Sensitivity analysis of the largest dependent eigenvalue functions of eigensystems

This paper presents and discusses the first order sensitivity of the largest eigenvalue functions of eigensystems. The analysis is based on investigating the entries of the matrices associated with the finite element formulations. These entries are continuously differentiable functions, but the dependent eigenvalue functions may be continuous nondifferentiable. The main emphasis of this contribution is the derivation and the investigation of this analysis under the assumption that the generalized dependent eigenvalue functions are continuous nondifferentiable.     

Bounding the spectrum of large Hermitian matrices

Estimating upper bounds of the spectrum of large Hermitian matrices has long been a problem with both theoretical and practical significance. Algorithms that can compute tight upper bounds with minimum computational cost will have applications in a variety of areas. We present a practical algorithm that exploits k-step Lanczos iteration with a safeguard step. The k is generally very small, say 5-8, regardless of the large dimension of the matrices. This makes the Lanczos iteration economical. The safeguard step can be realized with marginal cost by utilizing the theoretical bounds developed in this paper. The bounds establish the theoretical validity of a previous bound estimator that has been successfully used in various applications. Moreover, we improve the bound estimator which can now provide tighter upper bounds with negligible additional cost.     

Comment on Formulating and Generalizing Dirac's, Proca's, and Maxwell's Equations with Biquaternions or Clifford Numbers

Many difficulties of interpretation met by contemporary researchers attempting to recast or generalize Dirac's, Proca's, or Maxwell's theories using biquaternions or Clifford numbers have been encountered long ago by a number of physicists including Lanczos, Proca, and Einstien. In the modern approach initiated by Gürsey, these difficulties are solved by recognizing that most generalizations lead to theories describing superpositions of particles of different intrinsic spin and isospin, so that the correct interpretation emerges from the requirement of full Poincaré covariance, including space and time reversal, as well as reversion and gauge invariance. For instance, the doubling of the number of solutions implied by the simplest generalization of Dirac's equation (i.e., Lanczos's equation) can be interpreted as isospin. In this approach, biquaternions and Clifford numbers become powerful opportunities to formulate the Standard Model of elementary particles, as well as many of its possible generalizations, in very elegant and compact ways.     

On the “Equivalence” of the Maxwell and Dirac Equations

It is shown that Maxwell's equation cannot be put into a spinor form that is equivalent to Dirac's equation. First of all, the spinor in the representation of the electromagnetic field bivector depends on only three independent complex components whereas the Dirac spinor depends on four. Second, Dirac's equation implies a complex structure specific to spin 1/2 particles that has no counterpart in Maxwell's equation. This complex structure makes fermions essentially different from bosons and therefore insures that there is no physically meaningful way to transform Maxwell's and Dirac's equations into each other.     

Computation of Optimal Backward Perturbation Bounds for Large Sparse Linear Least Squares Problems

In this note we propose an algorithm based on the Lanczos bidiagonalization to approximate the backward perturbation bound for the large sparse linear squares problem. The algorithm requires ((m + n)l) operations where m and n are the size of the matrix under consideration and l <#60;<#60; min(m,n). The import of the proposed algorithm is illustrated by some examples coming from the Harwell-Boeing collection of test matrices.This revised version was published online in October 2005 with corrections to the Cover Date.     

Lanczos,Householder transformations,and implicit deflation for fast and reliable dominant singular subspace computation

Many applications, such as subspace‐based models in information retrieval and signal processing, require the computation of singular subspaces associated with the k dominant, or largest, singular values of an m×n data matrix A, where k?min(m,n). Frequently, A is sparse or structured, which usually means matrix–vector multiplications involving A and its transpose can be done with much less than ??(mn) flops, and A and its transpose can be stored with much less than ??(mn) storage locations. Many Lanczos‐based algorithms have been proposed through the years because the underlying Lanczos method only accesses A and its transpose through matrix–vector multiplications. We implement a new algorithm, called KSVD, in the Matlab environment for computing approximations to the singular subspaces associated with the k dominant singular values of a real or complex matrix A. KSVD is based upon the Lanczos tridiagonalization method, the WY representation for storing products of Householder transformations, implicit deflation, and the QR factorization. Our Matlab simulations suggest it is a fast and reliable strategy for handling troublesome singular‐value spectra. Copyright © 2001 John Wiley & Sons, Ltd.     

Application of the Lanczos algorithm for solving the linear systems that occur in continuation problems

We study the Lanczos method for solving symmetric linear systems with multiple right‐hand sides. First, we propose a numerical method of implementing the Lanczos method, which can provide all approximations to the solution vectors of the remaining linear systems. We also seek possible application of this algorithm for solving the linear systems that occur in continuation problems. Sample numerical results are reported. Copyright © 2002 John Wiley & Sons, Ltd.     

An adaptive block Lanczos algorithm

A generalization of the block Lanczos algorithm will be given, which allows the block size to be increased during the iteration process. In particular, the algorithm can be implemented with the block size chosen adaptively according to clustering of Ritz values. In this way, multiple and clustered eigenvalues can be found and the difficulty of choosing the block size is eased. Residual bounds for clustered eigenvalues are given. Numerical examples are presented to illustrate the adaptive algorithm.Research supported by a grant from Natural Sciences and Engineering Research Council of Canada.     

Preconditioned Lanczos method for generalized Toeplitz eigenvalue problems

We employ the sine transform-based preconditioner to precondition the shifted Toeplitz matrix _An−ρ_Bn$A_n-\rho B_n$ involved in the Lanczos method to compute the minimum eigenvalue of the generalized symmetric Toeplitz eigenvalue problem _Anx=λ_Bnx$A_{n}x=\lambda B_{n}x$, where _An$A_n$ and _Bn$B_n$ are given matrices of suitable sizes. The sine transform-based preconditioner can improve the spectral distribution of the shifted Toeplitz matrix and, hence, can speed up the convergence rate of the preconditioned Lanczos method. The sine transform-based preconditioner can be implemented efficiently by the fast transform algorithm. A convergence analysis shows that the preconditioned Lanczos method converges sufficiently fast, and numerical results show that this method is highly effective for a large matrix.     

A Convergence Result for Some Krylov–Tikhonov Methods in Hilbert Spaces

In this paper, we present a convergence result for some Krylov projection methods when applied to the Tikhonov minimization problem in its general form. In particular, we consider the method based on the Arnoldi algorithm and the one based on the Lanczos bidiagonalization process.