Toeplitz Solvers Research Articles

Acoustic multiple scattering using recursive algorithms

Acoustic multiple scattering by a cluster of cylinders in an acoustic medium is considered. A fast recursive technique is described which takes advantage of the multilevel Block Toeplitz structure of the linear system. A parallelization technique is described that enables efficient application of the proposed recursive algorithm for solving multilevel Block Toeplitz systems on high performance computer clusters. Numerical comparisons of CPU time and total elapsed time taken to solve the linear system using the direct LAPACK and TOEPLITZ libraries on Intel FORTRAN, show the advantage of the TOEPLITZ solver. Computations are optimized by multi-threading which displays improved efficiency of the TOEPLITZ solver with the increase of the number of scatterers and frequency.

Block pivoting implementation of a symmetric Toeplitz solver

Toeplitz matrices are characterized by a special structure that can be exploited in order to obtain fast linear system solvers. These solvers are difficult to parallelize due to their low computational cost and their closely coupled data operations. We propose to transform the Toeplitz system matrix into a Cauchy-like matrix since the latter can be divided into two independent matrices of half the size of the system matrix and each one of these smaller arising matrices can be factorized efficiently in multicore computers. We use OpenMP and store data in memory by blocks in consecutive positions yielding a simple and efficient algorithm. In addition, by exploiting the fact that diagonal pivoting does not destroy the special structure of Cauchy-like matrices, we introduce a local diagonal pivoting technique which improves the accuracy of the solution and the stability of the algorithm.

A direct method to solve block banded block Toeplitz systems with non-banded Toeplitz blocks

A fast solution algorithm is proposed for solving block banded block Toeplitz systems with non-banded Toeplitz blocks. The algorithm constructs the circulant transformation of a given Toeplitz system and then by means of the Sherman–Morrison–Woodbury formula transforms its inverse to an inverse of the original matrix. The block circulant matrix with Toeplitz blocks is converted to a block diagonal matrix with Toeplitz blocks, and the resulting Toeplitz systems are solved by means of a fast Toeplitz solver. The computational complexity in the case one uses fast Toeplitz solvers is equal to ξ ( m , n , k ) = O ( m n 3 ) + O ( k 3 n 3 ) flops, there are m block rows and m block columns in the matrix, n is the order of blocks, 2 k + 1 is the bandwidth. The validity of the approach is illustrated by numerical experiments.

Product-type block preconditioners for electromagnetic cavity problems

In a radar detecting system, the radar cross section (RCS) of a target is an important application in electromagnetics computation. Cavity prediction is usually required in the computation process. When wavelengths of electromagnetic field are large, cavity calculation is a challenging problem. The main aim of this paper is to employ the preconditioned GMRES method to solve linear systems arising in cavity calculation of the transverse magnetic and transverse electric problems. We develop new preconditioners based on fast transforms and Toeplitz solvers. Our numerical results show that the proposed preconditioners are quite efficient and effective, especially when the wavelength is large.

Book Review: An introduction to iterative Toeplitz solvers

Book Review: An introduction to iterative Toeplitz solvers

A Superfast Algorithm for Toeplitz Systems of Linear Equations

In this paper we develop a new superfast solver for Toeplitz systems of linear equations. To solve Toeplitz systems many people use displacement equation methods. With displacement structures, Toeplitz matrices can be transformed into Cauchy-like matrices using the FFT or other trigonometric transformations. These Cauchy-like matrices have a special property, that is, their off-diagonal blocks have small numerical ranks. This low-rank property plays a central role in our superfast Toeplitz solver. It enables us to quickly approximate the Cauchy-like matrices by structured matrices called sequentially semiseparable (SSS) matrices. The major work of the constructions of these SSS forms can be done in precomputations (independent of the Toeplitz matrix entries). These SSS representations are compact because of the low-rank property. The SSS Cauchy-like systems can be solved in linear time with linear storage. Excluding precomputations the main operations are the FFT and SSS system solve, which are both very efficient. Our new Toeplitz solver is stable in practice. Numerical examples are presented to illustrate the efficiency and the practical stability.

A Superfast Toeplitz Solver with Improved Numerical Stability

This paper describes a new O(n log3(n)) solver for the positive definite Toeplitz system Tx=b. Instead of computing generators for the inverse of T, the new algorithm adjoins b to T and applies a superfast Schur algorithm to the resulting augmented matrix. The generators of this augmented matrix and its Schur complements are used by a divide-and-conquer block back-substitution routine to complete the solution of the system. The goal is to avoid the well-known numerical instability inherent in explicit inversion. Experiments suggest that the algorithm is backward stable in most cases.

