Abstract. It is proved that the eigenvalues of the Jacobi Tau method for the second derivative operator with Dirichlet boundary conditions are real, negative and distinct for a range of the Jacobi parameters. Special emphasis is placed on the symmetric case of the Gegenbauer Tau method where the range of parameters included in the theorems can be extended and characteristic polynomials given by...

The linear finite element approximation of a general linear diffusion problem with arbitrary anisotropic meshes is considered. The conditioning of the resultant stiffness matrix and the Jacobi preconditioned stiffness matrix is investigated using a density function approach proposed by Fried in 1973. It is shown that the approach can be made mathematically rigorous for general domains and used ...

The Jacobi–Davidson method is a popular technique to compute a few eigenpairs of large sparse matrices. Its introduction, about a decade ago, was motivated by the fact that standard eigensolvers often require an expensive factorization of the matrix to compute interior eigenvalues. Such a factorization may be infeasible for large matrices as arise in today’s large-scale simulations. In the Jaco...

We consider a class of Jacobi matrices with unbounded entries in the so-called critical (double root, Jordan block) case. prove formula which relates spectral density matrix to asymptotics orthogonal polynomials associated it.

Jacobi-Davidson methods can efficiently compute a few eigenpairs of a large sparse matrix. Block variants of JacobiDavidson are known to be more robust than the standard algorithm, but they are usually avoided as the total number of floating point operations increases. We present the implementation of a block Jacobi-Davidson solver and show by detailed performance engineering and numerical expe...

Abstract -In this paper, the problem of H∞ filtering design is studied for nonlinear sampled-data systems using the Takagi-Sugeno (T-S) fuzzy model approach. Traditionally, the sufficient conditions for the existence of such H∞ filter are characterized in terms of the solution of a differential Hamilton-Jacobi inequality with jumps, which is equivalent to solving the partial differential inequa...

In latent semantic indexing, the addition of documents (or the addition of terms) to some already processed text collection leads to the updating of the best rank-k approximation of the term-document matrix. The computationally most intensive task in this updating is the computation of the singular value decomposition (SVD) of certain square matrix, which is upper or lower triangular, and conta...

We give a simple example of non-uniqueness in the inverse scattering for Jacobi matrices: roughly speaking S-matrix is analytic. Then, multiplying a reflection coefficient by an inner function, we repair this matrix in such a way that it does uniquely determine a Jacobi matrix of Szegö class; on the other hand the transmission coefficient remains the same. This implies the statement given in th...

which is obtained from eq. (1) by replacing Uk with A 2 k and rescaling t, while the cyclic boundary condition case is discussed in Goodman & Lax [2] for even N . Moser notes the existence of dN/2e polynomials of Ak that are invariant and states that they are independent. They are the coefficients of the characteristic equation of an N + 1 by N + 1 zero-diagonal symmetric Jacobi matrix whose te...

This paper studies the problem of Non-symmetric Joint Diagonalization (NsJD) of matrices, namely, jointly diagonalizing a set of complex matrices by one matrix multiplication from the left and one multiplication with possibly another matrix from the right. In order to avoid ambiguities in the solutions, these two matrices are restricted to lie in the complex oblique manifold. We derive a necess...