The Experts below are selected from a list of 321 Experts worldwide ranked by ideXlab platform

Martine Olivi - One of the best experts on this subject based on the ideXlab platform.

  • Balanced realization of lossless systems: Schur parameters, canonical forms and applications
    IFAC Proceedings Volumes, 2009
    Co-Authors: Ralf Peeters, Martine Olivi, Bernard Hanzon
    Abstract:

    Abstract Lossless systems have many applications in systems and control theory, including signal processing, filter design, system identification, system approximation, and the parameterization of classes of linear systems. In this survey paper we address the issue of parameterization of the space of rational lossless matrix functions by successfully combining two approaches. The first approach proceeds in state-space from balanced realizations and triangular pivot structures of reachability matrices. The second approach concerns interpolation theory with linear fractional transformations and the tangential Schur Algorithm. We construct balanced realizations (and canonical forms) in terms of the Schur parameters encountered in the tangential Schur Algorithm, and conversely, we interpret balanced realizations in discrete-time and in continuous-time in terms of Schur parameters. A number of application areas are discussed to illustrate the importance of this theory for a variety of topics.

  • lossless scalar functions boundary interpolation Schur Algorithm and ober s canonical form
    Conference on Decision and Control, 2008
    Co-Authors: Martine Olivi, Bernard Hanzon, Ralf Peeters
    Abstract:

    In [1] a balanced canonical form for continuous-time lossless systems was presented. This form has a tridiagonal dynamical matrix A and the useful property that the corresponding controllability matrix K is upper triangular. In [2], this structure is also derived from a LC ladder. In this paper, a connection is established between Ober?s canonical form and a Schur Algorithm built from angular derivative interpolation conditions. It provides a new interpretation of the parameters in Ober?s form as interpolation values at infinity and a recursive construction of the balanced realization.

  • Lossless scalar functions: boundary interpolation, Schur Algorithm and Ober's canonical form
    2008
    Co-Authors: Martine Olivi, Bernard Hanzon, Ralf L.m. Peeters
    Abstract:

    In Ober (1987) a balanced canonical form for continuous-time lossless systems was presented. This form has a tridiagonal dynamical matrix A and the useful property that the corresponding controllability matrix K is upper triangular. In this paper, a connection is established between Ober's canonical form and a Schur Algorithm builts from angular derivative interpolation conditions. It provides a new interpretation of the parameters in Ober's form, as interpolation values at infinity, and a recursive construction of the balanced realization.

  • CDC - Lossless scalar functions: Boundary interpolation, Schur Algorithm and Ober’s canonical form.
    2008 47th IEEE Conference on Decision and Control, 2008
    Co-Authors: Martine Olivi, Bernard Hanzon, Ralf Peeters
    Abstract:

    In [1] a balanced canonical form for continuous-time lossless systems was presented. This form has a tridiagonal dynamical matrix A and the useful property that the corresponding controllability matrix K is upper triangular. In [2], this structure is also derived from a LC ladder. In this paper, a connection is established between Ober?s canonical form and a Schur Algorithm built from angular derivative interpolation conditions. It provides a new interpretation of the parameters in Ober?s form as interpolation values at infinity and a recursive construction of the balanced realization.

  • Multipoint Schur's Algorithm, rational orthogonal functions, asymptotic properties and Schur rational approximation
    2008
    Co-Authors: Vincent Lunot, Laurent Baratchart, Stanislas Kupin, Martine Olivi
    Abstract:

    In a paper by Khrushchev, the connections between the Schur Algorithm, the Wall's continued fractions and the orthogonal polynomials are revisited and used to establish some nice convergence properties of the sequence of Schur functions associated with a Schur function. In this report, we generalize some of Krushchev's results to the case of a multipoint Schur Algorithm, that is a Schur Algorithm where all the interpolation points are not taken in 0 but anywhere in the open unit disk. To this end, orthogonal rational functions and a recent generalization of Geronimus theorem are used. Then, we consider the problem of approximating a Schur function by a rational function which is also Schur. This problem of approximation is very important for the synthesis and identification of passive systems. We prove that all strictly Schur rational function of degree $n$ can be written as the $2n$-th convergent of the Schur Algorithm if the interpolation points are correctly chosen. This leads to a parametrization using the multipoint Schur Algorithm. Some examples are computed by an $L^2$ norm optimization process and the results are validated by comparison with the unconstrained $L^2$ rational approximation.

