G. H. Golub and R. Underwood. The block lanczos method for computing eigenvalues. In J. R. Rice, editor, Mathematical Software III, pages 361--377. Academic Press, New York, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... end while Block methods improve robustness for difficult eigenproblems where the sought eigenvalues occur in multiplicities or are clustered together [21] In general, the total number of outer JD iterations reduces with larger block sizes, but the total number of matrix vector products increases [12, 22, 28]. However, larger blocks yield better cache efficiency, and better computation to communication ratio (coarser granularity) in parallel programs. The above block JD method is given in a data parallel (fine grain) form. The rows of each of the vectors x i , r i , z i , and i as well as of the ....

G. H. Golub and R. Underwood. The block Lanczos method for computing eigenvalues. In J. R. Rice, editor, Mathematical Software III, pages 361--377, New York, 1977. Academic Press. 7

.... Block methods improve robustness for dicult eigenproblems where the sought eigenvalues occur in multiplicities or are clustered (very close) together [25] In general, the total number of outer JD iterations reduce with larger block sizes, but the total number of matix vector operations increase [16, 28, 34]. However, larger blocks introduce repetitive patterns of computation yielding better cache eciency, and better computation to communication ratio (coarser granularity) in parallel programs. The above block JD method is given in a data parallel ( ne grain) form. The rows of each of the vectors x ....

G. H. Golub and R. Underwood. The block Lanczos method for computing eigenvalues. In J. R. Rice, editor, Mathematical Software III, pages 361-377, New York, 1977. Academic Press.

....of GMRES and Arnoldi, FGMRES [24] and Jacobi Davidson [28] are methods that allow for flexible preconditioning. Block algorithms may be used to further accelerate convergence on linear systems with multiple right hand sides [18] and eigenproblems with highly clustered or multiple eigenvalues [12]. They are also more robust because single vector methods can misconverge to unwanted eigenvalues in the presence of eigenvalue multiplicities. Although block methods improve cache performance, the number of matrix vector operations (and thus flops) performed is usually higher than their ....

G. H. Golub and R. Underwood. The block Lanczos method for computing eigenvalues. In J. R. Rice, editor, Mathematical Software III, pages 361--377, New York, 1977. Academic Press.

....to the block Krylov subspace K k (A; R 0 ) spanfR 0 ; AR 0 ; A R 0 g with R 0 = B Gamma AX 0 and X 0 2 IR is an initial guess. The correction Z k is determined by using a (semi)minimization or an orthogonality property. We note that block methods such as block Lanczos method [10], 9] and block Arnoldi method [20] are also used for solving large eigenvalue problems. The purpose of these block methods is to converge faster than their single right hand side counterparts. Another approach consists in projecting the initial residual matrix globally onto a matrix Krylov ....

G. H. Golub and R. R. Underwood, The block Lanczos method for computing eigenvalues, in Mathematical Software 3, J. R. Rice, ed., Academic Press, New York, 1977, pp. 364-377.

....fashion, where a new set of vectors (we call this internal set) is orthogonalized against a previously orthonormal set of vectors (we call this external) and then among themselves. This computation is typical in block Krylov methods, where the Krylov basis is expanded by a block of vectors [11, 12]. It is also typical when certain external orthogonalization constraints have to be applied to the vectors of an iterative method. Locking of converged eigenvectors in eigenvalue iterative methods is such an example [19, 22] The nature of these applications suggests that usually the internal set ....

G. H. Golub and R. Underwood. The block Lanczos method for computing eigenvalues. In J. R. Rice, editor, Mathematical Software III, pages 361-377, New York, 1977. Academic Press.

.... methods improve robustness for difficult eigenproblems where the sought eigenvalues occur in multiplicities or are clustered (very close) together [25] In general, the total number of outer JD iterations reduce with larger block sizes, but the total number of matix vector operations increase [16, 28, 34, 35]. However, larger blocks introduce repetitive patterns of computation yielding better cache efficiency, and better computation to communication ratio (coarser granularity) in parallel programs. The above block JD method is given in a data parallel (fine grain) form. The rows of each of the ....

G. H. Golub and R. Underwood. The block Lanczos method for computing eigenvalues. In J. R. Rice, editor, Mathematical Software III, pages 361--377, New York, 1977. Academic Press.

