Inverse Eigenvalue Problems for Symmetric Toeplitz Matrices

The inverse eigenvalue problem for symmetric matrices (IEP) is formulated as a system of two matrix equations. For solving the system a variation of Newton's method is used which has been proposed by Fusco and Zecca [Calcolo XXIII (1986), pp. 285–303] for the simultaneous computation of eigenvalues and eigenvectors of a given symmetric matrix. an iteration step of this method consists of a Newton step followed by an orthonormalization with the consequence that each iterate satisfies one of the given equations. The method converges locally y quadratically to regular solutions. The algorithm and some numerical examples are presented. In addition, it is shown that the so called Method III proposed by Friedland, Nocedal, and Overton [SIAM J. Numer. Anal.,24 (1987), pp. 634–667] for solving IEP may be constructed similarly to the method presented here

Forward stable eigenvalue decomposition of rank-one modifications of diagonal matrices

In this paper, the symmetric positive semidefinite Procrustes problem is considered. By using matrix inner product and matrix decomposition theory, an explicit expression of the solution is given. Based on the explicit expression given in this paper, it is easy to derive the explicit expression of the solution given by Woodgate [K.G. Woodgate. Least-squares solution of F = PG over positive semidefinite symmetric P . Linear Algebra Appl., 245:171–190, 1996.] and by Liao [A.P. Liao. On the least squares problem of a matrix equation. J. Comput. Math., 17:589–594, 1999.] for the Procrustes problem in some special cases. The explicit expression given in this paper also shows that the solution of the special inverse eigenvalue problem considered by Zhang [L. Zhang. A class of inverse eigenvalue problem for symmetric nonnegative definite matrices. J. Hunan Educational Inst., 2:11–17, 1995 (in Chinese).] can be computed exactly. Examples to illustrate the correctness of the theory results are given.

Precise asymptotic expansions for the eigenvalues of a Toeplitz matrix T_n(f), as the matrix size n tends to infinity, have recently been obtained, under suitable assumptions on the associated generating function f. A restriction is that f has to be polynomial, monotone, and scalar-valued. In this paper we focus on the case where mathbf {f} is an stimes s matrix-valued trigonometric polynomial with sge 1, and T_n(mathbf {f}) is the block Toeplitz matrix generated by mathbf {f}, whose size is N(n,s)=sn. The case s=1 corresponds to that already treated in the literature. We numerically derive conditions which ensure the existence of an asymptotic expansion for the eigenvalues. Such conditions generalize those known for the scalar-valued setting. Furthermore, following a proposal in the scalar-valued case by the first author, Garoni, and the third author, we devise an extrapolation algorithm for computing the eigenvalues of banded symmetric block Toeplitz matrices with a high level of accuracy and a low computational cost. The resulting algorithm is an eigensolver that does not need to store the original matrix, does not need to perform matrix-vector products, and for this reason is called matrix-less. We use the asymptotic expansion for the efficient computation of the spectrum of special block Toeplitz structures and we provide exact formulae for the eigenvalues of the matrices coming from the mathbb {Q}_p Lagrangian Finite Element approximation of a second order elliptic differential problem. Numerical results are presented and critically discussed.

In this note we review the technique for rational Chebyshev approximation as given by the generalization of the Remes algorithm and point out its connection to the generalized eigenvalue problem for symmetric Hankel matrices. By a classical example and some recent work it can be shown that interpolation with rational functions (even) on equidistant knots may lead to results that are superior to Chebyshev approximation by means of polynomials for comparable numerical work.

20aEA-10 Cavity analysis of the first eigenvalue problem of large symmetric random sparse matrices

We consider numerical methods for the extreme eigenvalue problem of large scale symmetric positive definite matrices. By the variational principle, the extreme eigenvalue can be obtained by minimizing some unconstrained optimization problem. Firstly, we propose two adaptive nonmonotone Barzilai-Borwein-like methods for the unconstrained optimization problem. Secondly, we prove the global convergence of the two algorithms under some conditions. Thirdly, we compare our methods with eigsand the power method for the standard test problems from the UF Sparse Matrix Collection. The primary numerical experiments indicate that the two algorithms are promising.

Bounds for the condition number are obtained which indicate the dependence of the conditioning of a symmetric positive definite Toeplitz matrix on the sign distribution of its reflection coefficients. This dependence is illustrated with numerical examples. Similar bounds for symmetric positive definite block Toeplitz matrices are also presented.