P. Van Dooren - One of the best experts on this subject based on the ideXlab platform.

  • The Generalized Schur Algorithm and Some Applications
    Axioms, 2018
    Co-Authors: Teresa Laudadio, Nicola Mastronardi, P. Van Dooren
    Abstract:

    The generalized Schur Algorithm is a powerful tool allowing to compute classical decompositions of matrices, such as the Q R and L U factorizations. When applied to matrices with particular structures, the generalized Schur Algorithm computes these factorizations with a complexity of one order of magnitude less than that of classical Algorithms based on Householder or elementary transformations. In this manuscript, we describe the main features of the generalized Schur Algorithm. We show that it helps to prove some theoretical properties of the R factor of the Q R factorization of some structured matrices, such as symmetric positive definite Toeplitz and Sylvester matrices, that can hardly be proven using classical linear algebra tools. Moreover, we propose a fast implementation of the generalized Schur Algorithm for computing the rank of Sylvester matrices, arising in a number of applications. Finally, we propose a generalized Schur based Algorithm for computing the null-space of polynomial matrices.

  • NAA - On the Stability of the Generalized Schur Algorithm
    Lecture Notes in Computer Science, 2001
    Co-Authors: Nicola Mastronardi, P. Van Dooren, Sabine Van Huffel
    Abstract:

    The generalized Schur Algorithm (GSA) is a fast method to compute the Cholesky factorization of a wide variety of structured matrices. The stability property of the GSA depends on the way it is implemented. In [15] GSA was shown to be as stable as the Schur Algorithm, provided one hyperbolic rotation in factored form [3] is performed at each iteration. Fast and efficient Algorithms for solving Structured Total Least Squares problems [14,15] are based on a particular implementation of GSA requiring two hyperbolic transformations at each iteration. In this paper the authors prove the stability property of such implementation provided the hyperbolic transformations are performed in factored form [3].

  • on the stability of the generalized Schur Algorithm
    International Conference on Numerical Analysis and Its Applications, 2000
    Co-Authors: Nicola Mastronardi, P. Van Dooren, Sabine Van Huffel
    Abstract:

    The generalized Schur Algorithm (GSA) is a fast method to compute the Cholesky factorization of a wide variety of structured matrices. The stability property of the GSA depends on the way it is implemented. In [15] GSA was shown to be as stable as the Schur Algorithm, provided one hyperbolic rotation in factored form [3] is performed at each iteration. Fast and efficient Algorithms for solving Structured Total Least Squares problems [14,15] are based on a particular implementation of GSA requiring two hyperbolic transformations at each iteration. In this paper the authors prove the stability property of such implementation provided the hyperbolic transformations are performed in factored form [3].

  • Stability Issues in the Factorization of Structured Matrices
    SIAM Journal on Matrix Analysis and Applications, 1997
    Co-Authors: Michael A. Stewart, P. Van Dooren
    Abstract:

    This paper provides an error analysis of the generalized Schur Algorithm of Kailath and Chun [SIAM J. Matrix Anal. Appl., 15 (1994), pp. 114--128]---a class of Algorithms which can be used to factorize Toeplitz-like matrices, including block-Toeplitz matrices, and matrices of the form $T^{T}T$, where $T$ is Toeplitz. The conclusion drawn is that if this Algorithm is implemented with hyperbolic transformations in the factored form which is well known to provide numerical stability in the context of Cholesky downdating, then the generalized Schur Algorithm will be stable. If a more direct implementation of the hyperbolic transformations is used, then it will be unstable. In this respect, the Algorithm is analogous to Cholesky downdating; the details of implementation of the hyperbolic transformations are essential for stability. An example which illustrates this instability is given. This result is in contrast to the ordinary Schur Algorithm for which an analysis by Bojanczyk, Brent, De Hoog, and Sweet [SIAM J. Matrix Anal. Appl., 16 (1995), pp. 40--57] shows that the sta- bility of the Algorithm is not dependent on the implementation of the hyperbolic transformations.

