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SPECTRAL K-WAY RATIO-CUT PARTITIONING AND CLUSTERING

IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS 1994年第9期13卷 1088-1096页

作者： CHAN, PK SCHLAG, MDF ZIEN, JY Computer Engineering Department University of California Santa Cruz Santa Cruz CA USA

Recent research on partitioning has focused on the ratio-cut cost metric, which maintains a balance between the cost of the edges cut and the sizes of the partitions without fixing the size of the partitions a priori. Iterative approaches and spectral approaches to two-way ratio-cut partitioning have yielded higher quality partitioning results. In this paper, we develop a spectral approach to multi-way ratio-cut partitioning that provides a generalization of the ratio-cut cost metric to k-way partitioning and a lower bound on this cost metric. Our approach involves finding the k smallest eigenvalue/eigenvector pairs of the Laplacian of the graph. The eigenvectors provide an embedding of the graph's n vertices into a k-dimensional subspace. We devise a time and space efficient clustering heuristic to coerce the points in the embedding into k partitions. Advancement over the current work is evidenced by the results of experiments on the standard benchmarks.

关键词： PARTITIONING MULTIWAY PARTITIONING RATIO-CUT METRIC SPECTRAL METHOD CLUSTERING EIGENVALUES lanczos algorithm

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A QUASI-MINIMAL RESIDUAL VARIANT OF THE BI-CGSTAB algorithm FOR NONSYMMETRIC SYSTEMS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 1994年第2期15卷 338-347页

作者： CHAN, TF GALLOPOULOS, E SIMONCINI, V SZETO, T TONG, CH SANDIA NATL LABS CTR COMP ENGN ALBUQUERQUE NM 87185 USA UNIV ILLINOIS DEPT COMP SCI URBANA IL 61801 USA UNIV ILLINOIS CTR SUPERCOMP RES & DEV URBANA IL 61801 USA

Motivated by a recent method of Freund [SIAM J. Sci. Comput., 14 (1993), pp. 470-4821, who introduced a quasi-minimal residual (QMR) version of the conjugate gradients squared (CGS) algorithm, a QMR variant of the biconjugate gradient stabilized (Bi-CGSTAB) algorithm of van der Vorst that is called QMRCGSTAB, is proposed for solving nonsymmetric linear systems. The motivation for both QMR variants is to obtain smoother convergence behavior of the underlying method. The authors illustrate this by numerical experiments that also show that for problems on which Bi-CGSTAB performs better than CGS, the same advantage carries over to QMRCGSTAB.

关键词： CONJUGATE GRADIENTS lanczos algorithm ITERATIVE METHODS BCG CGS QRM BI-CGSTAB NONSYMMETRICAL LINEAR SYSTEMS

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AN IMPLICIT SHIFT BIDIAGONALIZATION algorithm FOR ILL-POSED SYSTEMS

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BIT NUMERICAL MATHEMATICS 1994年第4期34卷 510-534页

作者： BJORCK, A GRIMME, E VANDOOREN, P LINKOPING UNIV DEPT MATH S-58183 LINKOPING SWEDEN UNIV ILLINOIS COORDINATED SCI LAB URBANA IL 61801 USA

Iterative methods based on lanczos bidiagonalization with full reorthogonalization (LBDR) are considered for solving large-scale discrete ill-posed linear least-squares problems of the form min(x)\\Ax - b\\(2). Methods for regularization in the Krylov subspaces are discussed which use generalized cross validation (GCV) for determining the regularization parameter. These methods have the advantage that no a priori information about the noise level is required. To improve convergence of the lanczos process we apply a variant of the implicitly restarted lanczos algorithm by Sorensen using zero shifts. Although this restarted method simply corresponds to using LBDR with a starting vector (AA(T))(p)b, it is shown that carrying out the process implicitly is essential for numerical stability. An LBDR algorithm is presented which incorporates implicit restarts to ensure that the global minimum of the CGV curve corresponds to a minimum on the curve for the truncated SVD solution. Numerical results are given comparing the performance of this algorithm with nonrestarted LBDR.

