Lower Triangular Toeplitz Research Articles

A Preconditioned MINRES Method for Block Lower Triangular Toeplitz Systems

A Preconditioned MINRES Method for Block Lower Triangular Toeplitz Systems

Efficient Implementation for SBL-Based Coherent Distributed mmWave Radar Imaging

In a distributed frequency-modulated continuous waveform (FMCW) radar system, the echo data collected are not continuous in the azimuth direction, so the imaging effect of the traditional range-Doppler (RD) algorithm is poor. Sparse Bayesian learning (SBL) is an optimization algorithm based on Bayesian theory that has been successfully applied to high-resolution radar imaging because of its strong robustness and high accuracy. However, SBL is highly computationally complex. Fortunately, with FMCW radar echo data, most of the time-consuming SBL operations involve a Toeplitz-block Toeplitz (TBT) matrix. In this article, based on this advantage, we propose a fast SBL algorithm that can be used to obtain high-angular-resolution images, in which the inverse of the TBT matrix can be transposed as the sum of the products of the block lower triangular Toeplitz matrix and the block circulant matrix by using a new decomposition method, and some of the matrix multiplications can be quickly computed using the fast Fourier transform (FFT), decreasing the computation time by several orders of magnitude. Finally, simulations and experiments were used to ensure the effectiveness of the proposed algorithm.

Polynomial Wiener LQG Controllers based on Toeplitz Matrices

This paper re-examines the discrete-time Linear Quadratic Gaussian (LQG) regulator problem. The normal approach to this problem is to use a Kalman filter state estimator and Kalman control state feedback. Though quite successful, an alternative approach in the frequency domain was employed later. That method used z-transfer functions or polynomials in the z-domain. The transfer function approach is similar to the method used in Wiener filtering and requires the use of Diophantine equations (sometimes bilateral) to find the optimal controller. The contribution here uses a similar approach but uses lower triangular Toeplitz matrices instead of polynomials to gain advantage of eliminating the use of Diophantine equations. This is because the single Diophantine equation approach fails when the system has non-relative prime polynomials and the need for bilateral Diophantine equations is computationally far more complex.

Block splitting preconditioner for time-space fractional diffusion equations

<abstract><p>For solving a block lower triangular Toeplitz linear system arising from the time-space fractional diffusion equations more effectively, a single-parameter two-step split iterative method (TSS) is introduced, its convergence theory is established and the corresponding preconditioner is also presented. Theoretical analysis shows that the original coefficient matrix after preconditioned can be expressed as the sum of the identity matrix, a low-rank matrix, and a small norm matrix. Numerical experiments show that the preconditioner improve the calculation efficiency of the Krylov subspace iteration method.</p></abstract>

The Cyclic Groups Generated by Catalan Matrices

Let $n$ a be positive integer. The Catalan matrix of order $n$, denoted by $\mathscr{C}_n[x]$, is a lower triangular Toeplitz matrix whose nonzero elements are expressions containing Catalan numbers and a real parameter $x$. In this paper, we investigate the multiplicative order of the cyclic group generated by $\mathscr{C}_n[x]$ (for $x\in\mathbb{N}$) modulo a positive integer $m$.

The smallest singular value of certain Toeplitz-related parametric triangular matrices

Abstract Let L be the infinite lower triangular Toeplitz matrix with first column (µ, a 1, a 2, ..., ap , a 1, ..., ap , ...) T and let D be the infinite diagonal matrix whose entries are 1, 2, 3, . . . Let A := L + D be the sum of these two matrices. Bünger and Rump have shown that if p = 2 and certain linear inequalities between the parameters µ, a 1, a 2, are satisfied, then the singular values of any finite left upper square submatrix of A can be bounded from below by an expression depending only on those parameters, but not on the matrix size. By extending parts of their reasoning, we show that a similar behaviour should be expected for arbitrary p and a much larger range of values for µ, a 1, ..., ap . It depends on the asymptotics in µ of the l 2-norm of certain sequences defined by linear recurrences, in which these parameters enter. We also consider the relevance of the results in a numerical analysis setting and moreover a few selected numerical experiments are presented in order to show that our bounds are accurate in practical computations.

