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Improving the Efficiency of a Nonlinear-System-Solver Using a Componentwise Newton Method
Abstract
A Componentwise Interval Newton Method: We give an efficient branch-and-prune algorithm for finding enclosures of all solutions of a system of nonlinear equations. It is based on a componentwise interval Newton operator that temporarily considers a function of the system of equations as a onedimensional real-valued function having interval coefficients. Using interval arithmetic and enhancing the componentwise method by several techniques, we present an algorithm that works rather efficiently, especially on many "real-world" problems. 1 Introduction We address the problem of reliably finding all solutions of the nonlinear system f i (x 1 ; x 2 ; : : : ; x n ) = 0; i = 1; : : : ; n; (1) where the variables x j are bounded by real intervals: x j 2 [x] j ; j = 1; : : : ; n: As usual, the set of real intervals is denoted by IIR, accordingly IIR n is the set real interval vectors. Thus, we denote the search area by [x] = ([x] 1 ; [x] 2 ; : : : ; [x] n ) ? 2 IIR n . When we write a ...
... Herbort & Ratz [7] introduced the problem in their attempt to develop a new componentwise Newton operator, using a univariate Newton iteration on a unary projection of f i onto one of the variables x 1 , . . . , x n . ...
... In this work, we propose a greedy algorithm which determines a transversal dynamically by exploring information not only from the incidence matrix (static) but also from the current subregion (dynamic). The algorithm has the advantage that it does not use any first order information, in contrast to the previous proposed algorithms [7,2,3,14]. From our point of view, the selection of an transversal can be seen as a matching problem on the bipartite graph G = (F, X , E) associated with the the incidence matrix of the nonlinear system. ...
... In this section, we introduce experimental results in order to demonstrate the acceletating of the efficiency of interval Gauss-Seidel method using our proposed algorithm as a preprocessing step (4M+Gauss-Seidel) and comparing it with the traditional method (Gauss Seidel) in a variate of benchmarks. The test problems have been taken from numerical [1,8,11] and interval analysis [7,10] papers. ...
Interval methods have been established for rigorously bounding all solutions of a nonlinear system of equations within a given region. In this paper, we introduce a new method for determining a good pivoting sequence for Gauss-Seidel method, based on a greedy algorithm, called 4M, solving assignment problems with worst case complexity O(n^2).
... Herbort and Ratz in [5] have been suggested a componentwise interval Newton method for finding enclosures of all solutions of a system of nonlinear equations. The proposed operator is applied on a temporarily univariate real-valued function of the system, which has interval coefficients. ...
... The following system is a representative of a set of examples given by Moore and Jones [5]. An analogous reduction is occurred in the above 10x10 polynomial system, using as starting searching box the interval [0, 2] 10 . ...
... The proposed heuristics do not constitute a new method but seems to optimize the performance of the using interval method. At the moment, we have considered the proposed technique as a preprocessing step, however, our major goal is to incorporate this reordering technique in the componentwise interval Newton method proposed in [5] by Herbort and Ratz. ...
... For our problem, previous experiments (see [11], [13]) have shown that the componentwise Newton operator (Ncmp) [7] and Gauss-Seidel (GS) operator [8] perform the best, but their performance varies highly on different problems. Indeed, when we look on Tables 1 and 2 in Section 4, below, the componentwise operator outperforms GS e.g. for Rheinboldt and Puma6 problems, but is much worse for TwoCirc, Hippopede or Puma7. ...
... the componentwise Newton operator with Herbort and Ratz heuristic to choose pairs equation-variable for reduction [7], -the Gauss-Seidel operator with a rectangular matrix; Gauss elimination with full pivoting used for elements choice and inverse-midpoint preconditioning [8], [11], [13]. ...
... the componentwise operator linearizes the function wrt (with respect to) one variable only at a time, while GS -wrt all of them, -the componentwise operator tries to narrow all variables using all equations (Herbort and Ratz heuristic [7]), while GS chooses m variables (out of n) and uses only one equation for each of them, -the componentwise operator -because of its essence -does not require any preconditioning or other matrix operations. ...