A Schur-based algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite Toeplitz matrix

This article refines a result of Delsarte, Genin, Kamp (Circuits Syst Signal Process 3:207–223, 1984), and Delsarte and Genin (Springer Lect Notes Control Inf Sci 58:194–213, 1984), regarding the number of zeros on the unit circle of eigenpolynomials of complex Hermitian Toeplitz matrices and generalized Caratheodory representations of such matrices. This is achieved by exploring a key observation of Schur (Uber einen Satz von C. Caratheodory. Sitzungsberichte der Koniglich Preusischen Akademie der Wissenschaften, pp. 4–15, 1912) stated in his proof of a famous theorem of Caratheodory (Rendiconti del Circolo Matematico di Palermo 32:193–217, 1911). In short, Schur observed that companion matrices corresponding to eigenpolynomials of Hermitian Toeplitz matrices H define isometries with respect to (spectrum shifted) submatrices of H. Looking at possible normal forms of these isometries leads directly to the results. This geometric, conceptual approach can be generalized to Hermitian or symmetric Toeplitz matrices over arbitrary fields. Furthermore, as a byproduct, Iohvidov’s law in the jumps of the ranks and the connection between the Iohvidov parameter and the Witt index are established for such Toeplitz matrices.

An extension of Levinson’s algorithm for solving linear systems with symmetric indefinite Toeplitz matrices is presented. This new algorithm is able to “look ahead” and, if necessary, use block Gaussian elimination to skip all ill-conditioned leading principal submatrices encountered during the recursive processing. This makes the new algorithm numerically stable for a broader class of symmetric Toeplitz matrices than the standard Levinson algorithm. In addition, a reliable condition number estimate is produced. The overhead is typically small.

This paper develops the geometry of locally bounded rational functions on non-singular real algebraic varieties. First various basic geometric and algebraic results regarding these functions are established in any dimension, culminating with a version of Łojasiewicz’s inequality. The geometry is further developed for the case of dimension 2, where it can be shown that there exist many of the usual correspondences between the algebra and geometry of these functions that one expects from complex algebraic geometry and from other classes of functions in real algebraic geometry such as regulous functions.

Centro-symmetric and centro-skewsymmetric Toeplitz matrices and Bezoutians

Real algebraic geometry deals with the solution set of (possibly quantified) systems of polynomial equations and/or inequalities over the real numbers, which arise frequently in science and engineering. Main concern in real algebraic geometry is to determine the properties of the solution sets such as non-emptiness, dimension and quantifier free description as a semi-algebraic set. Such tasks are carried out by symbolic and algebraic algorithms:cylindrical algebraic decomposition (CAD) or quantifier elimination (QE). Various algorithms and deep complexity results about CAD and QE have been studied during the last several decades. Moreover, practically efficient software systems of QE have been developed and also are applied to many nontrivial application problems. In this talk we explain several algorithms of CAD and QE together with their engineering applications.

Let A be an n x n banded block Toeplitz matrix of bandwidth k with m x m blocks having entries in a field F. We present algorithms for computing p(A) = det(A -XI) as well as the ratio p(A)/p'(A), where p'(A) is the first derivative of p(A) with respect to A, in roughly (3/2)fc2 logra + 0(k3) block multiplications.If the field F supports FFT, then the cost is reduced to 0((m2fclogfc + m3k) logn + k3m3) scalar multiplications.The algorithms generalize an algorithm given by W. Trench for computing p(A) in the case m = 1 in roughly A;logn + 0(k3) multiplications and rely on powering a companion matrix associated with the linear recurrence relation representing the original problem. Introduction.Toeplitz matrices, either with scalar entries or with blocks, arise in a variety of problems of Applied Mathematics; a wide literature deals with the problem of their inversion.Recently, some interest has been focused on the eigenvalue problem for banded Toeplitz matrices.In [1] and [2], spectral properties are expressed in terms of a suitable polynomial associated with a banded Toeplitz matrix A, and parallel algorithms have been devised for the computation of the characteristic polynomial p(A) = det(^4 -XI) at a point A as well as of the ratio p(X)/p'(X), where p'(A) is the first derivative of p(A).More recently, an algorithm for the computation of p(A) has been proposed by Trench [8]; its cost is roughly A:log2 n + 0(k3) multiplications, where n is the dimension of the matrix and k is its bandwidth.That algorithm works over any algebraically closed field (even though it was originally presented over the complex field), and it is assumed that the zeros of a suitable fcth degree polynomial, associated with the banded Toeplitz matrix, have been precomputed together with their multiplicities.In this paper we consider the eigenvalue problem for banded block Toeplitz matrices and present two new algorithms for the computation of p(A).The first algorithm, which is obtained dealing with the problem in terms of a linear difference ma-