Preconditioners for all-at-once system from the fractional mobile/immobile advection–diffusion model

The all-at-once system arising from fractional mobile/immobile advection–diffusion equations is studied. Firstly, the finite difference method with L1 formula is employed to discretize it. The resulting implicit scheme is a time-stepping scheme, which is not suitable for parallel computing. Based on this scheme, an all-at-once system is established, which will be suitable for parallel computing. Secondly, according to the block lower triangular Toeplitz structure of the all-at-once system, both the block bi-diagonal preconditioner and block stair preconditioner are designed. Finally, numerical examples are presented to show the performances of the proposed preconditioners.

Fast solvers for finite difference scheme of two-dimensional time-space fractional differential equations

Generally, solving linear systems from finite difference alternating direction implicit scheme of two-dimensional time-space fractional differential equations with Gaussian elimination requires $\mathcal {O}\left ({NM}_{1}M_{2}\left ({M_{1}^{2}}+{M_{2}^{2}}+NM_{1}M_{2}\right )\right )$ complexity and $\mathcal {O}\left ({N{M_{1}^{2}}{M_{2}^{2}}}\right )$ storage, where N is the number of temporal unknown and M1, M2 are the numbers of spatial unknown in x, y directions respectively. By exploring the structure of the coefficient matrix in fully coupled form, it possesses block lower-triangular Toeplitz structure and its blocks are block-dense Toeplitz matrices with dense-Toeplitz blocks. Based on this special structure and cooperating with time-marching or divide-and-conquer technique, two fast solvers with storage $\mathcal {O}\left ({NM}_{1}M_{2}\right )$ are developed. The complexity for the fast solver via time-marching is $\mathcal {O}\left ({NM}_{1}M_{2}\left (N+\log \left (M_{1}M_{2}\right )\right )\right )$ and the one via divide-and-conquer technique is $\mathcal {O}\left ({NM}_{1}M_{2}\left (\log ^{2} N+ \log \left (M_{1}M_{2}\right )\right )\right )$. It is worth to remark that the proposed solvers are not lossy. Some discussions on achieving convergence rate for smooth and non-smooth solutions are given. Numerical results show the high efficiency of the proposed fast solvers.

Truncated Fractional-Order Total Variation Model for Image Restoration

Fractional-order derivative is attracting more and more interest from researchers working on image processing because it helps to preserve more texture than total variation when noise is removed. In the existing works, the Grunwald–Letnikov fractional-order derivative is usually used, where the Dirichlet homogeneous boundary condition can only be considered and therefore the full lower triangular Toeplitz matrix is generated as the discrete partial fractional-order derivative operator. In this paper, a modified truncation is considered in generating the discrete fractional-order partial derivative operator and a truncated fractional-order total variation (tFoTV) model is proposed for image restoration. Hopefully, first any boundary condition can be used in the numerical experiments. Second, the accuracy of the reconstructed images by the tFoTV model can be improved. The alternating directional method of multiplier is applied to solve the tFoTV model. Its convergence is also analyzed briefly. In the numerical experiments, we apply the tFoTV model to recover images that are corrupted by blur and noise. The numerical results show that the tFoTV model provides better reconstruction in peak signal-to-noise ratio (PSNR) than the full fractional-order variation and total variation models. From the numerical results, we can also see that the tFoTV model is comparable with the total generalized variation (TGV) model in accuracy. In addition, we can roughly fix a fractional order according to the structure of the image, and therefore, there is only one parameter left to determine in the tFoTV model, while there are always two parameters to be fixed in TGV model.

