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A Harmonic Restarted Arnoldi Algorithm for Calculating Eigenvalues and Outlining the Spectrum of a Large Matrix
"... A restarted Arnoldi algorithm is given that computes eigenvalues and eigenvectors. It is related to implicitly restarted Arnoldi, but has a simpler restarting approach. Harmonic and regular RayleighRitz versions are possible. For multiple eigenvalues, an approach is proposed that first computes eig ..."
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A restarted Arnoldi algorithm is given that computes eigenvalues and eigenvectors. It is related to implicitly restarted Arnoldi, but has a simpler restarting approach. Harmonic and regular RayleighRitz versions are possible. For multiple eigenvalues, an approach is proposed that first computes eigenvalues with the new harmonic restarted Arnoldi algorithm, then uses random restarts to determine multiplicity. This avoids the need for a block method or for relying on roundoff error to produce the multiple copies.
Thickrestart Lanczos method for electronic structure calculations
 J. Comput. Phys
, 1999
"... This paper describes two recent innovations related to the classic Lanczos method for eigenvalue problems, namely the thickrestart technique and dynamic restarting schemes. Combining these two new techniques we are able to implement an ecient eigenvalue problem solver. This paper will demonstrat ..."
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This paper describes two recent innovations related to the classic Lanczos method for eigenvalue problems, namely the thickrestart technique and dynamic restarting schemes. Combining these two new techniques we are able to implement an ecient eigenvalue problem solver. This paper will demonstrate its eectiveness on one particular class of problems for which this method is well suited: linear eigenvalue problems generated from nonselfconsistent electronic structure calculations. 1 Introduction The Lanczos method is a very simple and yet eective algorithm for nding extreme eigenvalues of large matrices. Since it only needs to access the matrix through matrixvector multiplications, the user has the exibility of choosing the most appropriate matrixvector multiplication scheme to reduce computer memory usage and the computation time. There is never any need to explicitly store the full matrix which can be prohibitively large in many electronic structure calculations. There ar...
Restarted block GMRES with deflation of eigenvalues
 Appl. Numer. Math
"... Abstract. BlockGMRES is an iterative method for solving nonsymmetric systems of linear equations with multiple righthand sides. Restarting may be needed, due to orthogonalization expense or limited storage. We discuss how restarting affects convergence and the role small eigenvalues play. Then a v ..."
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Abstract. BlockGMRES is an iterative method for solving nonsymmetric systems of linear equations with multiple righthand sides. Restarting may be needed, due to orthogonalization expense or limited storage. We discuss how restarting affects convergence and the role small eigenvalues play. Then a version of restarted blockGMRES that deflates eigenvalues is presented. It is demonstrated that deflation can be particularly important for block methods.
Dealing with linear dependence during the iterations of the restarted block Lanczos methods
 Numer. Algorithms
"... Dedicated to Richard Varga on the occasion of his 70th birthday The Lanczos method can be generalized to block form to compute multiple eigenvalues without the need of any deflation techniques. The block Lanczos method reduces a general sparse symmetric matrix to a block tridiagonal matrix via a Gra ..."
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Dedicated to Richard Varga on the occasion of his 70th birthday The Lanczos method can be generalized to block form to compute multiple eigenvalues without the need of any deflation techniques. The block Lanczos method reduces a general sparse symmetric matrix to a block tridiagonal matrix via a GramSchmidt process. During the iterations of the block Lanczos method an offdiagonal block of the block tridiagonal matrix may become singular, implying that the new set of Lanczos vectors are linearly dependent on the previously generated vectors. Unlike the single vector Lanczos method, this occurrence of linearly dependent vectors may not imply an invariant subspace has been computed. This difficulty of a singular offdiagonal block is easily overcome in nonrestarted block Lanczos methods, see Golub and Underwood [12], and Ruhe [30]. The same schemes applied in nonrestarted block Lanczos methods can also be applied in restarted block Lanczos methods. This allows the largest possible subspace to be built before restarting. However, in some cases a modification of the restart vectors is required or a singular block will continue to reoccur. In this paper we examine the different schemes mentioned in [12, 30] for overcoming a singular block for the restarted block Lanczos methods, namely the restarted method reported in [12] and the Implicitly Restarted Block Lanczos (IRBL) method developed by Baglama et al. [3]. Numerical examples are presented to illustrate the different strategies discussed. block Lanczos method, eigenvalues, implicit restarting, singular block, polynomial acceleration.