  • A block Toeplitz look-ahead Schur Algorithm
    SVD and Signal Processing III, 1995
    Co-Authors: Kyle A. Gallivan, S. Thirumalai, P. Van Dooren
    Abstract:

    This paper gives a look-ahead Schur Algorithm for finding the symmetric factorization of a Hermitian block Toeplitz matrix. The method is based on matrix operations and does not require any relations with orthogonal polynomials. The simplicity of the matrix based approach ought to shed new light on other issues such as parallelism and numerical stability.

Bernard Hanzon - One of the best experts on this subject based on the ideXlab platform.

  • Balanced realization of lossless systems: Schur parameters, canonical forms and applications
    IFAC Proceedings Volumes, 2009
    Co-Authors: Ralf Peeters, Martine Olivi, Bernard Hanzon
    Abstract:

    Abstract Lossless systems have many applications in systems and control theory, including signal processing, filter design, system identification, system approximation, and the parameterization of classes of linear systems. In this survey paper we address the issue of parameterization of the space of rational lossless matrix functions by successfully combining two approaches. The first approach proceeds in state-space from balanced realizations and triangular pivot structures of reachability matrices. The second approach concerns interpolation theory with linear fractional transformations and the tangential Schur Algorithm. We construct balanced realizations (and canonical forms) in terms of the Schur parameters encountered in the tangential Schur Algorithm, and conversely, we interpret balanced realizations in discrete-time and in continuous-time in terms of Schur parameters. A number of application areas are discussed to illustrate the importance of this theory for a variety of topics.

  • lossless scalar functions boundary interpolation Schur Algorithm and ober s canonical form
    Conference on Decision and Control, 2008
    Co-Authors: Martine Olivi, Bernard Hanzon, Ralf Peeters
    Abstract:

    In [1] a balanced canonical form for continuous-time lossless systems was presented. This form has a tridiagonal dynamical matrix A and the useful property that the corresponding controllability matrix K is upper triangular. In [2], this structure is also derived from a LC ladder. In this paper, a connection is established between Ober?s canonical form and a Schur Algorithm built from angular derivative interpolation conditions. It provides a new interpretation of the parameters in Ober?s form as interpolation values at infinity and a recursive construction of the balanced realization.

  • Lossless scalar functions: boundary interpolation, Schur Algorithm and Ober's canonical form
    2008
    Co-Authors: Martine Olivi, Bernard Hanzon, Ralf L.m. Peeters
    Abstract:

    In Ober (1987) a balanced canonical form for continuous-time lossless systems was presented. This form has a tridiagonal dynamical matrix A and the useful property that the corresponding controllability matrix K is upper triangular. In this paper, a connection is established between Ober's canonical form and a Schur Algorithm builts from angular derivative interpolation conditions. It provides a new interpretation of the parameters in Ober's form, as interpolation values at infinity, and a recursive construction of the balanced realization.

  • CDC - Lossless scalar functions: Boundary interpolation, Schur Algorithm and Ober’s canonical form.
    2008 47th IEEE Conference on Decision and Control, 2008
    Co-Authors: Martine Olivi, Bernard Hanzon, Ralf Peeters
    Abstract:

    In [1] a balanced canonical form for continuous-time lossless systems was presented. This form has a tridiagonal dynamical matrix A and the useful property that the corresponding controllability matrix K is upper triangular. In [2], this structure is also derived from a LC ladder. In this paper, a connection is established between Ober?s canonical form and a Schur Algorithm built from angular derivative interpolation conditions. It provides a new interpretation of the parameters in Ober?s form as interpolation values at infinity and a recursive construction of the balanced realization.

