of J . Thus, I is approximate by ....

G. H. Golub, Some modified matrix eigenvalue problems, SIAM Rev., vol. 15 (1973), 318-334.

....throughout the remainder of this section assumes the matrix K is square, without any loss of generality. The singular values and singular vectors of M are related to the eigenvalues and eigenvectors of M T M = D zz T . For work on the rank one modification of the symmetric eigenproblem, see [12, 13, 23, 27, 38, 48, 66, 95]. The following result characterizes the singular values and singular vectors of M . It is assumed that the singular values of K are distinct and z i 6= 0 for i = 1; nq. Deflation may be used to reduce the original problem to a problem where these assumptions hold and is briefly discussed ....

....method safeguarded by bisection or by using the rational interpolation strategy in [12, 13] 4.3.1. Derivation of the Secular Equation This section derives the secular equation and the characterization of the singular vectors given in Lemma 9. The derivation is based on the results of Golub [38] and Bunch, Nielsen, and Sorensen [13] Cuppen [23] presents a similar derivation and Watkins [104, Section 6.3] presents a tutorial discussion of the results. Let M= 2 6 6 6 6 6 6 6 6 4 oe 1 . oe n z 1 Delta Delta Delta z n 3 7 7 7 7 7 7 7 7 5 ; z= z 1 Delta Delta Delta z n ....

G. H. Golub. Some modified matrix eigenvalue problems. SIAM Review, 15:318-- 334, 1973.

....algorithm, ScaLAPACK, LAPACK, distributed memory architecture AMS subject classifications. 65F15, 68C25 PII. S1064827598336951 1. Introduction. The divide and conquer algorithm for the symmetric tridiagonal eigenvalue problem was first developed by Cuppen [8] based on previous ideas of Golub [16] and Bunch, Nielsen, and Sorensen [5] for the solution of the secular equation. The algorithm was popularized as a practical parallel method by Dongarra and Sorensen [14] who implemented it on a shared memory machine. They concluded that divide and conquer algorithms, when properly implemented, ....

G. H. Golub, Some modified matrix eigenvalue problems, SIAM Rev., 15 (1973), pp. 318--334.

....eigenvalues of A are Gamma2, Gamma0:5 and 2. a) general case with the slope at also plotted; b) exact hard case; c) near hard case; d) detail of box in (c) where differentiation is with respect to , and x satisfies (A Gamma I)x = Gammag. The function OE appears in many contexts [2, 8, 18, 19] and Figure 1.a shows its typical behavior. It is worth noticing that the values of OE and OE 0 at an eigenvalue of B ff , are readily available and contain valuable information with respect to problem (1) as long as has a corresponding eigenvector with nonzero first component. Finding the ....

G.H. Golub. Some modified matrix eigenvalue problems, SIAM Rev., 15(2):318--334, 1973.

....the zero eigenvalue of T 1 . In [2] the hybrid Chebyshev Arnoldi algorithm [8] was applied to T 2 to find the leftmost eigenvalues of (1) Eigenvalue problems of the form (1) and (2) are intimately related to problems with constraints. If K is symmetric then (1) 2) is equivalent to the problem [3]: Find the extrema of u T Ku, subject to u T Mu = 1 and C T u = 0. In (1) 2) p 2 R m provides the Lagrange multipliers corresponding to the m constraints C T u = 0. This connection indicates that a natural setting for the eigenvalue problem (1) 2) is the subspace C : w ....

G.H. Golub. Some modified matrix eigenvalue problems. SIAM Review, 15:318-- 334, 1973. 7

.... case, the singular values of A 0 and M 1 are always well conditioned with respect to a perturbation, the singular vectors can be extremely sensitive to such perturbations [5] In both cases, the problem of updating the SVD has been considered by Bunch and Nielsen [1] using results from [2, 4]. Their scheme for finding the SVD of M and M 1 can be unstable [1, 2] And their algorithm takes about 2n min 2 (m; n) and 2mmin 2 (m; n) floating point operations to update the right and the left singular vector matrices, respectively. The lack of a fast algorithm for updating the SVD is ....

G. H. Golub, Some modified matrix eigenvalue problems, SIAM Review, 15 (1973), pp. 318-- 334.

.... U Omega Q 1 0 0 Q 2 U T : 1.5) In other words, the eigenvalues of A are the diagonal entries of Omega Gamma and the eigenvector matrix Q 1 0 0 Q 2 U (see Section 2) The computation of the eigendecomposition (1. 4) has been extensively discussed in the literature (see [5, 6, 8, 13, 10, 16, 18, 20]) The highly influential numerical linear algebra software package LAPACK [1] includes special subroutines for computing this eigendecomposition numerically stably in O(n 2 ) flops. Our algorithm is similar to the well known Cuppen s divide and conquer algorithm for the symmetric tridiagonal ....