The paper presents the author’s investigations in the field of improving the performance of a multithreaded interval solver of equations systems, targeted for underdetermined systems. New heuristics to choose the interval Newton operator variant and bisection direction are proposed. Numerical results for several benchmark problems are presented and analyzed. Also some other tools and future possible improvements are suggested.
... However, with large systems and large domains, algorithms that rely on matrix computation and linear approximations such as the Hansen-Sengupta method become computationally expensive and inefficient. As an alternative, Herbort and Ratz proposed a componentwise Newton method [7] that considers each equation fi(x1, . . . , xn) = 0 separately, using a unidimensional Newton iteration on a unary projection of fi onto one of the variables x1, . . . ...
... , fn, one considers the n unary projection constraints: and uses any unidimensional root-finding method to tighten the domain of each variable xi in turn. Using the unidimensional Newton method leads to the Gauss-Seidel-Newton method [12], whose extension to intervals is the Herbort-Ratz method [7]. Let HR be the elementary step performed by one unidimensional Newton step applied to a projection fi (j) , where i and j may be different. ...
... In the linear case, it is well-known that the efficiency of the Gauss-Seidel method may depend heavily on the ordering of the fis, that is the choice of a transversal in the coefficients matrix. The same holds true for nonlinear Gauss-Seidel methods and the choice of the pairs (fi, xj) to consider for generating the n projection constraints appears a key factor in their efficiency [7,1]. ...
We show that a classical interval constraint propagation algorithm enforcing box consistency may be interpreted as a free-steering nonlinear Gauss-Seidel procedure. This suggests that the choice of a transversal in the incidence matrix associated with the problem to solve is paramount to the efficiency of the algorithm. We present experimental evidences that it is indeed so, and we suggest an heuristics to compute good transversals. The improved interval constraint algorithm is compared with a classical one and with standard methods such as Hansen-Sengupta on some well-known benchmarks.
... For the linear case, it is well known that reordering equations and variables to select a transversal is paramount to the speed of convergence of first-order iterative methods such as Gauss-Seidel [3, 5]. Transversals may also be computed [14, 7, 4] in the nonlinear case when using nonlinear Gauss-Seidel methods [12] and when solving the linear systems arising in Newton methods (e.g., preconditioned Newton-Gauss-Seidel, aka Hansen-Sengupta's method [6]). Interval-based nonlinear Gauss-Seidel (INLGS) methods are of special importance because they constitute the basis for interval constraint algorithms [4] that often outperform extensions to intervals of numerical methods. ...
... and uses any unidimensional root-finding method to tighten the domain of each variable x i in turn. Unidimensional Newton leads to the Gauss-Seidel-Newton method [12], whose extension to intervals is the Herbort-Ratz method [7]. ...
... – Herbort and Ratz [7] compute the Jacobian J of System (1) w.r.t. the initial box D and select projections according to whether the corresponding entry in the Jacobian straddles zero or not. Their method is not completely static since they recompute the Jacobian after each iteration of INLGS (the choice of projections is not completely reconsidered, though). ...
This paper investigates the impact of the selection of a transversal on the speed of convergence of interval methods based
on the nonlinear Gauss-Seidel scheme to solve nonlinear systems of equations. It is shown that, in a marked contrast with
the linear case, such a selection does not speed up the computation in the general case; directions for researches on more
flexible methods to select projections are then discussed.
... However, to our knowledge, there is no sure-fire static method to select projections that ensures prompt convergence. What is more, it appears [9, 11, 8] that selecting more than n projections may sometime speed the solving process up. The bc3 algorithm [4] studied in this paper associates the good principles of first-order methods with a smart propagation algorithm devised by Mack- worth [19] to ensure consistency in a network of relations. ...
... and uses any unidimensional root-finding method to tighten the domain of each variable x i in turn. Using a unidimensional Newton-Raphson root-finder leads to the Gauss-Seidel-Newton method [23], whose extension to intervals is the Herbort-Ratz method [11]. Algorithm 1 Branch-and-Prune algorithm [BaP] in: F = (f 1 , . . . ...