A Stabilized Superfast Solver for Nonsymmetric Toeplitz Systems

We present a stabilized superfast solver for nonsymmetric Toeplitz systems Tx=b. An explicit formula for T-1 is expressed in such a way that the matrix-vector product T^-1b can be calculated via FFTs and Hadamard products. This inversion formula involves certain polynomials that can be computed by solving two linearized rational interpolation problems on the unit circle. The heart of our Toeplitz solver is a superfast algorithm to solve these interpolation problems. To stabilize the algorithm, i.e., to improve the accuracy, several techniques are used: pivoting, iterative improvement, downdating, and giving "difficult" interpolation points an adequate treatment. We have implemented our algorithm in Fortran 90. Numerical examples illustrate the effectiveness of our approach.

Stabilization by Perturbation of a 4n2 Toeplitz Solver

We consider a 4n2 -algorithm for computing the Gohberg--Semencul representation of the inverse of a nonsymmetric Toeplitz matrix and for obtaining the solution of a Toeplitz system. The original algorithm is unstable when one or more of the leading principal submatrices are ill-conditioned. In our stabilized version we slightly perturb thematrix when we encounter a division by a small number. As a consequence, the solution is also perturbed, and its accuracy can be improved by taking a small number of iterative refinement steps. We present a roundoff error analysis of the new algorithm as well as numerical results that support our analysis.

Scientific applications of iterative Toeplitz solvers

Recent research on using the preconditioned conjugate gradient method as an iterative method for solving Toeplitz systems has brought much attention. One of the main important results of this methodology is that the complexity of solving a large class of Toeplitz systems can be reduced toO (n logn) operations as compared to theO(n log2 n) operations required by fast direct Toeplitz solvers, provided that a suitable preconditioner is chosen under certain conditions on the Toeplitz operator. In this paper, we survery some applications of iterative Toeplitz solvers to Toeplitz-related problems arising from scientific applications. These applications include partial differential equations, queueing networks, signal and image processing, integral equations, and time series analysis.

Conjugate Gradient Methods for Toeplitz Systems

In this expository paper, we survey some of the latest developments in using preconditioned conjugate gradient methods for solving Toeplitz systems. One of the main results is that the complexity of solving a large class of $n$-by-$n$ Toeplitz systems is reduced to $O(n \log n)$ operations as compared to $O(n \log ^2 n)$ operations required by fast direct Toeplitz solvers. Different preconditioners proposed for Toeplitz systems are reviewed. Applications to Toeplitz-related systems arising from partial differential equations, queueing networks, signal and image processing, integral equations, and time series analysis are given.

Classical foundations of algorithms for solving positive definite Toeplitz equations

The ongoing development and analysis of efficient algorithms for solving positive definite Toeplitz equations is motivated to a large extent by the importance of these equations in signal processing applications. The role of positive definite Toeplitz matrices in this and other areas of mathematics and engineering stems from Schur's study of bounded analytic functions on the unit disk, and Szegő's theory of polynomials orthogonal on the unit circle. These ideas underlie several Toeplitz solvers, and provide a useful framework for understanding the relationships among these algorithms. In this paper we give an overview of several direct algorithms for solving positive definite Toeplitz systems of linear equations from this classical viewpoint.

A fast approach to stabilize two Toeplitz solvers of the Levinson type

Two algorithms of the Levinson type for the solution of general Toeplitz systems are considered. The second one is better suited for vector computers while the first one is faster on scalar machines. They can break down, or behave poorly, when some of the leading principal submatrices are singular, or ill conditioned, although the whole matrix is very well conditioned. A perturbation approach is shown to work well in this case for both algorithms. The number of the additional flops can be relatively small in comparison to the look-ahead strategies.

Fast Gaussian Elimination with Partial Pivoting for Matrices with Displacement Structure

Fast O(n 2 ) implementation of Gaussian elimination with partial pivoting is designed for matrices possessing Cauchy-like displacement structure. We show how Toeplitz-like, Toeplitz-plus-Hankel-like and Vandermonde-like matrices can be transformed into Cauchy-like matrices by using Discrete Fourier, Cosine or Sine Transform matrices. In particular this allows us to propose a new fast O(n 2 ) Toeplitz solver GKO, which incorporates partial pivoting. A large set of numerical examples showed that GKO demonstrated stable numerical behavior and can be recommended for solving linear systems, especially with nonsymmetric, indefinite and ill-conditioned positive definite Toeplitz matrices. It is also useful for block Toeplitz and mosaic Toeplitz ( Toeplitz-block ) matrices. The algorithms proposed in this paper suggest an attractive alternative to look-ahead approaches, where one has to jump over ill-conditioned leading submatrices, which in the worst case requires O(n 3 ) operations.