  • Canonical lossless state-space systems: Staircase forms and the Schur Algorithm
    Linear Algebra and its Applications, 2007
    Co-Authors: Ralf L.m. Peeters, Bernard Hanzon, Martine Olivi
    Abstract:

    A new finite atlas of overlapping balanced canonical forms for multivariate discrete-time lossless systems is presented. The canonical forms have the property that the controllability matrix is positive upper triangular up to a suitable permutation of its columns. This is a generalization of a similar balanced canonical form for continuous-time lossless systems. It is shown that this atlas is in fact a finite sub-atlas of the infinite atlas of overlapping balanced canonical forms for lossless systems that is associated with the tangential Schur Algorithm; such canonical forms satisfy certain interpolation conditions on a corresponding sequence of lossless transfer matrices. The connection between these balanced canonical forms for lossless systems and the tangential Schur Algorithm for lossless systems is a generalization of the same connection in the SISO case that was noted before. The results are directly applicable to obtain a finite sub-atlas of multivariate input-normal canonical forms for stable linear systems of given fixed order, which is minimal in the sense that no chart can be left out of the atlas without losing the property that the atlas covers the manifold.

Ralf Peeters - One of the best experts on this subject based on the ideXlab platform.

  • Balanced realization of lossless systems: Schur parameters, canonical forms and applications
    IFAC Proceedings Volumes, 2009
    Co-Authors: Ralf Peeters, Martine Olivi, Bernard Hanzon
    Abstract:

    Abstract Lossless systems have many applications in systems and control theory, including signal processing, filter design, system identification, system approximation, and the parameterization of classes of linear systems. In this survey paper we address the issue of parameterization of the space of rational lossless matrix functions by successfully combining two approaches. The first approach proceeds in state-space from balanced realizations and triangular pivot structures of reachability matrices. The second approach concerns interpolation theory with linear fractional transformations and the tangential Schur Algorithm. We construct balanced realizations (and canonical forms) in terms of the Schur parameters encountered in the tangential Schur Algorithm, and conversely, we interpret balanced realizations in discrete-time and in continuous-time in terms of Schur parameters. A number of application areas are discussed to illustrate the importance of this theory for a variety of topics.

  • lossless scalar functions boundary interpolation Schur Algorithm and ober s canonical form
    Conference on Decision and Control, 2008
    Co-Authors: Martine Olivi, Bernard Hanzon, Ralf Peeters
    Abstract:

    In [1] a balanced canonical form for continuous-time lossless systems was presented. This form has a tridiagonal dynamical matrix A and the useful property that the corresponding controllability matrix K is upper triangular. In [2], this structure is also derived from a LC ladder. In this paper, a connection is established between Ober?s canonical form and a Schur Algorithm built from angular derivative interpolation conditions. It provides a new interpretation of the parameters in Ober?s form as interpolation values at infinity and a recursive construction of the balanced realization.

  • CDC - Lossless scalar functions: Boundary interpolation, Schur Algorithm and Ober’s canonical form.
    2008 47th IEEE Conference on Decision and Control, 2008
    Co-Authors: Martine Olivi, Bernard Hanzon, Ralf Peeters
    Abstract:

    In [1] a balanced canonical form for continuous-time lossless systems was presented. This form has a tridiagonal dynamical matrix A and the useful property that the corresponding controllability matrix K is upper triangular. In [2], this structure is also derived from a LC ladder. In this paper, a connection is established between Ober?s canonical form and a Schur Algorithm built from angular derivative interpolation conditions. It provides a new interpretation of the parameters in Ober?s form as interpolation values at infinity and a recursive construction of the balanced realization.

  • Canonical Lossless State-Space Systems: Staircase Forms and the Schur Algorithm
    IFAC Proceedings Volumes, 2004
    Co-Authors: Ralf Peeters, Bernard Hanzon, Martine Olivi
    Abstract:

    Abstract A new finite atlas of overlapping balanced canonical forms for multivariate discrete-time lossless systems is presented. The canonical forms have the property that the controllability matrix is positive upper triangular up to a suitable permutation of its columns. This is a generalization of a similar balanced canonical form for continuous-time lossless systems. It is shown that this atlas is in fact a sub-atlas of the infinite atlas of overlapping balanced canonical forms for lossless systems that is associated with the tangential Schur Algorithm; such canonical forms satisfy certain interpolation conditions on a corresponding sequence of lossless transfer matrices. The connection between these balanced canonical forms for lossless systems and the tangential Schur Algorithm for lossless systems is a generalization of the same connection in the SISO case that was noted before. The results are directly applicable to obtain a finite atlas of multivariate input-normal canonical forms for stable linear systems of given fixed order, which is minimal in the sense that no chart can be left out of the atlas without losing the property that the atlas covers the manifold of stable linear systems of fixed given order.