G. H. Golub, Some modified matrix eigenvalue problems, SIAM Review, 15 (1973), pp. 318-- 334.

....Gauss Hermite quadrature rule. The Gauss Hermite quadrature rule is given by Z 1 Gamma1 g(x) 1 (2 ) 1=2 e Gammax 2 dx = m X i=1 w i g(x i ) where the equality holds for all polynomials of degree up to 2m Gamma 1 and the quadrature points x i and the weights are determined (e.g. see [9]) as follows. Let J be the symmetric tri diagonal matrix with zero diagonals and J i;i 1 = q i=2; 1 i m Gamma 1. Then fx i g are the eigenvalues of J and w i equal to j(v i ) 1 j 2 where (v i ) 1 is the first element of the i th normalized eigenvector of J . Thus, I is approximated by (3:3) ....

G. H. Golub, "Some modified matrix eigenvalue problems," SIAM Rev., vol. 15, pp. 318-334, 1973.

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G. Golub. Some modified matrix eigenvalue problem. SIAM Review, 15:318-- 334, 1973.

....that f (2n) #) 0 , #n ## a # b,andlet . Then, #N,## # [a, b] such that LG [f ] I [f ] 2N) A proof of this is given in [10] To obtain the Gauss Radau rule ( M =1in 3.1 3.2) we extend the matrix JN in 3. 4 in such a way that it has one prescribed eigenvalue, see [5]. For Gauss Radau, the remainder RGR is RGR [f] z 1 ) d#(#) Therefore, if we know the sign of the derivatives of f , we can bound the remainder. This is stated in the following theorem. Theorem 2 Suppose u = v and f is such that f (2n 1) #) 0, #n, ##, a # b. Let UGR ....

G.H. Golub, "Some modified matrix eigenvalue problems", SIAM Review v15 n2 (1973) pp 318--334.

....and denominators in (13) and (14) we need to estimate quadratic forms v T f(M)v, where M is some symmetric positive definite matrix and f( p # p =1# 2. There is extensive literature on the application of Gauss quadrature rules to bound bilinear forms# see papers by Golub and collaborators [3, 9, 11, 12, 14]. This paper applies these techniques to our blur identification problem. Let the eigendecomposition of M be given by M = Q T XiQ, where Q is an orthogonal matrix and Xi is a diagonal matrix of eigenvalues in increasing order. Then v T f(M)v = v T Q T f ( Xi)Qv = v T f ( Xi) v ....

G. Golub. Some modified matrix eigenvalue problem. SIAM Review, 15:318-- 334, 1973.

....however, have been proposed over the years including, among others, regularized least squares [4] ridge regression [4, 10] total 0 E mail addresses: shiv ece.ucsb.edu, golub sccm.stanford.edu, mgu math.ucla.edu, and sayed ee.ucla.edu. 1 2 CHANDRASEKARAN, GOLUB, GU, AND SAYED least squares [2, 3, 4, 7], and robust estimation [6, 9, 12, 14] These different formulations allow, in one way or another, incorporation of further a priori information about the unknown parameter into the problem statement. They are also more effective in the presence of data errors and incomplete statistical ....

G. H. Golub, Some modified matrix eigenvalue problems, SIAM Review, 15 (1973), pp. 318-- 344.

....Parameter estimation in the presence of data uncertainties is a problem of considerable practical importance, and many estimators have been proposed in the literature with the intent of handling modeling errors and measurement noise. Among the most notable is the total least squares method [1, 2, 3, 4], also known as orthogonal regression or errors in variables method in statistics and system identification [5] In contrast to the standard least squares problem, the TLS formulation allows for errors in the data matrix. Its performance may degrade in some situations where the effect of noise and ....

G. H. Golub, Some modified matrix eigenvalue problems, SIAM Review, 15 (1973), pp. 318-- 344. 22 CHANDRASEKARAN, GOLUB, GU, AND SAYED

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G. H. Golub. Some modified matrix eigenvalue problems. SIAM Review, 15:318--334, 1973. 149

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Golub G.H. (1973). Some modified matrix eigenvalue problems. Siam Review 15, 318 -- 344.

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G. H. GOLUB, Some modified matrix eigenvalue problems, Siam Review, 15 (1973), pp. 318-- 344. 15

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G. H. Golub, Some modified matrix eigenvalue problems, SIAM Review 15 (1973), pp. 318--334.

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