Fast Gaussian elimination with partial pivoting for matrices with displacement structure

Fast O ( n 2 ) O({n^2}) implementation of Gaussian elimination with partial pivoting is designed for matrices possessing Cauchy-like displacement structure. We show how Toeplitz-like, Toeplitz-plus-Hankel-like and Vandermonde-like matrices can be transformed into Cauchy-like matrices by using Discrete Fourier, Cosine or Sine Transform matrices. In particular this allows us to propose a new fast O ( n 2 ) O({n^2}) Toeplitz solver GKO, which incorporates partial pivoting. A large set of numerical examples showed that GKO demonstrated stable numerical behavior and can be recommended for solving linear systems, especially with nonsymmetric, indefinite and ill-conditioned positive definite Toeplitz matrices. It is also useful for block Toeplitz and mosaic Toeplitz (Toeplitz-block) matrices. The algorithms proposed in this paper suggest an attractive alternative to look-ahead approaches, where one has to jump over ill-conditioned leading submatrices, which in the worst case requires O ( n 3 ) O({n^3}) operations.

Efficient solution of block linear systems with Toeplitz entries using a channel decomposition technique

Abstract This paper is concerned with the development of efficient solvers for block linear systems with Toeplitz entries. Fast block Toeplitz solvers are required in a wide range of applications in the area of multichannel and multidimensional digital signal processing. The presentation and the derivation of all algorithms proposed in this paper are based on the context of multichannel FIR Wiener filtering. A novel channel decomposition technique is applied to obtain fast, order recursive algorithms that require scalar operations, only. The proposed algorithms can accommodate multichannel filter of different filter orders and manage to get free of matrix operations. Two basic algorithm structures are derived, the Levinson type and the Schur type. They are both based on suitable permutations that unravel the recursive nature of the pertinent matrices and enable the development of fast block linear system solvers. Normalized versions are also derived and a simple test for detecting the positive definiteness of a block matrix with Toeplitz entries, is obtained. An efficient lattice structure, that requires scalar operations, is also derived and is subsequently used to obtain Schur-type recursions. The Schur-type algorithm admits full parallelism and, if parallel processing environment is available, it reduces processing time by an order of magnitude. To illustrate the structure of the proposed algorithms, Matlab code is provided for the fast Levinson-type algorithm, as well as for the Schur counterpart.

The Use of Pivoting to Improve the Numerical Performance of Algorithms for Toeplitz Matrices

Classical $O( n^2 )$ Toeplitz solvers break down if any leading principal submatrices are singular, and as might be expected, the occurrence of nearly singular leading submatrices can cause a serious loss of accuracy in these solvers. In this paper, a pivoting scheme has been incorporated into the Toeplitz solver of Bareiss which allows near-singularities to be treated without significant loss of accuracy. As special cases, the pivoted Toeplitz solver also handles exact singularities and numerical singularities—the latter appear as near-singularities when finite-precision floating-point arithmetic is used. The Bareiss algorithm can also be used to compute the $LU$ factors of a Toeplitz matrix, and it is shown that a restricted version of the pivoted Toeplitz solver produces an $LU$ factorization of a row- and column-permuted version of the original Toeplitz matrix.Finally, it is shown how the inverter of Trench and Zohar can be derived from the multipliers produced by the Bareiss algorithm, and this conne...

Solution to the problem of instability in banded Toeplitz solvers

The author presents a numerically stable approach to solving banded Toeplitz systems of n linear equations. The algorithm is fast in that it requires only O(nq) operations, where q is the bandwidth of the matrix. An earlier version of the banded Toeplitz algorithm presented in the literature suffers from numerical instability. The author solves the instability problem by developing numerically stable alternatives to the back substitution process. These new algorithms have roughly the same number of calculations as simple back substitution, O(nq), and therefore can be used effectively in place of back substitution. The stabilization procedures described have been used to find accurate solutions to systems of order several thousand. >

Numerical experience with a superfast real Toeplitz solver

We briefly describe the generalized Schur algorithm for the superfast solution of positive definite Toeplitz systems of equations and its relationship with Schur's algorithm and the Szegö recursions. We then present some experimental results obtained with our FORTRAN implementation of this superfast Toeplitz solver. The algorithm displays favorable behavior in that the growth rates of the resulting residuals and errors are comparable with those of the Szegö recursions.

A fast but unstable orthogonal triangularization technique for Toeplitz matrices

D. Sweet's clever QR decomposition algorithm for Toeplitz matrices is considered. It requires only O( n 2) flops to factor an n × n matrix. We analyze the algorithm to point out its deficiencies, and compare it against two other Toeplitz solvers.