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(mn³)+O(k³n³) flops, there are m block rows and m block columns in the matrix, n is the order of blocks, 2k+1 is the bandwidth. The validity of the approach is illustrated by numerical experiments.     

Fast solution of unsymmetric banded Toeplitz systems by means of spectral factorizations and Woodbury's formula

A fast algorithm for solving systems of linear equations with banded Toeplitz matrices is studied. An important step in the algorithm is a novel method for the spectral factorization of the generating function associated with the Toeplitz matrix. The spectral factorization is extracted from the right deflating subspaces corresponding to the eigenvalues inside and outside the open unit disk of a companion matrix pencil constructed from the coefficients of the generating function. The factorization is followed by the Woodbury inversion formula and solution of several banded triangular systems. Stability of the algorithm is discussed and its performance is demonstrated by numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.     

Toeplitz型矩阵的逆矩阵的快速三角分解算法

针对有关“型”矩阵的三角分解问题 ,提出了一种 Toeplitz型矩阵的逆矩阵的快速三角分解算法 .首先假设给定 n阶非奇异矩阵 A,利用一组线性方程组的解 ,得到 A- 1的一个递推关系式 ,进而利用该关系式得到 A- 1的一种三角分解表达式 ,然后从 Toeplitz型矩阵的特殊结构出发 ,利用上述定理的结论 ,给出了Toeplitz型矩阵的逆矩阵的一种快速三角分解算法 ,算法所需运算量为 O( mn2 ) .最后 ,数值计算表明该算法的可靠性 .     

Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems

Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss–Borel factorization of two, left and a right, Cantero–Morales–Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. A block Gauss–Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix Szeg? polynomials, which can be expressed in terms of Schur complements of bordered truncations of the block moment matrix. The corresponding block extension of the Christoffel–Darboux theory is derived. Deformations of the quasi-definite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov–Shabat equations, bilinear equations and discrete flows — connected with Darboux transformations. We generalize the integrable flows of the Cafasso's matrix extension of the Toeplitz lattice for the Verblunsky coefficients of Szeg? polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel–Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomials.     

Structured mixed and componentwise condition numbers of some structured matrices

Structured matrices, such as Cauchy, Vandermonde, Toeplitz, Hankel, and circulant matrices, are considered in this paper. We apply a Kronecker product-based technique to deduce the structured mixed and componentwise condition numbers for the matrix inversion and for the corresponding linear systems.     

Generalized inversion of finite rank Hankel and Toeplitz operators with rational matrix symbols

The goal of the paper is a generalized inversion of finite rank Hankel operators and Hankel or Toeplitz operators with block matrices having finitely many rows. To attain it a left coprime fractional factorization of a strictly proper rational matrix function and the Bezout equation are used. Generalized inverses of these operators and generating functions for the inverses are explicitly constructed in terms of the fractional factorization.     

Randomized Preprocessing of Homogeneous Linear Systems of Equations

Our randomized preprocessing enables pivoting-free and orthogonalization-free solution of homogeneous linear systems of equations. In the case of Toeplitz inputs, we decrease the estimated solution time from quadratic to nearly linear, and our tests show dramatic decrease of the CPU time as well. We prove numerical stability of our approach and extend it to solving nonsingular linear systems, inversion and generalized (Moore-Penrose) inversion of general and structured matrices by means of Newton’s iteration, approximation of a matrix by a nearby matrix that has a smaller rank or a smaller displacement rank, matrix eigen-solving, and root-finding for polynomial and secular equations and for polynomial systems of equations. Some by-products and extensions of our study can be of independent technical intersest, e.g., our extensions of the Sherman-Morrison-Woodbury formula for matrix inversion, our estimates for the condition number of randomized matrix products, and preprocessing via augmentation.     

