A multivariate QD-like algorithm

The problem of constructing a univariate rational interpolant or Padé approximant for given data can be solved in various equivalent ways: one can compute the explicit solution of the system of interpolation or approximation conditions, or one can start a recursive algorithm, or one can obtain the rational function as the convergent of an interpolating or corresponding continued fraction.In case of multivariate functions general order systems of interpolation conditions for a multivariate rational interpolant and general order systems of approximation conditions for a multivariate Padé approximant were respectively solved in [6] and [9]. Equivalent recursive computation schemes were given in [3] for the rational interpolation case and in [5] for the Padé approximation case. At that moment we stated that the next step was to write the general order rational interpolants and Padé approximants as the convergent of a multivariate continued fraction so that the univariate equivalence of the three main defining techniques was also established for the multivariate case: algebraic relations, recurrence relations, continued fractions. In this paper a multivariate qd-like algorithm is developed that serves this purpose.     

Block Based Bivariate Blending Rational Interpolation via Symmetric Branched Continued Fractions

This paper constructs a new kind of block based bivariate blending rational interpolation via symmetric branched continued fractions. The construction process may be outlined as follows. The first step is to divide the original set of support points into some subsets (blocks). Then construct each block by using symmetric branched continued fraction. Finally assemble these blocks by Newton’s method to shape the whole interpolation scheme. Our new method offers many flexible bivariate blending rational interpolation schemes which include the classical bivariate Newton’s polynomial interpolation and symmetric branched continued fraction interpolation as its special cases. The block based bivariate blending rational interpolation is in fact a kind of tradeoff between the purely linear interpolation and the purely nonlinear interpolation. Finally, numerical examples are given to show the effectiveness of the proposed method.     

Representation of algebraic numbers by periodic branching continued fractions

The notion of a periodic branching continued fraction is a natural generalization of the notion of a periodic continued fraction. It is shown that for an arbitrary positive algebraic number one can construct a periodic branching continued fraction with natural elements convergent to this number.     

SEQUENTIAL SYSTEMS OF LINEAR EQUATIONS ALGORITHM FOR NONLINEAR OPTIMIZATION PROBLEMS-INEQUALITY CONSTRAINED PROBLEMS

AbstractIn this paper, a new superlinearly convergent algorithm of sequential systems of linear equations (SSLE) for nonlinear optimization problems with inequality constraints is proposed. Since the new algorithm only needs to solve several systems of linear equations having a same coefficient matrix per iteration, the computation amount of the algorithm is much less than that of the existing SQP algorithms per iteration. Moreover, for the SQP type algorithms, there exist so-called inconsistent problems, i.e., quadratic programming subproblems of the SQP algorithms may not have a solution at some iterations, but this phenomenon will not occur with the SSLE algorithms because the related systems of linear equations always have solutions. Some numerical results are reported.     

Sequential Systems of Linear Equations Algorithm for Nonlinear Optimization Problems with General Constraints

In Ref. 1, a new superlinearly convergent algorithm of sequential systems of linear equations (SSLE) for nonlinear optimization problems with inequality constraints was proposed. At each iteration, this new algorithm only needs to solve four systems of linear equations having the same coefficient matrix, which is much less than the amount of computation required for existing SQP algorithms. Moreover, unlike the quadratic programming subproblems of the SQP algorithms (which may not have a solution), the subproblems of the SSLE algorithm are always solvable. In Ref. 2, it is shown that the new algorithm can also be used to deal with nonlinear optimization problems having both equality and inequality constraints, by solving an auxiliary problem. But the algorithm of Ref. 2 has to perform a pivoting operation to adjust the penalty parameter per iteration. In this paper, we improve the work of Ref. 2 and present a new algorithm of sequential systems of linear equations for general nonlinear optimization problems. This new algorithm preserves the advantages of the SSLE algorithms, while at the same time overcoming the aforementioned shortcomings. Some numerical results are also reported.     

A SUFFICIENT CONDITION OF CONVERGENCE FOR CLIFFORD CONTINUED FRACTIONS

In this article, a sufficient condition for a Clifford continued fraction to be convergent is established, and some applications are given.     

A reduced Newton method for constrained linear least-squares problems

We propose an iterative method that solves constrained linear least-squares problems by formulating them as nonlinear systems of equations and applying the Newton scheme. The method reduces the size of the linear system to be solved at each iteration by considering only a subset of the unknown variables. Hence the linear system can be solved more efficiently. We prove that the method is locally quadratic convergent. Applications to image deblurring problems show that our method gives better restored images than those obtained by projecting or scaling the solution into the dynamic range.     