... In their paper, the static transversal thus obtained is then used in an interval Newton-Gauss-Seidel algorithm. Another approach uses a Gauss-Seidel-Newton method as presented in the introduction: Herbort and Ratz [11] (hr) compute the Jacobian J of the equation system w.r.t. the initial box D, and they select projections according to whether the corresponding entry in the Jacobian straddles zero or not. Their method is not completely static since they recompute the Jacobian after each outer step of Gauss-Seidel. ...
When solving systems of nonlinear equations with interval constraint methods, it has often been observed that many calls to contracting operators do not participate actively to the reduction of the search space. Attempts to statically select a subset of efficient contracting operators fail to offer reliable performance speed-ups. By embedding the recency-weighted average Reinforcement Learning method into a constraint propagation algorithm to dynamically learn the best operators, we show that it is possible to obtain robust algorithms with reliable performances on a range of sparse problems. Using a simple heuristic to compute initial weights, we also achieve significant performance speed-ups for dense problems.
... Herbort and Ratz in [5] have been suggested a componentwise interval Newton method for finding enclosures of all solutions of a system of nonlinear equations. The proposed operator is applied on a temporarily univariate real-valued function of the system, which has interval coefficients. ...
... The following system is a representative of a set of examples given by Moore and Jones [5]. An analogous reduction is occurred in the above 10x10 polynomial system, using as starting searching box the interval [0, 2] 10 . ...
... The proposed heuristics do not constitute a new method but seems to optimize the performance of the using interval method. At the moment, we have considered the proposed technique as a preprocessing step, however, our major goal is to incorporate this reordering technique in the componentwise interval Newton method proposed in [5] by Herbort and Ratz. ...
Methods of interval arithmetic can be used to reliably find with certainty all solutions to nonlinear systems of equations. In such methods, the system is transformed into a linear interval system and a preconditioned interval Gauss-Seidel method may then be used to compute such solution bounds. In this work, a new heuristic for solving polynomial systems is presented, called reordering technique. The proposed technique constitutes a preprocessing step to interval Gauss-Seidel method to improve the overall e#ciency of an interval Newton method. The key idea is to exploit some properties of the original polynomial system, expressed by two suitable permutation matrices, by reordering the resulted linearized system. Numerical experiments have been shown that the permuted system can be solved e#ciently when it is combined with an interval Newton method, like Hansen's algorithm.
... while (there is a quadruple in L, for which wid y ≥ ε y ) take this quadruple (y, L in , L bound , L unchecked ) from L; bisect y to y (1) and y (2) ; ...
... These are, in particular, the multicriterial variant of the monotonicity test and the componentwise Newton operator N cmp (see e.g. [2]), used to narrow/discard boxes that do not contain solutions of equations q k (x) − y k = 0, where k = 1, . . . , N and values y k (components of the inverted vector interval) are parameters for these equations. ...
Previous investigations of the authors surveyed the possibility of applying interval methods to seek the Pareto-front of a multicriterial nonlinear problem and an efficient algorithm has been proposed.
With the advent of multi-core computer architectures, algorithms could be modified to fit the multi-threading model.
This allows to decrease the execution time by running the computations in parallel on several cores.
The paper presents a multi-threaded variant of the previously developed algorithm, parallelized using the Pthreads library.
Numerical results for benchmark test problems are presented.
... Dans le cas non linéaire, cet ordre a aussi son importance, ce qui a conduit plusieurs chercheurs [15,5,8] à proposer des critères heuristiques permettant de choisir ces n projections. Nous avons cependant montré [6] que ces tentatives sont vouées à l'échec dans le cas général, les problèmes non-linéaires ne disposant pas toujours d'une transversale statique. ...
... Si cette approche évite les mauvais choix de transversale, elle s'avère rapidement impraticable pour les problèmes denses de grande taille ; en eet, le nombre de projections à considérer ralentit alors considérablement l'algorithme. Plusieurs auteurs ont proposé des heuristiques pour choisir statiquement [15,5,8] ou semi-statiquement [7] n projections parmi les n 2 possibles. Ces approches ont fait leurs preuves sur certains problèmes, mais chacune échoue sur d'autres problèmes, produisant des temps de résolution très important par rapport à BC3. ...