A look-ahead Bareiss algorithm for general Toeplitz matrices

Summary. The Bareiss algorithm is one of the classical fast solvers for systems of linear equations with Toeplitz coefficient matrices. The method takes advantage of the special structure, and it computes the solution of a Toeplitz system of order~ with only~ arithmetic operations, instead of~ operations. However, the original Bareiss algorithm requires that all leading principal submatrices be nonsingular, and the algorithm is numerically unstable if singular or ill-conditioned submatrices occur. In this paper, an extension of the Bareiss algorithm to general Toeplitz systems is presented. Using look-ahead techniques, the proposed algorithm can skip over arbitrary blocks of singular or ill-conditioned submatrices, and at the same time, it still fully exploits the Toeplitz structure. Implementation details and operations counts are given, and numerical experiments are reported. We also discuss special versions of the proposed look-ahead Bareiss algorithm for Hermitian indefinite Toeplitz systems and banded Toeplitz systems. Received August 27, 1993 / Revised version received March 1994     

Componentwise error analysis for the block LU factorization of totally nonnegative matrices

For the first time, perturbation bounds including componentwise perturbation bounds for the block LU factorization have been provided by Dopico and Molera (2005) [5]. In this paper, componentwise error analysis is presented for computing the block LU factorization of nonsingular totally nonnegative matrices. We present a componentwise bound on the equivalent perturbation for the computed block LU factorization. Consequently, combining with the componentwise perturbation results we derive componentwise forward error bounds for the computed block factors.     

Factorization of analytic functions by means of Koenig's theorem and Toeplitz computations

Summary. By providing a matrix version of Koenig's theorem we reduce the problem of evaluating the coefficients of a monic factor r(z) of degree h of a power series f(z) to that of approximating the first h entries in the first column of the inverse of an Toeplitz matrix in block Hessenberg form for sufficiently large values of n. This matrix is reduced to a band matrix if f(z) is a polynomial. We prove that the factorization problem can be also reduced to solving a matrix equation for an matrix X, where is a matrix power series whose coefficients are Toeplitz matrices. The function is reduced to a matrix polynomial of degree 2 if f(z) is a polynomial of degreeN and . These reductions allow us to devise a suitable algorithm, based on cyclic reduction and on the concept of displacement rank, for generating a sequence of vectors that quadratically converges to the vector having as components the coefficients of the factor r(z). In the case of a polynomial f(z) of degree N, the cost of computing the entries of given is arithmetic operations, where is the cost of solving an Toeplitz-like system. In the case of analytic functions the cost depends on the numerical degree of the power series involved in the computation. From the numerical experiments performed with several test polynomials and power series, the algorithm has shown good numerical properties and promises to be a good candidate for implementing polynomial root-finders based on recursive splitting strategies. Applications to solving spectral factorization problems and Markov chains are also shown. Received September 9, 1998 / Revised version received November 14, 1999 / Published online February 5, 2001     

Using FFT-based techniques in polynomial and matrix computations: recent advances and applicatons

Computations with univariate polynomials, like the evaluation of product, quotient, remainder, greatest common divisor, etc, are closely related to linear algebra computations performed with structured matrices having the Toeplitz-like or the Hankel-like structures.

The discrete Fourier transform, and the FFT algorithms for its computation, constitute a powerful tool for the design and analysis of fast algorithms for solving problems involving polynomials and structured matrices.

We recall the main correlations between polynomial and matrix computations and present two recent results in this field: in particular, we show how Fourier methods can speed up the solution of a wide class of problems arising in queueing theory where certain Markov chains, defined by infinite block Toeplitz matrices in generalized Hessenberg form, must be solved. Moreover, we present a new method for the numerical factorization of polynomials based on a matrix generalization of Koenig's theorem. This method, that relies on the evaluation/interpolation technique at the Fourier points, reduces the problem of polynomial factorization to the computation of the LU decomposition of a banded Toeplitz matrix with its rows and columns suitably permuted. Numerical experiments that show the effectiveness of our algorithms are presented     

QR factorization of Toeplitz matrices

Summary This paper presents a new algorithm for computing theQR factorization of anm×n Toeplitz matrix inO(mn) operations. The algorithm exploits the procedure for the rank-1 modification and the fact that both principal (m–1)×(n–1) submatrices of the Toeplitz matrix are identical. An efficient parallel implementation of the algorithm is possible.     