Thermodynamic Formalism and Multifractal Analysis of Conformal Infinite Iterated Function Systems

We develop the thermodynamic formalism for equilibrium states of strongly Hölder families of functions. These equilibrium states are supported on the limit set generated by iterating a system of infinitely many contractions. The theory of these systems was laid out in an earlier paper of the last two authors. The first five sections of this paper except Section 3 are devoted to developing the thermodynamic formalism for equilibrium states of Hölder families of functions. The first three sections provide us with the tools needed to carry out the multifractal analysis for the equilibrium states mentioned above assuming that the limit set is generated by conformal contractions. The theory of infinite systems of conformal contractions is laid out in [13]. The multifractal analysis is then given in Section 7. In Section 8 we apply this theory to some examples from continued fraction systems and Apollonian packing.     

THE LIMITING CASE OF THIELE'S INTERPOLATING CONTINUED FRACTION EXPANSION

1. IntroductionWhen we talk abode the interpolation by polynomials, it is natural for us to have at heatthe Lagrange interpolation, the Heedte interpolation aam the Newton interPOlation. Of theSeinterpolats, the Newton interpolating polyno~ is probably most favourite because of itsadVantages in carrying out compilations and performing ~s. As we know, a Newton interpolating polynondal is established on the basis of the divided ~, whose recursivecalculation maal it possible that the Newton i…     

A new sequential systems of linear equations algorithm of feasible descent for inequality constrained optimization

Based on a new efficient identification technique of active constraints introduced in this paper, a new sequential systems of linear equations （SSLE） algorithm generating feasible iterates is proposed for solving nonlinear optimization problems with inequality constraints. In this paper, we introduce a new technique for constructing the system of linear equations, which recurs to a perturbation for the gradients of the constraint functions. At each iteration of the new algorithm, a feasible descent direction is obtained by solving only one system of linear equations without doing convex combination. To ensure the global convergence and avoid the Maratos effect, the algorithm needs to solve two additional reduced systems of linear equations with the same coefficient matrix after finite iterations. The proposed algorithm is proved to be globally and superlinearly convergent under some mild conditions. What distinguishes this algorithm from the previous feasible SSLE algorithms is that an improving direction is obtained easily and the computation cost of generating a new iterate is reduced. Finally, a preliminary implementation has been tested.     

Integral estimation and adaptive stabilization of non-holonomic controlled systems

A method for the programmed stabilization of non-holonomic dynamic systems is proposed. The original problem is reduced to a constrained adaptive control problem with unknown perturbations, which are represented by the reactions of linear (not necessarily ideal) non-holonomic constraints. Effective control and parameter estimation algorithms are constructed for the exponential stabilization of the system. The method can be extended to non-holonomic systems whose parameters are not known in advance or undergo an unknown bounded drift with time.     

最优化两个拓广的SQP和SSLE算法模型及其超线性和二次收敛性

给出一般约束最优化的序列二次规划（SQP）和序列线性方程组（SSLE）算法两个拓广的模型,详细分析和论证两个模型的局部超线性收敛性及二次收敛性条件,其中并不需要严格互补条件,拓广的模型及其收敛速度结果具有更广泛的适用性,为SQP和SSLE算法收敛速度的研究提供了更为完善和便利的理论基础。     

An Improved Delay-Dependent Criterion for Asymptotic Stability of Uncertain Dynamic Systems with Time-Varying Delays

In this paper, the problem of stability analysis for uncertain dynamic systems with time-varying delays is considered. The parametric uncertainties are assumed to be bounded in magnitude. Based on the Lyapunov stability theory, a new delay-dependent stability criterion for the system is established in terms of linear matrix inequalities, which can be solved easily by various efficient convex optimization algorithms. Two numerical examples are illustrated to show the effectiveness of proposed method.     