  • Linear Fractional Transformations and Balanced Realization of Discrete-Time Stable All-Pass Systems
    IFAC Proceedings Volumes, 2001
    Co-Authors: Ralf Peeters, Bernard Hanzon, Martine Olivi
    Abstract:

    Abstract The tangential Schur Algorithm provides a means of constructing the class of multivariable discrete-time stable all-pass transfer functions of a fixed finite McMillan degree. In each iteration step a linear fractional transformation is employed which is associated with a J-inner rational matrix of McMillan degree 1. In this set-up, the emphasis is exclusively on transfer functions. In the present contribution we present a unified framework in which linear fractional transformations on transfer functions are represented by corresponding linear fractional transformations on state-space realization matrices. When applied to the case of the tangential Schur Algorithm, minimal balanced realizations of stable all-pass systems in terms of the parameters used are obtained. The balanced state-space approach of (Hanzon and Peeters, 2000) is incorporated as a special case.

Kyle A. Gallivan - One of the best experts on this subject based on the ideXlab platform.

  • High performance Algorithms for Toeplitz and block Toeplitz matrices
    Linear Algebra and its Applications, 1996
    Co-Authors: Kyle A. Gallivan, S. Thirumalai, P. Van Dooren, V. Vermaut
    Abstract:

    In this paper, we present several high performance variants of the classical Schur Algorithm to factor various Toeplitz matrices. For positive definite block Toeplitz matrices, we show how hyperbolic Householder transformations may be blocked to yield a block Schur Algorithm. This Algorithm uses BLAS3 primitives and makes efficient use of a memory hierarchy. We present three Algorithms for indefinite Toeplitz matrices. Two of these are based on look-ahead strategies and produce an exact factorization of the Toeplitz matrix. The third produces an inexact factorization via perturbations of singular principal miners. We also present an analysis of the numerical behavior of the third Algorithm and derive a bound for the number of iterations to improve the accuracy of the solution. For rank-deficient Toeplitz least-squares problems, we present a variant of the generalized Schur Algorithm that avoids breakdown due to an exact rank-deficiency. In the presence of a near rank-deficiency, an approximate rank factorization of the Toeplitz matrix is produced. Finally, we suggest an Algorithm to solve the normal equations resulting from a real Toeplitz least-squares problem based on transforming to Cauchy-like matrices. This Algorithm exploits both realness and symmetry in the normal equations.

  • High-performance Algorithms to solve toeplitz and block toeplitz matrices
    1996
    Co-Authors: Kyle A. Gallivan, Srikanth Thirumalai
    Abstract:

    Fast Algorithms to factor Toeplitz matrices have existed since the beginning of this century. The two most notable Algorithms to factor Toeplitz matrices are the Schur and the Levinson-Durbin. The former factors the Toeolitz matrix itself while the latter factors the inverse. In this thesis, we present several high performance variants of the classical Schur Algorithm to factor various Toeplitz matrices. For positive definite block Toeplitz matrices, we show how hyperbolic Householder transformations may be blocked to yield a block Schur Algorithm. This Algorithm uses BLAS3 primitives and makes efficient use of a memory hierarchy. We present three Algorithms for indefinite Toeplitz matrices. Two of these are based on look-ahead strategies and produce an exact factorization of the Toeplitz matrix. The third produces an inexact factorization via perturbations of singular principal minors. We also present an analysis of the numerical behavior of the third Algorithm and derive a bound for the number of iterations to improve the accuracy of the solution. Recently, there have been several Algorithms suggested to incorporate pivoting into the factorization of indefinite Toeplitz matrices by converting them to Cauchy-like matrices. We compare these Algorithms from a computational standpoint and suggest a few Algorithms that exploit properties such as realness and symmetry in the Toeplitz matrix while converting them to Cauchy-like matrices. In particular, we show how a Hermitian Toeplitz matrix may be converted to a real symmetric Cauchy-like matrix prior to factorization, yielding substantial savings in computation. For rank-deficient Toeplitz least-squares problems, we present a variant of the generalized Schur Algorithm that avoids breakdown due to an exact rank deficiency. In the presence of a near rank deficiency, an approximate rank factorization of the Toeplitz matrix is produced. Algorithms to solve real Toeplitz least-squares problems and to obtain rank-revealing QR factorizations of real Toeplitz matrices are also presented. We demonstrate the use of the Schur Algorithm in the construction of preconditioners to solve the problem of image deconvolution.

