Symmetric Positive Definite Toeplitz Systems Research Articles

The Parallel Algorithm for Solving Toeplitz Systems

This paper studies a parallel algorithm for real Toeplitz systems, which is proposed based on the block Jacobi iteration and GMRES method. The algorithm has the advantage of less float operations, fast convergence speed and especially suitable for parallel computating. In this paper, we first use the block Jacobi iterative method to obtain the iterative process, and then the GMRES method is nested to obtain the iterative sequences {xk}. Therefore, the parallel algorithm for solving symmetric positive definite Toeplitz systems is constructed. The convergence of the algorithm is also discussed simply in the paper. At the end, we give some numerical examples to illustrate the effectiveness of the parallel algorithm.

W Circulant Boundary Condition

AbstractWe use the preconditioned conjugate gradient (PCG) method to analyze the symmetric positive‐definite Toeplitz systems. Then we present a new embedding constructing preconditioner way and prove that many other ways of constructing preconditioners are generally the special cases of this method. We propose the w‐circulant boundary condition and make comparison with both the ordinary circulant boundary condition and helix one. Furthermore, we obtain a new type of boundary condition: the mixed boundary condition.

Iterative Toeplitz solvers with local quadratic convergence

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.

Circulant and Skewcirculant Matrices for Solving Toeplitz Matrix Problems

In recent papers, numerous authors studied the solutions of symmetric positive definite Toeplitz systems $Tx = b$ by the conjugate gradient method for different families of circulant preconditioners C. In this paper new circulant / skewcirculant approximations are introduced to T and their properties are studied. The main interest is directed to the skewcirculant case. Furthermore, algorithms for computing the eigenvalues of T are formulated, based on the Lanczos algorithm and Rayleigh quotient iteration. For some numerical examples the spectra of $C^{ - 1} T$ are compared and the behaviour of the introduced eigenvalue algorithms is displayed.

Design and analysis of Toeplitz preconditioners

The solution of symmetric positive definite Toeplitz systems Ax=b by the preconditioned conjugate gradient (PCG) method was recently proposed by Strang (1986) and analyzed by Chan (1989) and Chan and Strang (1989). The convergence rate of the PCG method depends on the choice of preconditioners for the given Toeplitz matrices. The authors present a general approach to the design of Toeplitz preconditioners based on the idea to approximate a partially characterized linear deconvolution with circular deconvolutions. All resulting preconditioners can therefore be inverted via various fast transform algorithms with O(N log N) operations. For a wide class of problems, the PCG method converges in a finite number of iterations independent of N so that the computational complexity for solving these Toeplitz systems is O(N log N). >

Arithmetic complexity and numerical properties of algorithms involving Toeplitz matrices

We consider both the algebraic complexity and numerical properties of problems involving Toeplitz matrices. From the algebraic point of view, we derive upper and lower bounds for number of multiplications required to compute the inverse or the product of Toeplitz matrices and consider several cases as well. The lower bounds for the general cases are in agreement with earlier results, but the specialized lower bounds and all the upper bounds are new. We also derive bounds for the number of multiplications needed to multiply certain rectangular Toeplitz matrices that are important in spectral estimation. From the numerical point of view, we study the numerical stability of transform-based circular deconvolution. We show that if the matrix being inverted is well conditioned then the computed solution is close to the exact solution. We then study the numerical stability of algorithms used to invert banded Toeplitz systems. We analyze the numerical behavior of several algorithms from the literature. We show that all of these algorithms are unstable. One algorithm is shown to be weakly stable when used to invert a symmetric banded Toeplitz matrix with a well-conditioned positive definite infinite extension. We present a new algorithm which is weakly stable under a more general condition and can be modified to invert certain Toeplitz-like matrices. Finally, we study the numerical solution of symmetric positive definite Toeplitz systems, A$\sb n$x = b, with iterative methods. We present a quadratically convergent algorithm based on steepest descent. Numerical experiments which indicate that the algorithm behaves better than a similar method based on residual correction are presented. We introduce a new class of preconditioners called p-extended circulant (PEC) preconditioners. We consider the case when the elements of A$\sb\infty$ are the Fourier coefficients of f, a positive function in the Wiener class. With PEC preconditioners we show that computing x to precision $\epsilon$ requires no more than $O(n \log n)$ + $d(f, \epsilon)$ operations, which is a record bound. We present efficient algebraic implementations of these new methods as well as previously proposed algorithms. These implementations reduce the arithmetic complexity of each iteration.

The Spectrum of a Family of Circulant Preconditioned Toeplitz Systems

The solutions of symmetric positive definite Toeplitz systems $Ax = b$ are studied by the preconditioned conjugate gradient method. The preconditioner is the circulant matrix C that minimizes the Frobenius norm $\| {C - A} \|_F $ [T. Chan, “An Optimal Circulant Preconditioner for Toeplitz Systems,” UCLA Department of Mathematics, CAM Report 87-06, June 1987]. The convergence rate of these iterative methods is known to depend on the distribution of the eigenvalues of $C^{ - 1} A$. For Toeplitz matrix A with entries which are Fourier coefficients of a positive function in the Wiener class, this paper establishes the invertibility of C, finds the asymptotic behaviour of the eigenvalues of the preconditioned matrix $C^{ - 1} A$ as the dimension increases and proves that they are clustered around 1.

Toeplitz Equations by Conjugate Gradients with Circulant Preconditioner

This paper studies the solution of symmetric positive definite Toeplitz systems $Ax = b$ by the preconditioned conjugate gradient method. The preconditioner is a circulant matrix C that copies the middle diagonals of A, and each iteration uses the Fast Fourier Transform. Convergence is governed by the eigenvalues of $C^{ - 1} A$–a Toeplitz-circulant eigenvalue problem—and it is fast if those eigenvalues are clustered. The limiting behavior of the eigenvalues is found as the dimension increases, and it is proved that they cluster around $\lambda = 1$. For a wide class of problems the error after q conjugate gradient steps decreases as $r^{q^2 } $.

Toeplitz Equations by Conjugate Gradients with Circulant Preconditioner

This paper studies the solution of symmetric positive definite Toeplitz systems $Ax = b$ by the preconditioned conjugate gradient method. The preconditioner is a circulant matrix C that copies the ...