Toeplitz Linear Systems Research Articles

Fast algorithms for the solution of perturbed symmetric Toeplitz linear system and its applications

Fast algorithms for the solution of perturbed symmetric Toeplitz linear system and its applications

Numerical algorithms for the fast and reliable solution of periodic tridiagonal Toeplitz linear systems

Numerical algorithms for the fast and reliable solution of periodic tridiagonal Toeplitz linear systems

Doubling Smith Method for a Class of Large-Scale Generalized Fractional Diffusion Equations

The implicit difference approach is used to discretize a class of generalized fractional diffusion equations into a series of linear equations. By rearranging the equations as the matrix form, the separable forcing term and the coefficient matrices are shown to be low-ranked and of nonsingular M-matrix structure, respectively. A low-ranked doubling Smith method with determined optimally iterative parameters is presented for solving the corresponding matrix equation. In comparison to the existing Krylov solver with Fast Fourier Transform (FFT) for the sequence Toeplitz linear system, numerical examples demonstrate that the proposed method is more effective on CPU time for solving large-scale problems.

A new algorithm for solving Toeplitz linear systems

In this paper, we are interested in solving the Toeplitz linear systems. By exploiting the special Toeplitz structure, we give a new decomposition form of the coefficient matrix. Based on this matrix decomposition form and combined with the Sherman–Morrison formula, we propose an efficient algorithm for solving the considered problem. A typical example is presented to illustrate the different steps of the proposed algorithm. In addition, numerical tests are given showing the efficiency of our algorithm.

Algorithms for solving a class of real quasi-symmetric Toeplitz linear systems and its applications

<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>

Numerical algorithms for corner-modified symmetric Toeplitz linear system with applications to image encryption and decryption

Numerical algorithms for corner-modified symmetric Toeplitz linear system with applications to image encryption and decryption

Tensorized low-rank circulant preconditioners for multilevel Toeplitz linear systems from high-dimensional fractional Riesz equations

The solution of d-dimensional fractional partial differential equations (FPDEs) is challenging in numerical computation. This paper proposes a low-rank preconditioned conjugate gradient (PCG) method for discretizations with a low-rank right-hand side stemming from the d-dimensional fractional Riesz equation. At each iteration step, a combination of low-rank in Tensor Train (TT) format technique and Fast Fourier Transform (FFT) is used to accelerate multilevel Toeplitz matrix-vector multiplications. Furthermore, a low-rank circulant preconditioner is constructed based on the spectrum of the coefficient matrix. The reported numerical results demonstrate the competitiveness of the new method compared with a state-of-the-art matrix-free approach based on FFT.

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>

A stable parallel algorithm for block tridiagonal toeplitz–block–toeplitz linear systems

In this paper, we present a direct parallel block WZ algorithm, named DPBWZA, for the solution of block tridiagonal toeplitz–block–toeplitz (TBT) linear system Ax=f. The algorithm is based on the proposed block WZ factorization of the coefficient matrix A. Existence of the block WZ factorization for block tridiagonal TBT block diagonally dominant matrix is proved. Error analysis of the parallel algorithm DPBWZA is presented and numerical stability of the algorithm is established. Numerical experiments are conducted to demonstrate the efficiency, stability and accuracy of the Direct Parallel Block WZ Algorithm on GPU platform. Forward and backward errors are computed and DPBWZA is found to be highly accurate, numerically stable and as efficient as the subroutine csrlsvlu of GPU accelerated cuSolverSP library.

Fast second-order implicit difference schemes for time distributed-order and Riesz space fractional diffusion-wave equations

In this paper, fast numerical methods are established to solve a class of time distributed-order and Riesz space fractional diffusion-wave equations. We derive new difference schemes by a weighted and shifted Grünwald formula in time and a fractional centered difference formula in space. The unconditional stability and second-order convergence in time, space and distributed-order of the difference schemes are analyzed. In the one-dimensional case, the Gohberg-Semencul formula utilizing a preconditioned Krylov subspace method is developed to solve the Toeplitz linear system derived from the proposed difference scheme. In the two-dimensional case, we also design a global preconditioned conjugate gradient method with a truncated preconditioner to solve the resulting Sylvester matrix equations. We prove that the spectrums of the preconditioned matrices in both cases are clustered around 1, such that the proposed numerical methods with preconditioners converge very quickly. Some numerical experiments are carried out to demonstrate the effectiveness of the proposed difference schemes and show that the performances of the proposed fast solution algorithms are better than other testing methods.

A fast method for solving quasi-pentadiagonal Toeplitz linear systems and its application to the Lax–Wendroff scheme

In this paper we present an algorithm for solving quasi-pentadiagonal Toeplitz linear systems, that is based on Du et al. method (Du et al., 2014) and the Sherman–Morrison–Woodbury inversion formula. All algorithms have been implemented in Matlab and numerical experiments are given in order to illustrate the validity and efficiency of our algorithm. An application of the proposed algorithm to the Lax–Wendroff scheme is also considered.

A fast algorithm for solving diagonally dominant symmetric quasi-pentadiagonal Toeplitz linear systems

In this paper, we develop a new algorithm for solving diagonally dominant symmetric quasi-pentadiagonal Toeplitz linear systems. Numerical experiments are given in order to illustrate the validity and efficiency of our algorithm.

