Orthogonal Symmetric Toeplitz Matrices

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

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.

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.

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

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.

We consider a symmetric Toeplitz matrix completion problem, of which the matrix possesses special row and column structures. It has wide applications in diverse areas and is well-known to be computationally NP-hard. This note presents an upper bound on the objective of minimizing the rank of the symmetric Toeplitz matrix in the completion problem based on conclusions from the trigonometric moment problem and the semi-infinite problem. We prove that the upper bound is less than twice the number of the active constraints of the associated semi-infinite problem. Moreover, it is less than twice the number of linear constraints of the problem.

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.

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.

Centro-symmetric and centro-skewsymmetric Toeplitz matrices and Bezoutians

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.

The set of Fortran subroutines discussed an implementation of the algorithm for finding the eigenvectors, x, and eigenvalues, lambda, such that Ax = lambdax, where A is a real skew-symmetric matrix or a real tridiagonal symmetric matrix with a constant diagonal. The algorithm uses only orthogonal similarity transformations and is believed to be the most efficient procedure available for computing all the eigenvalues or the complete eigensystem for the indicated classes of matrices. The three subroutines of the algorithm and their functions are described as follows: TRIZD--a subroutine that transforms an arbitrary real skew-symmetric matrix to skew-symmetric tridiagonal form by using orthogonal similarity transformations; IMZD--a subroutine that computes the eigenvalues and, optionally, the eigenvectors of a symmetric tridiagonal matrix with zeros on the diagonal or of a skew-symmetric tridiagonal matrix; TBAKZD--a subroutine that computes the eigenvectors of an arbitrary real skew-symmetric matrix by back-transforming the eigenvectors of the corresponding skew-symmetric tridiagonal matrix determined by TRIZD. The subroutines TRIZD, IMZD, and TBAKZD have been tested extensively on an IBM 360/91 computer using double precision arithmetic. Complete subroutine listings are available. (RWR)

We study an iterative, locally quadratically convergent algorithm for solving Toeplitz systems of equations from [R. P. Brent, F. G. Gustavson and D. Y. Y. Yun. “Fast solution of Toeplitz systems of equations and computation of Padé approximations”,J. Algorithms, 1:259–295, 1980]. We introduce a new iterative algorithm that is locally quadratically convergent when used to solve symmetric positive definite Toeplitz systems. We present a set of numerical experiments on randomly generated symmetric positive definite Toeplitz matrices. In these experiments, our algorithm performed significantly better than the previously proposed algorithm.

Several fast algorithms for the recovery of the impedance and reflection coefficients of a discrete asymmetric lossy medium are presented. The data required are the reflection response to an impulsive input on the surface, and the transmission response to an impulsive input incident from below. It is shown that these efficient algorithms are related to the generalized Levinson algorithm, the Schur algorithm, and the QR lattice algorithm for solutions of symmetric and asymmetric Toeplitz matrices equations. These unified, conceptual connections aid in the understanding of the symmetric and asymmetric Toeplitz matrix algorithms in the linear prediction literature. >