....being multiplied. All of the data mining algorithms in this paper multiply the sparse matrix times a set of vectors. In scientific computing, it also occurs in practice when there are multiple right hand sides in an iterative solver, or in blocked eigenvalue algorithms, such as block Lanczos [GU77, GLS94, Mar95] or block Arnoldi [Sad93, LM97] Another application is image segmentation in videos, where a set of vectors is used as the starting guess for a subsequent frame in the video [SM98] The use of multiple vectors in these problems essentially turns the kernel into a matrix matrix ....

G. H. Golub and R. Underwood. The Block Lanczos Method for Computing Eigenvalues. In J. R. Rice, editor, Mathematical Sotware III, pages 361--377. Academic Press, Inc., 1977.

.... these converged directions re enter the basis via round off effects and quickly cause a spurious copy of the previously computed eigenvalue to appear repeatedly in the spectrum of the projected matrix H k : The treatment of this basic difficulty has occupied a number of renowned researchers [29, 15, 7, 30, 32, 37, 4, 6, 5, 33, 34, 35, 17] since the early 1970 s. Within this context, restarting has proven to have important consequences for the development of numerical software based upon Arnoldi s method and this will be explored in the following section. 4.4 Restarting the Arnoldi Method An unfortunate aspect of the ....

....from the eigenvector expansion. An iteration is defined by a repeatly restarting until the current Arnoldi factorization contains the desired information. Saad s ideas were based on similar ones developed for the Lanczos process by Paige [29] Cullum and Donath [5] and Golub and Underwood [15]. It appears that Karush [17] proposed the first example of a restarted iteration. The ARPACK software is based upon another approach to restarting that offers a more efficient and numerically stable formulation. This approach called implicit restarting is a technique for combining the implicitly ....

G. H. Golub and R. Underwood. The block Lanczos method for computing eigenvalues. In J. R. Rice, editor, Mathematical Software III, pages 361--377, New York, 1977. Academic Press.

....singular values of an m Theta n matrix A and not their multiplicities. As shown in [11] these multiplicities can be recovered along with the suppression of zero eigenvalues of the matrix B caused by m 6= n. Following the block Lanczos recursion for the sparse symmetric eigenvalue problem ( 45] [22]) 33) may be represented in matrix form as A T U k = V k J T k Z k ; A V k = U k J k Z k ; 34) 23 where U k = u 1 ; Delta Delta Delta ; u k ] V k = v 1 ; Delta Delta Delta ; v k ] J k is a k Theta k bidiagonal matrix with J k [j; j] fi 2j and J k [j; j 1] ....

G. H. Golub, and R. R. Underwood. The block Lanczos method for computing eigenvalues. In Mathematical Software III, Academic Press, New York, 361--377, 1977.

....section, we present a block Lanczos method for the standard eigenvalue problem (1) and show how to compute eigenvalues and eigenvectors. The Lanczos method was originally proposed by Lanczos (1950) The block Lanczos method was developed for computing clustered eigenvalues (Cullum and Donath 1974, Golub and Underwood 1977, Scott 1979, Cullum and Willoughby 1985a, Cullum and Willoughby 1985b, Grimes et al. 1986, Grimes et al. 1994) Starting from an orthonormal set of b initial vectors V 1 2 R n Thetab , the Lanczos method builds an orthonormal basis for the Krylov space K k;b = spanfV 1 ; AV 1 ; A 2 V 1 ; ....

Golub, G. and Underwood, R. (1977), The block Lanczos method for computing eigenvalues, in J. Rice, ed., `Mathematical Software III', Academic Press, New York.

....fashion, where a new set of vectors (we call this internal set) is orthogonalized against a previously orthonormal set of vectors (we call this external) and then among themselves. This computation is typical in block Krylov methods, where the Krylov basis is expanded by a block of vectors [12, 11]. It is also typical when certain external orthogonalization constraints have to be applied to the vectors of an iterative method. Locking of converged eigenvectors in eigenvalue iterative methods is such an example [19, 22] This problem differs from the classical QR factorization in that the ....