Preconditioners for multilevel Toeplitz linear systems from steady-state and evolutionary advection-diffusion equations

In this paper, we study preconditioners for multilevel Toeplitz linear systems arising from discretization of steady-state and evolutionary advection-diffusion equations, in which upwind scheme and central difference scheme are employed to discretize first-order and second-order terms, respectively. For the steady-state case, the preconditioner is constructed by replacing each of the discrete advection terms with a square root of the negative of discrete Laplacian matrix and the so constructed preconditioner is diagonalizable by a sine transform. Due to its diagonalizability, the preconditioner can be applied in a two-sided way. We prove that the GMRES solver for the preconditioned linear system has a linear convergence rate independent of discretization step-sizes. The sum of the time discretization and the steady-state preconditioner constitutes the evolutionary preconditioner. A fast implementation is proposed for the evolutionary preconditioner. Moreover, for the evolutionary case, we prove that the modulus of the eigenvalues of the preconditioned matrix is lower and upper bounded by positive constants independent of discretization step-sizes. We test the proposed preconditioners with several Krylov subspace solvers on some advection-dominated advection-diffusion problems and compare their performance with other preconditioners to show its efficiency.

Fast solvers for tridiagonal Toeplitz linear systems

Let A be a tridiagonal Toeplitz matrix denoted by $$A = {\text {Tritoep}} (\beta , \alpha , \gamma )$$ . The matrix A is said to be: strictly diagonally dominant if $$|\alpha |>|\beta |+|\gamma |$$ , weakly diagonally dominant if $$|\alpha |\ge |\beta |+|\gamma |$$ , subdiagonally dominant if $$|\beta |\ge |\alpha |+|\gamma |$$ , and superdiagonally dominant if $$|\gamma |\ge |\alpha |+|\beta |$$ . In this paper, we consider the solution of a tridiagonal Toeplitz system $$A\mathbf {x}= \mathbf {b}$$ , where A is subdiagonally dominant, superdiagonally dominant, or weakly diagonally dominant, respectively. We first consider the case of A being subdiagonally dominant. We transform A into a block $$2\times 2$$ matrix by an elementary transformation and then solve such a linear system using the block LU factorization. Compared with the LU factorization method with pivoting, our algorithm takes less flops, and needs less memory storage and data transmission. In particular, our algorithm outperforms the LU factorization method with pivoting in terms of computing efficiency. Then, we deal with superdiagonally dominant and weakly diagonally dominant cases, respectively. Numerical experiments are finally given to illustrate the effectiveness of our algorithms.

Preconditioned GMRES method for a class of Toeplitz linear systems in fractional eigenvalue problems

In this paper, we consider the solution of a class of Toeplitz linear systems coming from the fractional eigenvalue problems. We construct the Strang circulant matrix as a preconditioner to solve the Toeplitz linear systems, and analyze the properties of eigenvalues of the preconditioned coefficient matrix. We also propose the preconditioned generalized minimal residuals method for solving this linear systems, and give the computational costs of this algorithm. The numerical examples show the effecticiency of our method.

A GPIU method for fractional diffusion equations

The fractional diffusion equations can be discretized by applying the implicit finite difference scheme and the unconditionally stable shifted Grünwald formula. Hence, the generating linear system has a real Toeplitz structure when the two diffusion coefficients are non-negative constants. Through a similarity transformation, the Toeplitz linear system can be converted to a generalized saddle point problem. We use the generalization of a parameterized inexact Uzawa (GPIU) method to solve such a kind of saddle point problem and give a new algorithm based on the GPIU method. Numerical results show the effectiveness and accuracy for the new algorithm.

迭代精化下求解三对角Toeplitz线性方程组的快速算法

迭代精化下求解三对角Toeplitz线性方程组的快速算法

GMRES on tridiagonal block Toeplitz linear systems

GMRES on tridiagonal block Toeplitz linear systems

A medium-shifted splitting iteration method for a diagonal-plus-Toeplitz linear system from spatial fractional Schr\xf6dinger equations

The centered difference discretization of the spatial fractional coupled nonlinear Schrödinger equations obtains a discretized linear system whose coefficient matrix is the sum of a real diagonal matrix D and a complex symmetric Toeplitz matrix T̃ which is just the symmetric real Toeplitz T plus an imaginary identity matrix iI. In this study, we present a medium-shifted splitting iteration method to solve the discretized linear system, in which the fast algorithm can be utilized to solve the Toeplitz linear system. Theoretical analysis shows that the new iteration method is convergent. Moreover, the new splitting iteration method naturally leads to a preconditioner. Analysis shows that the eigenvalues of the corresponding preconditioned matrix are tighter than those of the original coefficient matrix A. Finally, compared with the other algorithms by numerical experiments, the new method is more effective.

A Simplified Architecture of the Zhang Neural Network for Toeplitz Linear Systems Solving

In this paper, we derive a compact implementation of the discrete time Zhang neural network for computing the solution of an $$n\times n$$ Toeplitz linear system. In the proposed scheme, the multiplication of the weight matrix by the input vector is performed efficiently in $$O(n\log n)$$ operations using the fast Fourier transform instead of $$O(n^{2})$$ in the original Zhang neural network. We have investigated also the use of the proposed neural network to compute the Toeplitz matrices inversion; it consists of dividing the system into n Toeplitz sub-systems requiring n independent subnets similar to the one used for solving the Toeplitz system. In another way, the computation of the first and the last columns of the matrix inverse is sufficient to compute the matrix inverse via the Gohberg–Semencul formulas, in this case only two subnets are needed to define all elements of the matrix inverse.