It is well known that the discretization of fractional diffusion equations with fractional derivatives α∈(1,2)$$ \alpha \in \left(1,2\right) $$, using the so‐called weighted and shifted Grünwald ...formula, leads to linear systems whose coefficient matrices show a Toeplitz‐like structure. More precisely, in the case of variable coefficients, the related matrix sequences belong to the so‐called generalized locally Toeplitz class. Conversely, when the given FDE has constant coefficients, using a suitable discretization, we encounter a Toeplitz structure associated to a nonnegative function ℱα$$ {\mathcal{F}}_{\alpha } $$, called the spectral symbol, having a unique zero at zero of real positive order between one and two. For the fast solution of such systems by preconditioned Krylov methods, several preconditioning techniques have been proposed in both the one‐ and two‐dimensional cases. In this article we propose a new preconditioner denoted by 𝒫ℱα which belongs to the τ$$ \tau $$‐algebra and it is based on the spectral symbol ℱα$$ {\mathcal{F}}_{\alpha } $$. Comparing with some of the previously proposed preconditioners, we show that although the low band structure preserving preconditioners are more effective in the one‐dimensional case, the new preconditioner performs better in the more challenging multi‐dimensional setting.
In this letter, we devise an efficient approach for estimating the directions-of-arrival (DOAs) of coherent signals using coprime arrays. Specifically, we firstly derive an augmented uniform linear ...array (ULA) via coprime array interpolation. Subsequently, we define a Toeplitz matrix that is formed using the correlation information of the interpolated ULA theoretical outputs, and recover the Toeplitz matrix by solving a nuclear norm minimization problem. After the low-rank Toeplitz matrix is recovered, the coherent signals are well resolved by the MUSIC algorithm. Simulation results demonstrate the advantages of the proposed approach over various existing methods when dealing with coherent signals.
In this paper, we present a general method to construct q-analogues and other generalizations of Pascal-like matrices. Our matrices are obtained as functions of strictly lower triangular matrices and ...include several types of generalized Pascal-like matrices and matrices related with modified Hermite polynomials of two variables and other polynomial sequences. We find explicit expressions for products, powers, and inverses of the matrices and also some factorization formulas using this method.
•Toeplitz matrix-based deconvolution technique is proposed for hologram reconstruction.•A procedure to avoid the aliasing effect in the reconstructed images is discussed.•The reconstruction technique ...is applicable in digital in-line holography.•High computational efficiency is offered by proposed method compared to FFT.
A new hologram reconstruction algorithm is proposed for digital in-line holography configuration using the Toeplitz matrix based deconvolution. The variable separable property of the convolution kernel associated with hologram recording and reconstruction process allows the matrix formulation of deconvolution. A procedure to avoid the aliasing effect in the reconstructed images is discussed. The proposed method is found to provide improved computational efficiency in comparison to the conventional FFT based deconvolution approach without compromising the reconstruction quality. Simulation and experimental results substantiate the practical applicability of the proposed method.
Distributed matrix computations - matrix-matrix or matrix-vector multiplications - are well-recognized to suffer from the problem of stragglers (slow or failed worker nodes). Much of prior work in ...this area is (i) either sub-optimal in terms of its straggler resilience, or (ii) suffers from numerical problems, i.e., there is a blow-up of round-off errors in the decoded result owing to the high condition numbers of the corresponding decoding matrices. Our work presents a convolutional coding approach to this problem that removes these limitations. It is optimal in terms of its straggler resilience, and has excellent numerical robustness as long as the workers' storage capacity is slightly higher than the fundamental lower bound. Moreover, it can be decoded using a fast peeling decoder that only involves add/subtract operations. Our second approach has marginally higher decoding complexity than the first one, but allows us to operate arbitrarily close to the storage capacity lower bound. Its numerical robustness can be theoretically quantified by deriving a computable upper bound on the worst case condition number over all possible decoding matrices by drawing connections with the properties of large block Toeplitz matrices. All above claims are backed up by extensive experiments done on the AWS cloud platform.
In this paper, we study the matrix period and the competition period of Toeplitz matrices over a binary Boolean ring B={0,1}. Given subsets S and T of {1,…,n−1}, an n×n Toeplitz matrix A=Tn〈S;T〉 is ...defined to have 1 as the (i,j)-entry if and only if j−i∈S or i−j∈T. We show that if maxS+minT≤n and minS+maxT≤n, then A has the matrix period d/d′ and the competition period 1 where d=gcd(s+t|s∈S,t∈T) and d′=gcd(d,minS). Moreover, it is shown that the limit of the matrix sequence {Am(AT)m}m=1∞ is a directed sum of matrices of all ones except zero diagonal. In many literatures we see that graph theoretic method can be used to prove strong structural properties about matrices. Likewise, we develop our work from a graph theoretic point of view.
We compute an asymptotic formula for the supremum of the resolvent norm ||(ζ−T)−1|| over |ζ|≥1 and contractions T acting on an n-dimensional Hilbert space, whose spectral radius does not exceed a ...given r∈(0,1). We prove that this supremum is achieved on the unit circle by an analytic Toeplitz matrix.
A linear code with complementary dual (or an LCD code) is defined to be a linear code which intersects its dual code trivially. Let I be an identity matrix and T be a Toeplitz matrix of the same ...order over a finite field. A Double Toeplitz code (or a DT code) is a linear code generated by a generator matrix of the form (I,T). In 2021, Shi et al. obtained necessary and sufficient conditions for a Double Toeplitz code to be LCD when T is symmetric and tridiagonal. In this paper, by using a result on factoring Dickson polynomials over finite fields, we determine when a Double Toeplitz code is LCD for T being a skew symmetric and tridiagonal matrix. In addition, using a concatenation, we construct LCD codes with arbitrary minimum distance from DT codes over extension fields, provided the length of which is increased if necessary.
The possible instability of partial indices is one of the important constraints in the creation of approximate methods for the factorization of matrix functions. This paper is devoted to a study of a ...specific class of triangular matrix functions given on the unit circle with a stable and unstable set of partial indices. Exact conditions are derived that guarantee a preservation of the unstable set of partial indices during a perturbation of a matrix within the class. Thus, even in this probably simplest of cases, when the factorization technique is well developed, the structure of the parametric space (guiding the types of matrix perturbations) is non-trivial.
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