G. H. Golub and R. Underwood. The block Lanczos method for computing eigenvalues. In J. R. Rice, editor, Mathematical Software III, pages 361--377, New York, 1977. Academic Press.

....the residual vector f m is orthogonal to the columns of Xm . The matrix Hm = X T m AXm is the orthogonal projection of A onto the column space of Xm j Km (A; x 1 ) The idea of restarting is based on similar approaches used for the Lanczos process by Paige (1971) Cullum and Donath (1974) and Golub and Underwood (1977). The first example of a restarted iteration is attributed to Karush (1951) A relatively recent variant was developed by Sorensen (1992) as a more efficient and numerically stable way to implement restarting. One of the benefits of this implicitly restarted Arnoldi iteration (ira iteration) is ....

G. H. Golub and R. Underwood. The block Lanczos method for computing eigenvalues. In J. R. Rice, ed., `Mathematical Software III', pp. 361--377, 1977.

....also discussed by Smith [122] and Joly [81] Mathematically, all these methods are equivalent. They only differ in the way of implementation. We may also use the block CG method proposed by O Leary [101] and discussed by Nikishin and Yeremin [100] Similarly, one may use the block Lanczos method [62] but we then have to face the storage problem again. Simoncini and Gallopoulos [119, 120] combined the idea of the seed method and hybrid techniques to solve nonsymmetric linear systems. Among all these methods, we observed that the class of Galerkin projection methods are very efficient which ....

....method. Then there exists ff 0, independent of the m p s, such that the residual of the non seed systems after dk=se Gamma 1 restarts satisfies k r dk=se;j 0 k ff dk=se X p=1 k fi mp 1 k; j = k 1; N; where fi mp 1 comes from the recurrence of the block Lanczos algorithm [62], AV mp = V mp T mp U mp 1 fi mp 1 E T mp ; p = 1; dk=se: Finally, if b(t) is continuous, we can bound the residual in the same way as in Theorem 9.3. Theorem 9.8 If b(t) is a continuous function in t on [t 1 ; t N ] then given ffl 0, there exist integers k and ff 0 such ....

G. H. Golub and R. Underwood. The block lanczos method for computing eigenvalues. In J. R. Rice, editor, Mathematical Software III, pages 361--377, New York, 1977. Academic Press.

....to compute the desired Ritz value approximations. An explicitly restarted Arnoldi iteration (era iteration) was introduced by Saad [30] to overcome these difficulties. The idea is based on similar ones developed for the Lanczos process by Paige [25] Cullum and Donath [10] and Golub and Underwood [17]. Karush proposed the first example of a re started iteration in [21] DEFLATION TECHNIQUES FOR IMPLICIT RESTARTING 3 After k steps, the Arnoldi algorithm computes a truncated factorization AV k = V k H k f k e T k ; 2.1) of A 2 R n Thetan into upper Hessenberg form where V T k V k = I k ....

G. H. Golub and R. Underwood, The block Lanczos method for computing eigenvalues, in Mathematical Software III, J. R. Rice, ed., 1977, pp. 361--377.

....vectors. For eigenvalue computations of a matrix A with multiple or clusters of eigenvalues, it is usually preferable to employ a Lanczos type method that iterates on A LANCZOS TYPE METHOD FOR MULTIPLE STARTING VECTORS 3 blocks of, say m, vectors, rather than on single vectors; see, e.g. [10, 21, 35]. Such a procedure then involves m right and m left starting vectors. Some applications require the repeated solution of linear systems (1:1) with the same matrix A, but different right hand sides, b 1 ; b 2 ; bm , that are all available simultaneously. These m systems can be summarized in ....

....particular, block Lanczos algorithms are restricted to the special case that p = m and that possible deflation occurs simultaneously in the right and left block Krylov subspaces. Block Lanczos algorithms for Hermitian matrices were first proposed by Cullum and Donath [9] and Golub and Underwood [21, 39]. Further and more recent work for the Hermitian case is described in [10, 30, 35] and the references given therein. We remark that only the algorithms in [9, 10, 30] and Ruhe s algorithm [35] include a proper deflation procedure. For non Hermitian matrices, O Leary with her block BCG algorithm ....

G. H. Golub and R. Underwood, The block Lanczos method for computing eigenvalues, Mathematical Software III (J. R. Rice, ed.), Academic Press, New York, 1977, pp. 361--377.

