Inversion of polynomial matrices by interpolation

Known polynomial interpolation methods to polynomial matrices are generalized to obtain new algorithms for the computation of the inverse of such matrices. The algorithms use numerically stabilizable manipulations of constant matrices. Among the three methods investigated Lagrange's interpolation seems especially suitable for the purpose     

Modelling of an arch dam by polynomial interpolation

Polynomial curves controlled by points as well as by the tangents in them allow for a great margin of freedom, far from the conditions implied by the use of conventional curves, in the design of free-form curves.The lofting technique, in turn, permits us to generate surfaces that include a series of given curves.The algorithms propitiated to generate these curves and surfaces are very suitable for the geometric design of the arch dam, since they provide for a design of the curves forming the horizontal arches in which the dam can conceivably be divided taking into account, besides, the angle among the tangents to the arch at its ends as well as the hillsides of the closed arch. Once these curves are designed, the facings of a dam can be generated as the surfaces involving both series of previous curves.This paper presents the geometric calculations behind the generation of both surfaces and their computerized processing, in addition to a practical example of the use of the proposed methodology.     

On the decidability of sparse univariate polynomial interpolation

We consider the problem of determining whether or not there exists a sparse univariate polynomial that interpolates a given setS={(x _i ,y _i )} of points. Several important cases are resolved, e.g., the case when thex _i's are all positive rational numbers. But the general problem remains open.     

On interpolation by generalized planar splines I: The polynomial case

Complex planar splines were introduced by Opfer and Puri [4] and further investigated by several authors, cf. [3, 5, 6, 8]. These papers were mainly concerned with the properties of piecewise linear or quadratic polynomials. In the present paper polynomial planar splines of arbitrary degree are investigated. Our results are mainly concerned with their interpolatory properties on polygonal regions and contain those of [3, 4, 5] for triangular and rectangular regions as special cases.     

Size reduction by interpolation in fuzzy rule bases

Fuzzy control is at present still the most important area of real applications for fuzzy theory. It is a generalized form of expert control using fuzzy sets in the definition of vague/linguistic predicates, modeling a system by If...then rules. In the classical approaches it is necessary that observations on the actual state of the system partly match (fire) one or several rules in the model (fired rules), and the conclusion is calculated by the evaluation of the degrees of matching and the fired rules. Interpolation helps reduce the complexity as it allows rule bases with gaps. Various interpolation approaches are shown. It is proposed that dense rule bases should be reduced so that only the minimal necessary number of rules remain still containing the essential information in the original base, and all other rules are replaced by the interpolation algorithm that however can recover them with a certain accuracy prescribed before reduction. The interpolation method used for demonstration is the Lagrange method supplying the best fitting minimal degree polynomial. The paper concentrates on the reduction technique that is rather independent from the style of the interpolation model, but cannot be given in the form of a tractable algorithm. An example is shown to illustrate possible results and difficulties with the method.     

A matrix triangularization algorithm for the polynomial point interpolation method

A novel matrix triangularization algorithm (MTA) is proposed to overcome the singularity problem in the point interpolation method (PIM) using the polynomial basis, and to ensure stable and reliable construction of PIM shape functions. The present algorithm is validated using several examples, and implemented in the local point interpolation method (LPIM) that is a truly meshfree method based on a local weak form. Numerical examples demonstrate that LPIM using the present MTA are very easy to implement, and very robust for solving problems of computational mechanics. It is shown that PIM with the present MTA is very effective in constructing shape functions. Most importantly, PIM shape functions possess Kronecker delta function properties. Parameters that influence the performance of them are studied in detail. The convergence and efficiency of them are thoroughly investigated.     

并行格型结构实现——一维块时间递归实值离散Gabor变换

为了有效和快速地计算实值离散Gabor变换,本文提出了在临界抽样条件下,一维块时间递归实值离散Gabor变换系数求解算法和由变换系数重建原信号算法,并研究了并行格型结构实现这两种算法的方法。     

Column and diagonal‐reduction of polynomial matrices by orthogonal transformation

Many of the applications of polynomial matrices in real world systems require column‐ or diagonally‐reduced polynomial matrices. If a given polynomial matrix is not column‐ or diagonally‐reduced, Callier or Wolowich algorithms, which use unimodular transformations, can be applied for column‐ or diagonal‐reduction, respectively, as a pre‐processing step in the applications. However, Callier and Wolowich algorithms may be unstable, from a numerical viewpoint, because they use elementary column and row operations. The purpose of this paper is to present sufficient conditions for existence of a constant orthogonal transformation of the given polynomial matrix so that it becomes column‐ or diagonally‐reduced. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Development of stability-preserving time-limited model reduction framework for 2-D and 1-D models with error bound

Due to their complex structure, 2-Dmodels are challenging towork with; additionally, simulation, analysis, design, and control get increasingly difficult as the order of the model grows. Moreover, in particular time intervals, Gawronski and Juang’s timelimited model reduction schemes produce an unstable reduced-order model for the 2-D and 1-D models. Researchers revealed some stability preservation solutions to address this key flaw which ensure the stability of 1-D reduced-order systems; nevertheless, these strategies result in large approximation errors. However, to the best of the authors’ knowledge, there is no literature available for the stability preserving time-limited-interval Gramian-based model reduction framework for the 2-D discrete-time systems. In this article, 2-D models are decomposed into two separate sub-models (i.e., two cascaded 1-D models) using the condition of minimal rank-decomposition. Model reduction procedures are conducted on these obtained two 1-D sub-models using limited-time Gramian. The suggestedmethodology works for both 2-D and 1-Dmodels. Moreover, the suggested methodology gives the stability of the reduced model as well as a priori error-bound expressions for the 2-D and 1-D models. Numerical results and comparisons between existing and suggested methodologies are provided to demonstrate the effectiveness of the suggested methodology.     

