An upper bound on the minimum rank of a symmetric Toeplitz matrix completion problem

Computer Solution of Large Linear Systems

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.

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

A derivation of the Levinson algorithm for solving linear systems with symmetric positive definite Toeplitz matrix

We show that the number of orthogonal and symmetric Toeplitz matrices of a given order is finite and determine all these matrices. In this way we also obtain a description of the set of all symmetric Toeplitz matrices whose spectrum is a prescribed doubleton.

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.

The inverse eigenvalue problems for symmetric Toeplitz matrices with complex-valued (IEPSCTM) and real-valued (IEPSRTM) entries are studied. The main tools are complex and real algebraic geometry. In the complex case it is shown that the IEPSCTM is solvable for most spectra and always solvable for $n\leqq 4$. In the real case the natural decomposition of the space of all $n \times n$ real symmetric Toeplitz matrices to a finite number of connected components of matrices with a simple spectrum is given. It is then shown that the solvability of the IEPSRTM for all spectra can be deduced if the corresponding map to the IEPSRTM has a nonzero degree for at least one component. This is the case for $n\leqq 4$, which gives an alternative proof to Delsarte and Genin’s results. The IEPSRTM for odd Toeplitz matrices with real-valued entries is also considered.

Many natural phenomena can be modeled by a second-order dynamical system , where stands for an appropriate state variable and M, C, K are time-invariant, real and symmetric matrices. In contrast to the classical inverse vibration problem where a model is to be determined from natural frequencies corresponding to various boundary conditions, the inverse mode problem concerns the reconstruction of the coefficient matrices (M, C, K) from a prescribed or observed subset of natural modes. This paper set forth a mathematical framework for the inverse mode problem and resolves some open questions raised in the literature. In particular, it shows that given merely the desirable structure of the spectrum, namely given the cardinalities of real or complex eigenvalues but not of the actual eigenvalues, the set of eigenvectors can be completed via solving an under-determined nonlinear system of equations. This completion suffices to construct symmetric coefficient matrices (M, C, K) whereas the underlying system can have arbitrary eigenvalues. Generic conditions under which the real symmetric quadratic inverse mode problem is solvable are discussed. Applications to important tasks such as updating models without spill-over or constructing models with positive semi-definite coefficient matrices are discussed.

Stabilizing Trench's algorithm to invert symmetric Toeplitz matrices

In this correspondence, a fast computational method is proposed to approximate the Karhunen-Loeve transform (KLT) for the covariance matrix of the autoregressive process. A fast algorithm which reduces the computation of eigenvalues of an N/spl times/N symmetric Toeplitz matrix from O(N/sup 3/) in KLT to N/sup 2/ is further developed. Experimental results demonstrate that the performance of the fast algorithm is very close to the KLT in eigenvalue computation and in energy constrained signal subspace speech enhancement for speech recognition in a car environment.

Despite the fact that symmetric Toeplitz matrices can have arbitrary eigenvalues, the numerical construction of such a matrix having prescribed eigenvalues remains to be a challenge. A two-step method using the continuation idea is proposed in this paper. The first step constructs a centro-symmetric Jacobi matrix with the prescribed eigenvalues in finitely many steps. The second step uses the Cayley transform to integrate flows in the linear subspace of skew-symmetric and centro-symmetric matrices. No special geometric integrators are needed. The convergence analysis is illustrated for the case of n = 3. Numerical examples are presented.

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.

2-Adic valuations of certain ratios of products of factorials and applications

Diagonalization of complex symmetric matrices: Generalized Householder reflections, iterative deflation and implicit shifts

A novel method for computing the minimal eigenvalue of a symmetric positive-definite Toeplitz matrix is presented. Similar to the algorithm of Cybenko and Van Loan, it is a combination of bisection and a root finding method. Both phases of the method are accelerated considerably by rational Hermitian interpolation of the secular equation. For randomly generated test problems of dimension 800 the average number of linear systems which must be solved to determine the smallest eigenvalue is 6.6, which reduces the computational cost of the method of Cybenko and Van Loan to approximately 35%. The method includes a rigorous error bound.