  • High performance Algorithms for Toeplitz and block Toeplitz matrices
    Linear Algebra and its Applications, 1996
    Co-Authors: Kyle A. Gallivan, S. Thirumalai, P. Van Dooren, V. Vermaut
    Abstract:

    AbstractIn this paper, we present several high performance variants of the classical Schur Algorithm to factor various Toeplitz matrices. For positive definite block Toeplitz matrices, we show how hyperbolic Householder transformations may be blocked to yield a block Schur Algorithm. This Algorithm uses BLAS3 primitives and makes efficient use of a memory hierarchy. We present three Algorithms for indefinite Toeplitz matrices. Two of these are based on look-ahead strategies and produce an exact factorization of the Toeplitz matrix. The third produces an inexact factorization via perturbations of singular principal minors. We also present an analysis of the numerical behavior of the third Algorithm and derive a bound for the number of iterations to improve the accuracy of the solution. For rank-deficient Toeplitz least-squares problems, we present a variant of the generalized Schur Algorithm that avoids breakdown due to an exact rank-deficiency. In the presence of a near rank-deficiency, an approximate rank factorization of the Toeplitz matrix is produced. Finally, we suggest an Algorithm to solve the normal equations resulting from a real Toeplitz least-squares problem based on transforming to Cauchy-like matrices. This Algorithm exploits both realness and symmetry in the normal equations

  • NORTH- HOLLAND High Performance Algorithms for Toeplitz and Block Toeplitz Matrices
    1996
    Co-Authors: Kyle A. Gallivan, S. Thirumalai, P. Van Dooren, V. Vermaut
    Abstract:

    In this paper, we present several high performance variants of the classical Schur Algorithm to factor various Toeplitz matrices. For positive definite block Toeplitz matrices, we show how hyperbolic Householder transformations may be blocked to yield a block Schur Algorithm. This Algorithm uses BLAS3 primitives and makes efficient use of a memory hierarchy. We present three Algorithms for indefinite Toeplitz matrices. Two of these are based on look-ahead strategies and produce an exact factorization of the Toeplitz matrix. The third produces an inexact faetorization via perturbations of singular principal minors. We also present an analysis of the numerical behavior of the third Algorithm and derive a bound for the number of iterations to improve the accuracy of the solution. For rank-deficient Toeplitz least-squares problems, we present a variant of the generalized Schur Algorithm that avoids breakdown due to an exact rank-deficiency. In the presence of a near rank-deficiency, an approximate rank factorization of the Toeplitz matrix is produced. Finally, we suggest an Algorithm to solve the normal equations resulting from a real Toeplitz least-squares problem based on transforming to Cauehy-like matrices. This Algorithm exploits both realness and symmetry in the normal equations.

  • A block Toeplitz look-ahead Schur Algorithm
    SVD and Signal Processing III, 1995
    Co-Authors: Kyle A. Gallivan, S. Thirumalai, P. Van Dooren
    Abstract:

    This paper gives a look-ahead Schur Algorithm for finding the symmetric factorization of a Hermitian block Toeplitz matrix. The method is based on matrix operations and does not require any relations with orthogonal polynomials. The simplicity of the matrix based approach ought to shed new light on other issues such as parallelism and numerical stability.