595 research outputs found
Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors
Motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors
and their generating vectors in this paper. Hilbert tensors are symmetric
Cauchy tensors. An even order symmetric Cauchy tensor is positive semi-definite
if and only if its generating vector is positive. An even order symmetric
Cauchy tensor is positive definite if and only if its generating vector has
positive and mutually distinct entries. This extends Fiedler's result for
symmetric Cauchy matrices to symmetric Cauchy tensors. Then, it is proven that
the positive semi-definiteness character of an even order symmetric Cauchy
tensor can be equivalently checked by the monotone increasing property of a
homogeneous polynomial related to the Cauchy tensor. The homogeneous polynomial
is strictly monotone increasing in the nonnegative orthant of the Euclidean
space when the even order symmetric Cauchy tensor is positive definite.
Furthermore, we prove that the Hadamard product of two positive semi-definite
(positive definite respectively) symmetric Cauchy tensors is a positive
semi-definite (positive definite respectively) tensor, which can be generalized
to the Hadamard product of finitely many positive semi-definite (positive
definite respectively) symmetric Cauchy tensors. At last, bounds of the largest
H-eigenvalue of a positive semi-definite symmetric Cauchy tensor are given and
several spectral properties on Z-eigenvalues of odd order symmetric Cauchy
tensors are shown. Further questions on Cauchy tensors are raised
The structured distance to singularity of a symmetric tridiagonal Toeplitz matrix
This paper is concerned with the distance of a symmetric tridiagonal Toeplitz
matrix to the variety of similarly structured singular matrices, and with
determining the closest matrix to in this variety. Explicit formulas are
presented, that exploit the analysis of the sensitivity of the spectrum of
with respect to structure-preserving perturbations of its entries.Comment: 16 pages, 5 Figure
Preconditioned Lanczos Methods for the Minimum Eigenvalue of a Symmetric Positive Definite Toeplitz Matrix
In this paper, we apply the preconditioned Lanczos (PL) method to compute the minimum eigenvalue of a symmetric positive definite Toeplitz matrix. The sine transform-based preconditioner is used to speed up the convergence rate of the PL method. The resulting method involves only Toeplitz and sine transform matrix-vector multiplications and hence can be computed efficiently by fast transform algorithms. We show that if the symmetric Toeplitz matrix is generated by a positive -periodic even continuous function, then the PL method will converge sufficiently fast. Numerical results including Toeplitz and non-Toeplitz matrices are reported to illustrate the effectiveness of the method.published_or_final_versio
Estimates for the spectral condition number of cardinal B-spline collocation matrices
The famous de Boor conjecture states that the condition of the polynomial B-spline collocation matrix at the knot averages is bounded independently of the knot sequence, i.e., it depends only on the spline degree.
For highly nonuniform knot meshes, like geometric meshes, the conjecture is known to be false. As an effort towards finding an answer for uniform meshes, we investigate the spectral condition number of cardinal B-spline collocation matrices. Numerical testing strongly suggests that the conjecture is true for cardinal B-splines
Matrix algebras in optimal preconditioning
AbstractThe theory and the practice of optimal preconditioning in solving a linear system by iterative processes is founded on some theoretical facts understandable in terms of a class V of spaces of matrices including diagonal algebras and group matrix algebras. The V-structure lets us extend some known crucial results of preconditioning theory and obtain some useful information on the computability and on the efficiency of new preconditioners. Three preconditioners not yet considered in literature, belonging to three corresponding algebras of V, are analyzed in detail. Some experimental results are included
Eigenfilters: A new approach to least-squares FIR filter design and applications including Nyquist filters
A new method of designing linear-phase FIR filters is proposed by minimizing a quadratic measure of the error in the passband and stopband. The method is based on the computation of an eigenvector of an appropriate real, symmetric, and positive-definite matrix. The proposed design procedure is general enough to incorporate both time- and frequency-domain constraints. For example, Nyquist filters can be easily designed using this approach. The design time for the new method is comparable to that of Remez exchange techniques. The passband and stopband errors in the frequency domain can be made equiripple by an iterative process, which involves feeding back the approximation error at each iteration. Several numerical design examples and comparisons to existing methods are presented, which demonstrate the usefulness of the present approach
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