IMPROVED SEED METHODS FOR SYMMETRIC POSITIVE DEFINITE LINEAR EQUATIONS WITH MULTIPLE RIGHTHAND
, 810
"... Abstract. We consider symmetric positive definite systems of linear equations with multiple righthand sides. The seed conjugate gradient method solves one righthand side with the conjugate gradient method and simultaneously projects over the Krylov subspace thus developed for the other righthand ..."
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Abstract. We consider symmetric positive definite systems of linear equations with multiple righthand sides. The seed conjugate gradient method solves one righthand side with the conjugate gradient method and simultaneously projects over the Krylov subspace thus developed for the other righthand sides. Then the next system is solved and used to seed the remaining ones. Rounding error in the conjugate gradient method limits how much the seeding can improve convergence. We propose three changes to the seed conjugate gradient method: only the first righthand side is used for seeding, this system is solved past convergence, and the roundoff error is controlled with some reorthogonalization. We will show that results are actually better with only one seeding, even in the case of related righthand sides. Controlling rounding error gives the potential for rapid convergence for the second and subsequent righthand sides.
RESTARTING THE NONSYMMETRIC LANCZOS ALGORITHM for Eigenvalues . . .
"... A restarted nonsymmetric Lanczos algorithm is given for computing eigenvalus and both right and left eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Restarting also makes it possible to deal with roundoff error in new ways. We give a scheme for avoiding nea ..."
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A restarted nonsymmetric Lanczos algorithm is given for computing eigenvalus and both right and left eigenvectors. The restarting limits the storage so that finding eigenvectors is practical. Restarting also makes it possible to deal with roundoff error in new ways. We give a scheme for avoiding nearbreakdown and discuss maintaining biorthogonality. A system of linear equations can be solved simultaneously with the eigenvalue computations. Deflation from the presence of the eigenvectors allows the linear equations to generally have good convergence in spite of the restarting. The right and left eigenvectors generated while solving the linear equations can be used to help solve systems with multiple righthand sides.
An Evaluation of the Parallel ShiftandInvert Lanczos Method
 Las Vegas USA
, 1999
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Distributed, LargeScale Latent Semantic Analysis by Index Interpolation
"... Latent semantic analysis [12] is a wellknown technique to extrapolate concepts from a set of documents; it discards noise by reducing the rank of (a variant of) the term/document matrix of a document collection by singular value decomposition. The latter is performed by solving an equivalent symmet ..."
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Latent semantic analysis [12] is a wellknown technique to extrapolate concepts from a set of documents; it discards noise by reducing the rank of (a variant of) the term/document matrix of a document collection by singular value decomposition. The latter is performed by solving an equivalent symmetric eigenvector problem on a related matrix. Scaling to large set of documents, however, is problematic because every vectormatrix multiplication required by iterative solvers requires a number of multiplications equal to twice the number of postings of the collection. We show how to combine standard searchengine algorithmic tools in such a way to compute (reasonably) quickly the cooccurrence matrix C of a large document collection, and solve directly the associated symmetric eigenvector problem. Albeit the size of C is quadratic in the number of terms, we can distribute its computation among any number of computational unit without increasing the overall number of multiplications. Moreover, our approach is advantageous when the document collection is large, because the number of terms over which latent semantic analysis has to be performed is inherently limited by the size of a language lexicon. We present experiments over a collection with 3.6 billions of postings—two orders of magnitudes larger than any published experiment in the literature.
An Evaluation of the Parallel ShiftandInvert Lanczos Methody
"... Abstract When the Lanczos method is used to compute eigenvalues, it is often restarted or used with the shiftandinvert scheme. The restarted scheme usually uses less memory but the shiftandinvert scheme is more robust. In addition, the shiftandinvert Lanczos method requires accurate solutions of ..."
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Abstract When the Lanczos method is used to compute eigenvalues, it is often restarted or used with the shiftandinvert scheme. The restarted scheme usually uses less memory but the shiftandinvert scheme is more robust. In addition, the shiftandinvert Lanczos method requires accurate solutions of a series of linear systems. Parallel software packages suitable for these linear systems are only started to become available. In this talk, we will present our evaluation of two such packages and briefly exam when it is necessary to use the shiftandinvert scheme.