A non-oscillatory and conservative semi-Lagrangian scheme with fourth-degree polynomial interpolation for solving the Vlasov equation

A conservative semi-Lagrangian scheme for the numerical solution of the Vlasov equation is developed based on the fourth-degree polynomial interpolation. Then, a numerical filter is implemented that preserves positivity and non-oscillatory. The numerical results of both one-dimensional linear advection and two-dimensional Vlasov–Poisson simulations show that the numerical diffusion with the fourth-degree polynomial interpolation is suppressed more than with the cubic polynomial interpolation. It is also found that inherent conservation properties of the Vlasov equation can be improved by combining numerical fluxes of the upwind-biased and central fourth-degree polynomial interpolations.     

Chebyshev approximation of functions by the sum of a polynomial and an expression with a nonlinear parameter and endpoint interpolation

Sufficient existence conditions are established for the uniform Chebyshev (minimax) approximation of a function by the sum of a polynomial and an expression with a nonlinear parameter with the minimum absolute error and interpolation at the interval endpoints. An algorithm for determining the parameters of such an approximation using the Remez algorithm is proposed. The application of the iterative method to calculating the nonlinear parameter is substantiated. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 64–75, January–February 2009.     

An efficient algorithm for computing the coefficients of the characteristic polynomial of 2-D state-space models

For 2-D systems described by a state-space structure, two different methods for computing the coefficients of their characteristic polynomial are introduced. The relative efficiencies of these methods are compared to that of Faddeeva's method. An efficient method is then chosen which can be used to compute the coefficients of the characteristic polynomial with a minimum number of multiplications.     

General model of n-D system with variable coefficients and its reduction to 1-D system with variable structure

A general model of an n-D linear discrete system with variable coefficients and its solution are presented. A method for reduction of this model to an equivalent 1-D system with a variable structure is given.     

Multisignal 1-D compression by F-transform for wireless sensor networks applications

In wireless sensor networks a large amount of data is collected for each node. The challenge of transferring these data to a sink, because of energy constraints, requires suitable techniques such as data compression. Transform-based compression, e.g. Discrete Wavelet Transform (DWT), are very popular in this field. These methods behave well enough if there is a correlation in data. However, especially for environmental measurements, data may not be correlated. In this work, we propose two approaches based on F-transform, a recent fuzzy approximation technique. We evaluate our approaches with Discrete Wavelet Transform on publicly available real-world data sets. The comparative study shows the capabilities of our approaches, which allow a higher data compression rate with a lower distortion, even if data are not correlated.     

MIMO l 1 optimal control problems via the polynomial equations approach

The general MIMO multi-block l 1 -optimal control problem is considered. A novel solution is derived by resorting to polynomial techniques and simple algebraic conditions on the free parameter of the YJBK parameterization. As a result, we arrive at unconstrained LP formulations, less affected by redundancy and solvable by standard LP solvers. As usual, because the optimal controller is in general infinite-dimensional, a solution scheme based on solving sequences of increasing larger finite dimensional sub/super-optimal optimization problem is proposed, viz. sequences of finite dimensional optimization problems whose solutions provide lower and, respectively, upper bounds to the optimum, monotonically converging to it. Finally, an example is presented in order to exemplify the theory and show the effectiveness of the method.     

Learning the dynamics of objects by optimal functional interpolation

Many areas of science and engineering rely on functional data and their numerical analysis. The need to analyze time-varying functional data raises the general problem of interpolation, that is, how to learn a smooth time evolution from a finite number of observations. Here, we introduce optimal functional interpolation (OFI), a numerical algorithm that interpolates functional data over time. Unlike the usual interpolation or learning algorithms, the OFI algorithm obeys the continuity equation, which describes the transport of some types of conserved quantities, and its implementation shows smooth, continuous flows of quantities. Without the need to take into account equations of motion such as the Navier-Stokes equation or the diffusion equation, OFI is capable of learning the dynamics of objects such as those represented by mass, image intensity, particle concentration, heat, spectral density, and probability density.     

Estimation of 3-D peak L5/S1 joint moment during asymmetric lifting tasks with cubic spline interpolation of segment Euler angles

Previous research proposed a method using interpolation of the joint angles in key frames extracted from a field-survey video to estimate the dynamic L5/S1 joint loading for symmetric lifting tasks. The advantage of this method is that there is no need to use unwieldy equipment for capturing full body movement for the lifting tasks. The current research extends this method to asymmetric lifting tasks. The results indicate that 4-point cubic spline interpolation of segment Euler angles combined with a biomechanical model can provide a good estimation of 3-D peak L5/S1 joint moments for asymmetric lifting tasks. The average absolute error in the coronal, sagittal, and transverse planes with respect to the local pelvis axes was 16Nm, 22Nm, and 11Nm, respectively. It was also found that the dynamic component of the peak L5/S1 joint moment was not monotonously convergent when the number of interpolation points was increased. These results can be helpful for developing applied ergonomic field-survey tools such as video bases systems for estimating L5/S1 moments of manual materials handling tasks.