On the Toeplitz embedding of an arbitrary matrix

It has been shown by Delosme and Morf that an arbitrary block matrix can be embedded into a block Toeplitz matrix; the dimension of this embedding depends on the complexity of the matrix structure compared to the block Toeplitz structure. Due to the special form of the embedding matrix, the algebra of matrix polynomials relative to block Toeplitz matrices can be interpreted directly in terms of the original matrix and therefore can be extended to arbitrary matrices. In fact, these polynomials turn out to provide an appropriate framework for the recently proposed generalized Levinson algorithm solving the general matrix inversion problem.     

On the analytical solution of the linear‐fractional Riemann problem

The linear‐fractional problem is a generalization of the linear Riemann problem that includes the (non‐linear) factorization problem. In case of normal type it can be equivalently reduced to a family of homogeneous linear vector Riemann problems by space foliation and adequate substitutions. Moreover these are equivalent to systems of non‐homogeneous Toeplitz equations with special data. The reduced problem is solved by matrix factorization in various cases. Procedures for reduction to these cases are exposed. Various modified problems and generalizations are pointed out. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Approximate inversion method for time‐fractional subdiffusion equations

The 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.     

On explicit factorization and applications

This paper is devoted to two topics connected with factorization of triangular 2 by 2 matrix functions. The first application is concerned with explicit factorization of a class of matrices of Daniel-Khrapkov type and the second is related to inversion of finite Toeplitz matrices. In the first section we present the scheme of factorization of triangular 2 by 2 matrix functions.     

Complexity reduction of least squares problems involving special vandermonde matrices

This paper presents a new QRD factorization of a rectangular Vandermonde matrix for a special point distribution, including the symmetric case, based on ak-dimensional block decomposition of the matrix and some properties of the Kronecker product. The computational reduction factor with respect to any QR method isk ², in the general case, and 4 in the symmetric one. By the resulting matrix factorization, new formulas are devised for the least squares system solution, whose implementation produces an algorithm of reduced computational cost and computer storage. Finally the perturbation bounds of this new factorization are devised.     

Block toeplitz products of block toeplitz matrices

Necessary and sufficient conditions for the product of two block Toeplitz matrices to be block Toeplitz are obtained. In the special case of two Toeplitz matrices, the conditions simplify considerably and, when combined with known necessary and sufficient conditions for a nonsingular Toeplitz matrix to have a Toeplitz inverse, provide a simple characterization of the additional matrix structure required by a subclass of Toeplitz matrices in order for it to be closed with respect to both inversion and multiplication.     

A note on the solution of not balanced banded Toeplitz systems

A direct algorithm is presented for the solution of linear systems having banded Toeplitz coefficient matrix with unbalanced bandwidths. It is derived from the cyclic reduction algorithm, it makes use of techniques based on the displacement rank and it relies on the Morrison–Sherman–Woodbury formula. The algorithm always equals and sometimes outperforms the already known direct ones in terms of asymptotic computational cost. The case where the coefficient matrix is a block banded block Toeplitz matrix in block Hessenberg form is analyzed as well. The algorithm is numerically stable if applied to M‐matrices that are point diagonally dominant by columns. Copyright © 2007 John Wiley & Sons, Ltd.     

Symmetric centrosymmetric matrix–vector multiplication

We present a method for the multiplication of an arbitrary vector by a symmetric centrosymmetric matrix, requiring floating-point operations, rather than the 2n² operations needed in the case of an arbitrary matrix. Combining this method with Trench's algorithm for Toeplitz matrix inversion yields a method for solving Toeplitz systems with the same complexity as Levinson's algorithm.