Data-driven recursive least squares methods for non-affined nonlinear discrete-time systems

Aiming at identifying nonlinear systems, one of the most challenging problems in system identification, a class of data-driven recursive least squares algorithms are presented in this work. First, a full form dynamic linearization based linear data model for nonlinear systems is derived. Consequently, a full form dynamic linearization-based data-driven recursive least squares identification method for estimating the unknown parameter of the obtained linear data model is proposed along with convergence analysis and prediction of the outputs subject to stochastic noises. Furthermore, a partial form dynamic linearization-based data-driven recursive least squares identification algorithm is also developed as a special case of the full form dynamic linearization based algorithm. The proposed two identification algorithms for the nonlinear nonaffine discrete-time systems are flexible in applications without relying on any explicit mechanism model information of the systems. Additionally, the number of the parameters in the obtained linear data model can be tuned flexibly to reduce computation complexity. The validity of the two identification algorithms is verified by rigorous theoretical analysis and simulation studies.     

用链式模型讨论圣文南原理

用泛函分析的双空间理论为计算力学构造卫一个严密的背景理论,以此在链式模型上讨论圣南原理,同时将传统的连分数扩展为算子连分式作为链式模型的本征关系式,平衡力系的影响在链式模型上由近及远扩衰减受算子边分式的收敛性的控制,所以南原理的合理成分体现为算子连分式的收敛性,发散的算子连分式对应着平衡力系的明显非零的影响可以传达到无穷远的场合,所以“圣南原理”并不是普遍成立的原理。     

Feasible Direction Interior-Point Technique for Nonlinear Optimization

We propose a feasible direction approach for the minimization by interior-point algorithms of a smooth function under smooth equality and inequality constraints. It consists of the iterative solution in the primal and dual variables of the Karush–Kuhn–Tucker first-order optimality conditions. At each iteration, a descent direction is defined by solving a linear system. In a second stage, the linear system is perturbed so as to deflect the descent direction and obtain a feasible descent direction. A line search is then performed to get a new interior point and ensure global convergence. Based on this approach, first-order, Newton, and quasi-Newton algorithms can be obtained. To introduce the method, we consider first the inequality constrained problem and present a globally convergent basic algorithm. Particular first-order and quasi-Newton versions of this algorithm are also stated. Then, equality constraints are included. This method, which is simple to code, does not require the solution of quadratic programs and it is neither a penalty method nor a barrier method. Several practical applications and numerical results show that our method is strong and efficient.     

The local convergence of ABS methods for nonlinear algebraic equations

Summary In this paper we consider an extension to nonlinear algebraic systems of the class of algorithms recently proposed by Abaffy, Broyden and Spedicato for general linear systems. We analyze the convergence properties, showing that under the usual assumptions on the function and some mild assumptions on the free parameters available in the class, the algorithm is locally convergent and has a superlinear rate of convergence (per major iteration, which is computationally comparable to a single Newton's step). Some particular algorithms satisfying the conditions on the free parameters are considered.     

Study of a Generalized Nonlinear Euler-Poisson-Darboux System: Numerical and Bessel Based Solutions

In this paper a nonlinear Euler-Poisson-Darboux system is considered. In a first part, we proved the genericity of the hypergeometric functions in the development of exact solutions for such a systemin some special cases leading to Bessel type differential equations. Next, a finite difference scheme in two-dimensional case has been developed. The continuous system is transformed into an algebraic quasi linear discrete one leading to generalized Lyapunov-Sylvester operators. The discrete algebraic system is proved to be uniquely solvable, stable and convergent based on Lyapunov criterion of stability and Lax-Richtmyer equivalence theorem for the convergence. A numerical example has been provided at the end to illustrate the efficiency of the numerical scheme developed in section 3. The present method is thus proved to be more accurate than existing ones and lead to faster algorithms.     

The Borel—Bernstein Theorem for multidimensional continued fractions

A central result in the metric theory of continued fractions, the Borel—Bernstein Theorem gives statistical information on the rate of increase of the partial quotients. We introduce a geometrical interpretation of the continued fraction algorithm; then, using this set-up, we generalize it to higher dimensions. In this manner, we can define known multidimensional algorithms such as Jacobi—Perron, Poincaré, Brun, Rauzy induction process for interval exchange transformations, etc. For the standard continued fractions, partial quotients become return times in the geometrical approach. The same definition holds for the multidimensional case. We prove that the Borel—Bernstein Theorem holds for recurrent multidimensional continued fraction algorithms. Supported by a grant from the CNP _q -Brazil, 301456/80, and FINEP/CNP _q /MCT 41.96.0923.00 (PRONEX).     

Branched continued fractions for double power series

A branched continued fraction (BCF) is defined and some of their properties are shown. This branched continued fraction corresponds to the double power series. One theorem of Van Vleck is transformed for the case of double power series and BCF.