Condition Numbers of Toeplitz and Block Toeplitz Matrices

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

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.

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

Various preconditioners for symmetric positive-definite (SPD) Toeplitz matrices in circulant matrix form have recently been proposed. The spectral properties of the preconditioned SPD Toeplitz matrices have also been studied. In this research, Strang's preconditioner SN and our preconditioner KN are applied to an N N nonsymmetric (or nonhermitian) Toeplitz system TNX b. For a large class of Toeplitz matrices, it is proved that the singular values of SITN and KvlTN are clustered around unity except for a fixed number independent ofN. IfTN is additionally generated by a rational function, the eigenvalues of SITN and KITN can be characterized directly. Let the eigenvalues ofSITN andKITN be classified into the outliers and the clustered eigenvalues dependingon whether they converge to I asymptotically. Then, the number ofoutliers depends on the order of the rational generating function, and the clustering radius is proportional to the magnitude of the last elements in the generating sequence used to construct the preconditioner. Numerical experiments are provided to illustrate our theoretical study.

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 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.

In this paper we will adapt a known method for diagonal scaling of symmetric positive definite tridiagonal matrices towards the semiseparable case. Based on the fact that a symmetric, positive definite tridiagonal matrix $$T$$ satisfies property A, one can easily construct a diagonal matrix $$\hat{D}$$ such that $$\hat{D}T\hat{D}$$ has the lowest condition number over all matrices $$DTD$$ , for any choice of diagonal matrix $$D$$ . Knowing that semiseparable matrices are the inverses of tridiagonal matrices, one can derive similar properties for semiseparable matrices. Here, we will construct the optimal diagonal scaling of a semiseparable matrix, based on a new inversion formula for semiseparable matrices. Some numerical experiments are performed. In a first experiment we compare the condition numbers of the semiseparable matrices before and after the scaling. In a second numerical experiment we compare the scalability of matrices coming from the reduction to semiseparable form and matrices coming from the reduction to tridiagonal form.

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 problem of completing a partially prescribed Toeplitz matrix to a positive definite Toeplitz matrix is considered. The main result is a characterization of the patterns of specified entries that guarantee the existence of a positive definite Toeplitz completion of a partial positive definite Toeplitz matrix. This characterization resolves affirmatively a conjecture of C. R. Johnson and L. Rodman.

In this paper, we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations. The finite difference method is employed to approximate the multi-dimensional Riesz fractional derivatives, which will generate symmetric positive definite ill-conditioned multi-level Toeplitz matrices. The preconditioned conjugate gradient method with a preconditioner based on the sine transform is employed to solve the resulting linear system. Theoretically, we prove that the spectra of the preconditioned matrices are uniformly bounded in the open interval (1/2,3/2) and thus the preconditioned conjugate gradient method converges linearly. The proposed method can be extended to multi-level Toeplitz matrices generated by functions with zeros of fractional order. Our theoretical results fill in a vacancy in the literature. Numerical examples are presented to demonstrate our new theoretical results in the literature and show the convergence performance of the proposed preconditioner that is better than other existing preconditioners.

Centro-symmetric and centro-skewsymmetric Toeplitz matrices and Bezoutians

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.

The paper is concerned with a sequence of constants which appear in several problems. These problems include the minimal eigenvalue of certain positive definite Toeplitz matrices, the minimal eigenvalue of some higher-order ordinary differential operators, the norm of the Green kernels of these operators, the best constant in a Wirtinger-Sobolev inequality, and the conditioning of a special least squares problem. The main result of the paper gives the asymptotics of this sequence.

<abstract><p>In this paper, fast numerical methods for solving the real quasi-symmetric Toeplitz linear system are studied in two stages. First, based on an order-reduction algorithm and the factorization of Toeplitz matrix inversion, a sequence of linear systems with a constant symmetric Toeplitz matrix are solved. Second, two new fast algorithms are employed to solve the real quasi-symmetric Toeplitz linear system. Furthermore, we show a fast algorithm for quasi-symmetric Toeplitz matrix-vector multiplication. In addition, the stability analysis of the splitting symmetric Toeplitz inversion is discussed. In mathematical or engineering problems, the proposed algorithms are extraordinarily effective for solving a sequence of linear systems with a constant symmetric Toeplitz matrix. Fast matrix-vector multiplication and a quasi-symmetric Toeplitz linear solver are proven to be suitable for image encryption and decryption.</p></abstract>

Symmetric positive definite (SPD) matrices used as feature descriptors in image recognition are usually high dimensional. Traditional manifold learning is only applicable for reducing the dimension of high-dimensional vector-form data.For high-dimensional SPD matrices, directly using manifold learning algorithms to reduce the dimension of matrix-form data is impossible. The SPD matrix must first be transformed into a long vector, and then the dimension of this vector must be reduced. However, this approach breaks the spatial structure of the SPD matrix space. To overcome this limitation, we propose a new dimension reduction algorithm on SPD matrix space to transform high-dimensional SPD matrices into low-dimensional SPD matrices. Our work is based on the fact that the set of all SPD matrices with the same size has a Lie group structure, and we aim to transform the manifold learning to the SPD matrix Lie group. We use the basic idea of the manifold learning algorithm called locality preserving projection (LPP) to construct the corresponding Laplacian matrix on the SPD matrix Lie group. Thus, we call our approach Lie-LPP to emphasize its Lie group character. We present a detailed algorithm analysis and show through experiments that Lie-LPP achieves effective results on human action recognition and human face recognition.