R. H. Bartels

Bio: R. H. Bartels is an academic researcher from University of Texas at Austin. The author has contributed to research in topics: Matrix differential equation & Centrosymmetric matrix. The author has an hindex of 1, co-authored 1 publications receiving 1647 citations.

Solution of the matrix equation AX + XB = C [F4]

Abstract: and Canada) or $18.00 (elsewhere). If the user sends a small tape (wt. less than 1 lb.) the algorithm will be copied on it and returned to him at a charge of $10.O0 (U.S. only). All orders are to be prepaid with checks payable to ACM Algorithms. The algorithm is re corded as one file of BCD 80 character card images at 556 B.P.I., even parity, on seven ~rack tape. We will supply the algorithm at a density of 800 B.P.I. if requested. The cards for the algorithm are sequenced starting at 10 and incremented by 10. The sequence number is right justified in colums 80. Although we will make every attempt to insure that the algorithm conforms to the description printed here, we cannot guarantee it, nor can we guarantee that the algorithm is correct.-L.D.F. Descdption The following programs are a collection of Fortran IV sub-routines to solve the matrix equation AX-.}-XB = C (1) where A, B, and C are real matrices of dimensions m X m, n X n, and m X n, respectively. Additional subroutines permit the efficient solution of the equation ArX + xa = C, (2) where C is symmetric. Equation (1) has applications to the direct solution of discrete Poisson equations [2]. It is well known that (1) has a unique solution if and only if the One proof of the result amounts to constructing the solution from complete systems of eigenvalues and eigenvectors of A and B, when they exist. This technique has been proposed as a computational method (e.g. see [1 ]); however, it is unstable when the eigensystem is ill conditioned. The method proposed here is based on the Schur reduction to triangular form by orthogonal similarity transformations. Equation (1) is solved as follows. The matrix A is reduced to lower real Schur form A' by an orthogonal similarity transformation U; that is A is reduced to the real, block lower triangular form.

Principal component analysis in linear systems: Controllability, observability, and model reduction

Abstract: Kalman's minimal realization theory involves geometric objects (controllable, unobservable subspaces) which are subject to structural instability. Specifically, arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces. This situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization. Structural instability associated with geometric theories is not unique to control; it arises in the theory of linear equations as well. In this setting, the computational problems have been studied for decades and excellent tools have been developed for coping with the situation. One of the main goals of this paper is to call attention to principal component analysis (Hotelling, 1933), and an algorithm (Golub and Reinsch, 1970) for computing the singular value decompositon of a matrix. Together they form a powerful tool for coping with structural instability in dynamic systems. As developed in this paper, principal component analysis is a technique for analyzing signals. (Singular value decomposition provides the computational machinery.) For this reason, Kalman's minimal realization theory is recast in terms of responses to injected signals. Application of the signal analysis to controllability and observability leads to a coordinate system in which the "internally balanced" model has special properties. For asymptotically stable systems, this yields working approximations of X_{c}, X_{\bar{o}} , the controllable and unobservable subspaces. It is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces.

A Schur method for solving algebraic Riccati equations

Abstract: In this paper a new algorithm for solving algebraic Riccati equations (both continuous-time and discrete-time versions) is presented. The method studied is a variant of the classical eigenvector approach and uses instead an appropriate set of Schur vectors thereby gaining substantial numerical advantages. Complete proofs of the Schur approach are given as well as considerable discussion of numerical issues. The method is apparently quite numerically stable and performs reliably on systems with dense matrices up to order 100 or so, storage being the main limiting factor. The description given below is a considerably abridged version of a complete report given in [0].

A Hessenberg-Schur method for the problem AX + XB= C

Abstract: One of the most effective methods for solving the matrix equation AX+XB=C is the Bartels-Stewart algorithm. Key to this technique is the orthogonal reduction of A and B to triangular form using the QR algorithm for eigenvalues. A new method is proposed which differs from the Bartels-Stewart algorithm in that A is only reduced to Hessenberg form. The resulting algorithm is between 30 and 70 percent faster depending upon the dimensions of the matrices A and B . The stability of the new method is demonstrated through a roundoff error analysis and supported by numerical tests. Finally, it is shown how the techniques described can be applied and generalized to other matrix equation problems.

Zero-Shot Learning - A Comprehensive Evaluation of the Good, the Bad and the Ugly

Abstract: Due to the importance of zero-shot learning, i.e. classifying images where there is a lack of labeled training data, the number of proposed approaches has recently increased steadily. We argue that it is time to take a step back and to analyze the status quo of the area. The purpose of this paper is three-fold. First, given the fact that there is no agreed upon zero-shot learning benchmark, we first define a new benchmark by unifying both the evaluation protocols and data splits of publicly available datasets used for this task. This is an important contribution as published results are often not comparable and sometimes even flawed due to, e.g. pre-training on zero-shot test classes. Moreover, we propose a new zero-shot learning dataset, the Animals with Attributes 2 (AWA2) dataset which we make publicly available both in terms of image features and the images themselves. Second, we compare and analyze a significant number of the state-of-the-art methods in depth, both in the classic zero-shot setting but also in the more realistic generalized zero-shot setting. Finally, we discuss in detail the limitations of the current status of the area which can be taken as a basis for advancing it.