关键词： ILL-POSED PROBLEMS lanczos algorithm REGULARIZATION LEAST SQUARES

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SIMULATION OF CONSTRAINED FERMIONS N LOW-DIMENSIONAL SYSTEMS

SIMULATION OF CONSTRAINED FERMIONS N LOW-DIMENSIONAL SYSTEMS

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作者： Matthias TROYER Swiss Federal Institute of Technology in Zurich

学位级别：博士

Several numerical methods have been improved and eciently implemented to an- swer some important questions about constrained fermion systems. We have de- veloped an improved version of the quantum transfer matrix algorithm to study quasi one-dimensional systems such as chains or ladders. The extreme eigenval- ues and eigenvectors of the quantum transfer matrix are calculated by the recently developed look-ahead lanczos algorithm. The phase diagram of the one-dimensional Kondo lattice model has been in- vestigated using the quantum Monte Carlo world line algorithm and quantum transfer matrix techniques. In the strong coupling region ferromagnetic ordering is found even at large band llings. In the weak coupling region the system shows a Ruderman-Kittel-Kasuya-Yoshida (RKKY) like behaviour. The one-dimensional t-J model with additional density-density repulsive inter- actions has been studied by exact diagonalisation and quantum Monte Carlo meth- ods. A short-range repulsion pushes phase separation to larger values of J/t, and leads to a widened precursor region in which a spin gap and strengthened super- conducting correlations appear. The correlation exponents and the phase diagram are calculated. The new quantum transfer matrix method has been applied to the antiferro- magnetic Heisenberg S = 1/2 ladder (two coupled chains). The temperature de- pendence of the magnetic susceptibility, specic heat, correlation length and nuclear spin relaxation rate 1/T1 are calculated. The results are compared to experiments on ladder-type compounds. Exact diagonalisation studies for a doped t-J ladder show hole pairing in the ground state. The excitation spectrum separates into a limited number of quasi- particles which carry charge +|e| and spin 1 and a triplet mode. At half-lling the 2 former vanish but the latter evolves continuously into the triplet band of the spin liquid. At low doping the quasiparticles form a dilute Fermi gas with a strong at- traction but simultaneously t

关键词： Quantum transfer matrix lanczos algorithm eigenvectors Low-Dimensional numerical methods Kondo lattice Extreme Eigenvalues CONSTRAINED FERMIONS look-ahead

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A Completed Theory of the Unsymmetric lanczos Process and Related algorithms. Part II

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SIAM Journal on Matrix Analysis and Applications 1994年第1期15卷 15-58页

作者： Martin H. Gutknecht

This paper is a continuation of Part I [M. H. Gutknecht, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 594–639], where the theory of the “unsymmetric” lanczos biorthogonalization (BO) algorithm and the corresponding iterative method BIORES for non-Hermitian linear systems was extended to the nongeneric case. The analogous extension is obtained here for the biconjugate gradient (or BIOMIN) method and for the related BIODIR method. Here, too, the breakdowns of these methods can be cured. As a preparation, mixed recurrence formulas are derived for a pair of sequences of formal orthogonal polynomials belonging to two adjacent diagonals in a nonnormal Padé table, and a matrix interpretation of these recurrences is developed. This matrix interpretation leads directly to a completed formulation of the progressive qd algorithm, valid also in the case of a nonnormal Padé table. Finally, it is shown how the cure for exact breakdown can be extended to near-breakdown in such a way that (in exact arithmetic) the well-conditioned formal orthogonal polynomials and the corresponding Krylov space vectors do not depend on the threshold specifying the near-breakdown.

关键词： 65F10 30E05 41A21 65F15 lanczos algorithm biconjugate gradient algorithm BIOMIN BIODIR breakdown formal orthogonal polynomial recurrence Padé approximation staircase quotient difference algorithm qd algorithm

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VARIANTS OF BICGSTAB FOR MATRICES WITH COMPLEX SPECTRUM

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 1993年第5期14卷 1020-1033页

作者： GUTKNECHT, MH

Recently Van der Vorst [SLAM J. Sci. Statist. Comput., 13 (1992), pp. 631-644] proposed for solving nonsymmetric linear systems Az = b a biconjugate gradient (BICG)-based Krylov space method called BICGSTAB that, like the biconjugate gradient squared (BICGS) method of Sonneveld, does not require matrix-vector multiplications with the transposed matrix A(T), and that has typically a much smoother convergence behavior than BICG and BICGS. Its nth residual polynomial is the product of the one of BICG (i.e., the nth lanczos polynomial) with a polynomial of the same degree with real zeros. Therefore, nonreal eigenvalues of A are not approximated well by the second polynomial factor. Here, the author presents for real nonsymmetric matrices a method BICGSTAB2 in which the second factor may have complex conjugate zeros. Moreover, versions suitable for complex matrices are given for both methods.