A limited-memory block bi-diagonal Toeplitz preconditioner for block lower triangular Toeplitz system from time–space fractional diffusion equation

A block lower triangular Toeplitz system arising from time-space fractional diffusion equation is discussed. For efficient solutions of such the linear system, the preconditioned biconjugate gradient stabilized method and flexible general minimal residual method are exploited. The main contribution of this paper has two aspects: (i) A block bi-diagonal Toeplitz preconditioner is developed for the block lower triangular Toeplitz system, whose storage is of $\mathcal{O}(N)$ with $N$ being the spatial grid number; (ii) A new skew-circulant preconditioner is designed to fast calculate the inverse of the block bi-diagonal Toeplitz preconditioner multiplying a vector. Numerical experiments are given to demonstrate the efficiency of our preconditioners.

Restrictions matrices for Platonic solids invariance and applications to space–time energetic BEM

Geometrical symmetry is found in many linear field problems. Group representation theory is the only valuable tool for exploiting this property from a computational point of view. In this context, here we apply a technique for taking into account equivariance in the numerical treatment of space–time boundary integral equations, which are invariant under a finite group G of congruences of R 3 , each related to one of the so-called Platonic solids. This technique is based upon suitable restriction matrices strictly related to a system of unitary, irreducible, pairwise not-equivalent matrix representations of G . The main development is expounded in the framework of space–time energetic boundary element method applied to Neumann exterior wave propagation problems, where the discretization matrices have a block lower triangular Toeplitz structure, and the diagonal block, to be inverted at each time step, is typically dense. Several numerical results, related to computational saving, are presented.

On the solutions of the equation AXB = C under Toeplitz‐like and Hankel matrices constraint

In this paper, we are mainly concerned with 2 types of constrained matrix equation problems of the form AXB=C, the least squares problem and the optimal approximation problem, and we consider several constraint matrices, such as general Toeplitz matrices, upper triangular Toeplitz matrices, lower triangular Toeplitz matrices, symmetric Toeplitz matrices, and Hankel matrices. In the first problem, owing to the special structure of the constraint matrix , we construct special algorithms; necessary and sufficient conditions are obtained about the existence and uniqueness for the solutions. In the second problem, we use von Neumann alternating projection algorithm to obtain the solutions of problem. Then we give 2 numerical examples to demonstrate the effectiveness of the algorithms.

Using q-calculus to study LDLt factorization of a certain Vandermonde matrix

We use tools from $q$-calculus to study $LDL^T$ decomposition of the Vandermonde matrix $V_q$ with coefficients $v_{i,j}=q^{ij}$. We prove that the matrix $L$ is given as a product of diagonal matrices and the lower triangular Toeplitz matrix $T_q$ with coefficients $t_{i,j}=1/(q;q)_{i-j}$, where $(z;q)_k$ is the q-Pochhammer symbol. We investigate some properties of the matrix $T_q$, in particular, we compute explicitly the inverse of this matrix.

Approximate inversion method for time‐fractional subdiffusion equations

SummaryThe finite‐difference method applied to the time‐fractional subdiffusion equation usually leads to a large‐scale linear system with a block lower triangular Toeplitz coefficient matrix. The approximate inversion method is employed to solve this system. A sufficient condition is proved to guarantee the high accuracy of the approximate inversion method for solving the block lower triangular Toeplitz systems, which are easy to verify in practice and have a wide range of applications. The applications of this sufficient condition to several existing finite‐difference schemes are investigated. Numerical experiments are presented to verify the validity of theoretical results.