.... these converged directions re enter the basis via round off effects and quickly cause a spurious copy of the previously computed eigenvalue to appear repeatedly in the spectrum of the projected matrix H k : The treatment of this basic difficulty has occupied a number of renowned researchers [29, 15, 7, 30, 32, 37, 4, 6, 5, 33, 34, 35, 17] since the early 1970 s. Within this context, restarting has proven to have important consequences for the development of numerical software based upon Arnoldi s method and this will be explored in the following section. 4.4 Restarting the Arnoldi Method An unfortunate aspect of the ....

....from the eigenvector expansion. An iteration is defined by a repeatly restarting until the current Arnoldi factorization contains the desired information. Saad s ideas were based on similar ones developed for the Lanczos process by Paige [29] Cullum and Donath [5] and Golub and Underwood [15]. It appears that Karush [17] proposed the first example of a restarted iteration. The ARPACK software is based upon another approach to restarting that offers a more efficient and numerically stable formulation. This approach called implicit restarting is a technique for combining the implicitly ....

G. H. Golub and R. Underwood. The block Lanczos method for computing eigenvalues. In J. R. Rice, editor, Mathematical Software III, pages 361--377, New York, 1977. Academic Press.

....the application builder and the code generator. 7 Multiple Vectors The performance of matrix vector multiplication is inherently lower than that of matrix matrix multiplication because there is no reuse of element of matrix A in matrix vector multiplication. Algorithms such as block Lanczos [GU77] which compute a set of eigenvalues and associated eigenvectors, require multiplication of matrix to a set of vectors. Since there is potential for much higher performance, we extend the idea of register blocking to multiplication of sparse matrix and a set of vectors. Figure 9 shows the ....

G. H. Golub and R. Underwood. The Block Lanczos Method for Computing Eigenvalues. In J. R. Rice, editor, Mathematical Sotware III, pages 361--377. Academic Press, Inc., 1977.

....better understanding about the convergence of the Lanczos method. 2 Convergence of the block Lanczos method The literature contains two basically different types of block Lanczos procedures, iterative and non iterative. For iterative procedures see Cullum and Donath [14] and Golub and Underwood [15, 16], who replace the single vector x 0 by a system of r independent vectors (x 1 ; x 2 ; Delta Delta Delta ; x r ) For non iterative procedure see Lewis [17] Ruhe [18] and Scott [19] who mimic the single vector Lanczos procedures. Chain of blocks are generated, the length of the chain depends ....

....We have proposed the new theoretical error bounds on the rate of convergence of the Lanczos method in the previous section. Here the block generalization of Lanczos method can be treated as a system U 0 of r vectors U 0 = x 1 ; x 2 ; Delta Delta Delta ; x r ) instead of a single vector x 0 [14, 16, 18]. We will also follow Saad s notation suggested in [1] Theorem 2 in [1] has shown that there is no loss of generality in assuming that the eigenvalues of A are of multiplicity not exceed r. The largest k positive eigenvalues of A under consideration will therefore be numbered in decreasing order ....

[Article contains additional citation context not shown here]

G. H. Golub and R. Underwood. The block Lanczos method for computing eigenvalues. In J. Rice, editor, Mathematical Software, volume 3, pages 364--377. Academic Press, New York, 1977.

....when a large number of steps are required for convergence. The explicitly restarted Arnoldi iteration (era iteration) was introduced by Saad [32] to overcome these difficulties, based on similar ideas developed for the Lanczos process by Paige [27] Cullum and Donath [7] and Golub and Underwood [11]. Karush [14] proposes what appears to This work was supported in part by ARPA (U.S. Army ORA4466.01) by the U.S. Department of Energy (Contracts DE FG0f 91ER25103 and W 31 109 Eng 38) and by the National Science Foundation (Cooperative agreement CCR 9120008) y Mathematics and Computer ....

G. H. Golub and R. Underwood, The block Lanczos method for computing eigenvalues, in Mathematical Software III, J. R. Rice, ed., 1977, pp. 361--377.