关键词： lanczos algorithm BICONJUGATE GRADIENT algorithm CONJUGATE GRADIENT SQUARED algorithm BICGSTAB FORMAL ORTHOGONAL POLYNOMIAL NONSYMMETRIC LINEAR SYSTEM KRYLOV SPACE METHOD

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A COMPLETED THEORY OF THE UNSYMMETRIC lanczos PROCESS AND RELATED algorithmS .1.

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 1992年第2期13卷 594-639页

作者： GUTKNECHT, MH

The theory of the "unsymmetric" lanczos biorthogonalization (BO) algorithm, which has so far been restricted to an essentially generic situation (characterized by the nonsingularity of the leading principal submatrices of the associated moment matrix or by the existence of a full set of regular formal orthogonal polynomials) is extended to the nongeneric case. The "serious" breakdowns due to the occurrence of two orthogonal right and left iteration vectors x(n) and y(n) can be overcome. For an operator of finite rank N the nongeneric BO algorithm, which generalizes the look-ahead lanczos algorithm of Parlett, Taylor, and Liu [Math. Comp., 44 (1985), pp. 105-124], terminates regularly in at most N steps, except when a very special situation depending on the initial vectors occurs;but even then the algorithm produces in at most N steps a block tridiagonal matrix whose blocks are either small or sparse and whose characteristic polynomial is the minimal polynomial of the restriction of the operator to an invariant subspace. Formulas are also derived for a nongeneric version of the corresponding linear equation solver BIORES (brief for BIORTHORES or lanczos/ORTHORES). The whole theory is developed as a consequence of known corresponding results on formal orthogonal polynomials and Pade approximants, for many of which new and simpler derivations are given.

关键词： lanczos algorithm BIORTHOGONALIZATION algorithm BO algorithm BIORES BICONJUGATE GRADIENT algorithm FORMAL ORTHOGONAL POLYNOMIAL RECURRENCE PADE APPROXIMATION CONTINUED FRACTION QUOTIENT DIFFERENCE algorithm QD algorithm

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REDUCTION TO TRIDIAGONAL FORM AND MINIMAL REALIZATIONS

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 1992年第2期13卷 567-593页

作者： PARLETT, BN UNIV CALIF BERKELEY DEPT ELECT ENGN & COMP SCI DIV COMP SCI BERKELEY CA 94720 USA UNIV OXFORD COMP LAB NUMER ANAL GRP OXFORD ENGLAND

This paper presents the theoretical background relevant to any method for producing a tridiagonal matrix similar to an arbitrary square matrix. Gragg's work on factoring Hankel matrices and the Kalman-Gilbert structure theorem from systems theory both find a place in the development. Tridiagonalization is equivalent to the application of the generalized Gram-Schmidt process to a pair of Krylov sequences. In Euclidean space proper normalization allows one to monitor a tight lower bound on the condition number of the transformation. The various possibilities for breakdown find a natural classification by the ranks of certain matrices. The theory is illustrated by some small examples and some suggestions for restarting are evaluated.

关键词： lanczos algorithm LINEAR SYSTEMS THEORY TRIDIAGONAL FORM MINIMAL REALIZATIONS

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CIRCULANT AND SKEWCIRCULANT MATRICES FOR SOLVING TOEPLITZ MATRIX PROBLEMS

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 1992年第3期13卷 767-777页

作者： HUCKLE, T

In recent papers, numerous authors studied the solutions of symmetric positive definite Toeplitz systems Tx = b by the conjugate gradient method for different families of circulant preconditioners C. In this paper new circulant/skewcirculant approximations are introduced to T and their properties are studied. The main interest is directed to the skewcirculant case. Furthermore, algorithms for computing the eigenvalues of T are formulated, based on the lanczos algorithm and Rayleigh quotient iteration. For some numerical examples the spectra of C-1 T are compared and the behaviour of the introduced eigenvalue algorithms is displayed.

关键词： TOEPLITZ MATRIX CIRCULANT MATRIX PRECONDITIONED CONJUGATE GRADIENT METHOD lanczos algorithm RAYLEIGH QUOTIENT ITERATION

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A CONVERGENCE ANALYSIS FOR NONSYMMETRIC lanczos algorithmS

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MATHEMATICS OF COMPUTATION 1991年第194期56卷 677-691页

作者： YE, Q Department of Mathematics and Statistics University of Calgary

A convergence analysis for the nonsymmetric lanczos algorithm is presented. By using a tridiagonal structure of the algorithm, some identities concerning Ritz values and Ritz vectors are established and used to derive approximation bounds. In particular, the analysis implies the classical results for the symmetric lanczos algorithm.

关键词： lanczos algorithm CONVERGENCE BOUND TRIDIAGONAL MATRICES

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