Superregular Lower Triangular Toeplitz Matrices for Low Delay Wireless Streaming

A matrix is termed superregular if all of its possible submatrices are non-singular. Superregular lower triangular Toeplitz matrices are useful for MDS convolutional codes and (sequential) network codes. In this paper, we present the explicit matrix constructions for superregular lower triangular Toeplitz matrices in GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sup> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k×k</sup> , k ≤ 5. For k > 5 we provide a greedy algorithm, which (over sufficiently large fields) is guaranteed to find a superregular lower triangular Toeplitz matrix. We introduce (product preserving) joint superregularity, and extend our explicit matrix constructions to these cases. We provide methods for deriving the exact symbol loss probability and delay for any deterministic block code. We derive the exact symbol loss probability and delay for codes using a superregular lower triangular matrix and for codes using two (product preserving) jointly superregular lower triangular matrices. We then compare these results with those obtained from both simulations and our practical implementation, and for each case we also compare with random-based codes. Furthermore, our experiments show a gain in coding throughput above 40% for superregular lower triangular Toeplitz matrices over random matrices.

A fast numerical method for block lower triangular Toeplitz with dense Toeplitz blocks system with applications to time-space fractional diffusion equations

Based on the circulant-and-skew-circulant representation of Toeplitz matrix inversion and the divide-and-conquer technique, a fast numerical method is developed for solving N-by-N block lower triangular Toeplitz with M-by-M dense Toeplitz blocks system with \(\mathcal {O}(MN\log N(\log N+\log M))\) complexity and \(\mathcal {O}(NM)\) storage. Moreover, the method is employed for solving the linear system that arises from compact finite difference scheme for time-space fractional diffusion equations with significant speedup. Numerical examples are given to show the efficiency of the proposed method.

A Petrov--Galerkin Spectral Method of Linear Complexity for Fractional Multiterm ODEs on the Half Line

We present a new tunably accurate Laguerre Petrov--Galerkin spectral method for solving linear multiterm fractional initial value problems with derivative orders at most one and constant coefficients on the half line. Our method results in a matrix equation of special structure which can be solved in $\mathcal{O}(N \log N)$ operations. We also take advantage of recurrence relations for the generalized associated Laguerre functions (GALFs) in order to derive explicit expressions for the entries of the stiffness and mass matrices, which can be factored into the product of a diagonal matrix and a lower-triangular Toeplitz matrix. The resulting spectral method is efficient for solving multiterm fractional differential equations with arbitrarily many terms, which we demonstrate by solving a fifty-term example. We apply this method to a distributed order differential equation, which is approximated by linear multiterm equations through the Gauss--Legendre quadrature rule. We provide numerical examples demonstra...

A fast accurate approximation method with multigrid solver for two-dimensional fractional sub-diffusion equation

A fast accurate approximation method with multigrid solver is proposed to solve a two-dimensional fractional sub-diffusion equation. Using the finite difference discretization of fractional time derivative, a block lower triangular Toeplitz matrix is obtained where each main diagonal block contains a two-dimensional matrix for the Laplacian operator. Our idea is to make use of the block ϵ-circulant approximation via fast Fourier transforms, so that the resulting task is to solve a block diagonal system, where each diagonal block matrix is the sum of a complex scalar times the identity matrix and a Laplacian matrix. We show that the accuracy of the approximation scheme is of O(ϵ). Because of the special diagonal block structure, we employ the multigrid method to solve the resulting linear systems. The convergence of the multigrid method is studied. Numerical examples are presented to illustrate the accuracy of the proposed approximation scheme and the efficiency of the proposed solver.

Lower triangular Toeplitz–Ramanujan systems whose solution yields the Bernoulli numbers

We prove that the Bernoulli numbers satisfy some special lower triangular Toeplitz systems of linear equations. One of these systems has a strong link with eleven Ramanujan's linear equations satisfied by the first eleven Bernoulli numbers.

A fast algorithm for solving banded Toeplitz systems

A fast algorithm for solving systems of linear equations with banded Toeplitz matrices is presented. This new approach is based on extending the given matrix with several rows on the top and several columns on the right and to assign zeros and some nonzero constants in each of these rows and columns in such a way that the augmented matrix has a lower triangular Toeplitz structure. Stability of the algorithm is discussed and its performance is showed by numerical experiments.