The lanczos algorithm is most commonly used in approximating a small number of extreme eigenvalues and eigenvectors for symmetric large sparse matrices. Main memory accesses for shared memory systems or global communi...
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The lanczos algorithm is most commonly used in approximating a small number of extreme eigenvalues and eigenvectors for symmetric large sparse matrices. Main memory accesses for shared memory systems or global communications (synchronizations) in message passing systems decrease the computation speed. In this paper, the standard lanczos algorithm is restructured so that only one synchronization point is required;that is, one global communication in a message passing distributed-memory machine or one global memory sweep in a shared-memory machine per each iteration is required. We also introduce the s-step lanczos method for finding a few eigenvalues of symmetric large sparse matrices in a similar way to the s-step Conjugate Gradient method [2], and we prove that the s-step method generates reduction matrices which are similar to reduction matrices generated by the standard method. One iteration of the s-step lanczos algorithm corresponds to s iterations of the standard lanczos algorithm. The s-step method has improved data locality, minimized global communication and has superior parallel properties to the standard method. These algorithms are implemented on a 64-node NCUBE/seven hypercube and a CRAY-2, and performance results are presented.
The weighted checksum scheme has been proposed as a low-cost fault tolerant procedure for parallel matrix computations. To guarantee multiple error detection and correction, the chosen weight vectors must satisfy some...
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The weighted checksum scheme has been proposed as a low-cost fault tolerant procedure for parallel matrix computations. To guarantee multiple error detection and correction, the chosen weight vectors must satisfy some very specific properties about linear independence. However, previous weight generating methods that fulfill the independence criteria have troubles with numerical overflow. We will present a new scheme that generates weight vectors via Chebyshev polynomials to meet the requirements about independence and to avoid the difficulties with overflow.
A lanczos-type method is presented for nonsymmetric sparse linear systems as arising from discretisations of elliptic partial differential equations. The method is based on a polynomial variant of the conjugate gradie...
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A lanczos-type method is presented for nonsymmetric sparse linear systems as arising from discretisations of elliptic partial differential equations. The method is based on a polynomial variant of the conjugate gradients algorithm. Although related to the so-called bi-conjugate gradients (Bi-CG) algorithm, it does not involve adjoint matrix-vector multiplications, and the expected convergence rate is about twice that of the Bi-CG algorithm. Numerical comparison is made with other solvers, testing the method on a family of convection diffusion equations, on various grids, and with the use of two different preconditioning methods. Upwind as well as central differencing is used in the experiments.
An iterative method based on lanczos bidiagonalization is developed for computing regularized solutions of large and sparse linear systems, which arise from discretizations of ill-posed problems in partial differentia...
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An iterative method based on lanczos bidiagonalization is developed for computing regularized solutions of large and sparse linear systems, which arise from discretizations of ill-posed problems in partial differential or integral equations. Determination of the regularization parameter and termination criteria are discussed. Comments are given on the computational implementation of the algorithm.
An algorithm for computing the eigenvectors corresponding to the m algebraically smallest or largest eigenvalues of an $n \times n$ symmetric matrix ${\bf A}$ is described. The algorithm consists of repeated applicati...
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An algorithm for computing the eigenvectors corresponding to the m algebraically smallest or largest eigenvalues of an $n \times n$ symmetric matrix ${\bf A}$ is described. The algorithm consists of repeated applications of the Rayleigh-Ritz procedure to a sequence of subspaces of dimension $m + 1$ which converges to the desired subspace. The method is closely related to the lanczos method, but requires a constant amount of computation at each iteration. Applications of the algorithm include the adaptive covariance eigenstructure computation, in which the matrix ${\bf A}$ can change while the algorithm is in progress.
We present methods for computing the controllable space of a Linear Dynamic System. The methods are based on orthogonalizing Krylov sequences and can use much less storage than other methods with little or no loss of ...
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We present methods for computing the controllable space of a Linear Dynamic System. The methods are based on orthogonalizing Krylov sequences and can use much less storage than other methods with little or no loss of numerical stability.
This paper discusses the use of the linear conjugate-gradient method (developed via the lanczos method) in the solution of large-scale unconstrained minimization problems. At each iteration of a Newton-type method, th...
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This paper discusses the use of the linear conjugate-gradient method (developed via the lanczos method) in the solution of large-scale unconstrained minimization problems. At each iteration of a Newton-type method, the direction of search is defined as the solution of a quadratic subproblem. When the number of variables is very large, this subproblem may be solved using the linear conjugate-gradient method of Hestenes and Stiefel. We show how the equivalent lanczos characterization of the linear conjugate-gradient method may be exploited to define a modified Newton method which can be applied to problems that do not necessarily have positive-definite Hessian matrices at all points of the region of interest. This derivation also makes it possible to compute a negative-curvature direction at a stationary point.
This paper presents a fast method for estimating dominant harmonics in a sequence of data. In a stochastic sense, the proposed method finds the autoregressive scheme with a pure point spectrum that best describes the ...
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This paper presents a fast method for estimating dominant harmonics in a sequence of data. In a stochastic sense, the proposed method finds the autoregressive scheme with a pure point spectrum that best describes the data, while from a deterministic point of view, the method is a special case of the lanczos algorithm for finding eigenvalues of a symmetric matrix. Eigenvalue approximations come into play because every circulant matrix is diagonalized by the discrete Fourier transform matrix, and so using the lanczos algorithm with the given data as the initial vector on a simple circulant matrix, the eigenvalues that are first approximated are the eigenvalues corresponding to eigenvectors which are dominant in the initial vector. It is shown that this method is related to “lattice methods” for linear prediction and to Prony’s method for exponential approximation.
algorithms are presented for computing some of the eigenvalues and their associated eigenvectors of the quadratic $\lambda $-matrix$M\lambda ^2 + C\lambda + K$. M, C and K are assumed to have special symmetrytype pr...
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algorithms are presented for computing some of the eigenvalues and their associated eigenvectors of the quadratic $\lambda $-matrix$M\lambda ^2 + C\lambda + K$. M, C and K are assumed to have special symmetrytype properties which insure that theory analogous to the standard symmetric eigenproblem exists. The algorithms are based on a generalization of the Rayleigh quotient and the, lanczos method for computing eigenpairs of standard symmetric eigenproblems. Monotone quadratic convergence of the basic method is proved. Test examples are presented.
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