....in iterative methods for 3D problems in which 2D subproblems are solved accurately in each step. Several solution methods for systems of the form (1.1) are available. They include block conjugate gradient and block Lanczos methods, as well as modifications of the vector Lanczos method 1 ; see [12, 17, 18, 19, 23, 25] and references therein. The Chebyshev and Richardson iteration methods can be implemented effectively on vector and parallel computers and have therefore received renewed attention; see [6, 24] When applying these methods one typically requires that an interval [a; b] 0 a b 1, that ....

.... whose eigenvalues are Ritz values of A; see, e.g. Golub and Van Loan [13] Gragg [14] and Parlett [20] The block Lanczos algorithm determines, from an initial matrix R 0 2 R n Thetas , a symmetric block tridiagonal matrix H j , whose eigenvalues are Ritz values of A; see Golub and Underwood [12], Parlett [20] and references therein. This section presents a block version of the modified Chebyshev algorithm that determines the matrix H j from the residual matrices R 0 ; R 1 ; R 2j Gamma1 given by (1.3) and from recursion coefficients ffi 0 ; ffi 1 ; ffi 2j Gamma1 of the ....

Golub, G.H., Underwood, R. (1977): The block Lanczos method for computing eigenvalues. In: Rice, J.R., ed., Mathematical Software III, pp. 361--377. Academic Press, New York

....proceed toward a solution. We discussed the real nonsymmetric problem in [36] The methods we present here are different from those of [36] even though the objectives are the same. Block methods have been investigated in several contexts: 25, 26, 35] discuss the case of real linear systems; [14, 17, 31] the eigenvalue problem; and [2, 15] for block methods in control. Our paper is structured as follows. In Sections 2.1 and 2.2 the block quasi minimum residual implementation of CG like and Lanczos methods are presented; a modified version of the block Lanczos algorithm is described in Section ....

G. Golub and R. Underwood. The block Lanczos method for computing eigenvalues. In J. R. Rice, editor, Mathematical Software III, pages 364--377. Academic Press, New York, 1977.

....are derived, and illustrated with numerical experiments. The role of invariant subspaces in the effectiveness of block methods is also discussed. 1. INTRODUCTION AND SUMMARY Block iterative methods have been proposed as an attractive approach for handling eigenvalue problems and linear systems [10, 21, 39]. They promise favorable convergence properties and effective exploitation of parallel computer architectures [21, 22] Block methods are natural candidates Work performed at the Center for Supercomputing Research and Development, University of Illinois at Urbana Champaign, with support from ....

....for nonsymmetric A [40] Sadkane [30] and Simon and Yeremin [33] on block Arnoldi. Another interesting contribution is that of Jia, who discussed the use of block Lanczos type algorithms, including block Arnoldi, for nonsymmetric eigenvalue problems [12] and thus generalized earlier results [10, 28] and 3 provided useful implementation discussions; and a recent effort by Broyden toward a comprehensive theory for block methods [3] We remark that the effectiveness of block methods applied to nonsymmetric systems is still under investigation. For example, it was shown in [36] that in the ....

G. Golub and R. Underwood, The block Lanczos method for computing eigenvalues, in Mathematical Software III, J. R. Rice, ed., Academic Press, New York, 1977, pp. 364--377.

....(the Ritz vector) corresponding to the desired eigenvalue, it has been suggested to restart with a vector that is a mix of Ritz vectors, corresponding to relevant Ritz values. See [65] Ch.VII and [61] for such strategies. These strategies are related to an earlier approach suggested in [29]. These strategies may have the advantage that the new starting vector also contains information for nearby eigenvalues. The main problem is that in this kind of restart we try to catch the information for an approximate subspace in one single vector, and apart from this, it is not easy to find ....

G. H. Golub and R. Underwood. The block Lanczos method for computing eigenvalues. In J.R. Rice, editor, Mathematical Software III, pages 361--377. Academic Press, New York, 1977.

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G. H. Golub and R. Underwood. The block lanczos method for computing eigenvalues. In J. R. Rice, editor, Mathematical Software III, pages 361--377. Academic Press, New York, 1977.

No context found.

G. H. Golub, R. Underwood, The block Lanczos method for computing eigenvalues, in: J. R. Rice (Ed.), Mathematical Software III, Academic Press, New York, 1977, pp. 361-377.

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