Durant la résolution de systèmes d'équations non linéaires, de nombreuses projections sont utilisées sans résultat. Les heuristiques visant à sélectionner a priori les meilleures projections ne donnent pas toujours de bons résultats ; en fait, nous avons démontré récemment qu'aucune telle heuristique ne peut fonctionner en général car l'intérêt d'une projection varie en cours de résolution. Dans cet article, nous considérons le problème de la sé- lection dynamique des projections comme un problème de bandit-manchot non stationnaire ; nous montrons que l'utilisation de méthodes d'apprentissage conduit à un nouvel algorithme de propagation qui surpasse les algorithmes standards sur plusieurs problèmes, et surtout ore des performances stables sur l'ensemble des probl èmes de notre banc d'essai.
... The definition. According to [9] we define the interval componentwise Newton operator as follows. ...
... Known properties. In [9] two simple properites of the N cmp operator are proven: ...
This paper considers two main topics.
The first one is a new interval global optimization algorithm, using some symbolic transformations of the optimality conditions. The theory of Groebner bases and the idea of componentwise interval Newton method are used.
The second topic is the description of an optimization problem connected with access control to a computer server. This optimization problem is solved by using a new algorithm and, for comparison, by using a classical interval branch-and-bound algorithm.
... the componentwise Newton operator [7], -the Gauss-Seidel (GS) operator with rectangular matrix [9]. ...
... Pairs equation-variable can be chosen using several heuristics. Our implementation uses the strategy of S. Herbort and D. Ratz [7] and tries to use all possible pairs subsequently. ...
The paper presents an extension of a previously developed interval method for solving multi-criteria problems [13]. The idea is to use second order information (i.e., Hesse matrices of criteria and constraints) in a way analogous to global optimization (see e.g. [6], [9]). Preliminary numerical results are presented and parallelization of the algorithm is considered.
... Additionally, the algorithm exploits the feasible intervals of monomials and polynomials to update the interval of variables within them. BICP involves both Boolean constraint propagation and numerical computation algorithms, a typical example being Newton's method for interval arithmetic [14]. During the propagation, BICP efficiently contracts the feasible interval of integer variables. ...
Satisfiability Modulo Theories on arithmetic theories have significant applications in many important domains. Previous efforts have been mainly devoted to improving the techniques and heuristics in sequential SMT solvers. With the development of computing resources, a promising direction to boost performance is parallel and even distributed SMT solving. We explore this potential in a divide-and-conquer view and propose a novel dynamic parallel framework with variable-level partitioning. To the best of our knowledge, this is the first attempt to perform variable-level partitioning for arithmetic theories. Moreover, we enhance the interval constraint propagation algorithm, coordinate it with Boolean propagation, and integrate it into our variable-level partitioning strategy. Our partitioning algorithm effectively capitalizes on propagation information, enabling efficient formula simplification and search space pruning. We apply our method to three state-of-the-art SMT solvers, namely CVC5, OpenSMT2, and Z3, resulting in efficient parallel SMT solvers. Experiments are carried out on benchmarks of linear and non-linear arithmetic over both real and integer variables, and our variable-level partitioning method shows substantial improvements over previous partitioning strategies and is particularly good at non-linear theories.
... Solving algebraic equations is a basic and important problem [2,3,11,15]. There are a large number of problems in practice that can be transformed into finding real roots of algebraic equations [7,12,13] (especially the multivariable nonlinear algebraic equations). ...
Solving algebraic equations over GF(2) is a problem which has a wide range of applications, including NP-Hard problems and problems related to cryptography. The existing mature algorithms are difficult to solve large-scale problems. Inspired by Schöning’s algorithm and its quantum version, we apply related methods to solve algebraic equations over GF (2). The new algorithm we proposed has a significant improvement of solving efficiency in large-scale and sparse algebraic equations. As a hybrid algorithm, the new algorithm can not only run on a classic computer alone, but also use small-scale quantum devices to assist acceleration. And the new algorithm can be seen as an example of solving a large-scale problem on a small-scale quantum device.
... • Interval constraint propagation (ICP ) [27,34] uses interval arithmetic to contract given variable domains under the assumption of certain constraints. For example, if x ∈ [0, 2] and y ∈ [1, 3] and x = y should hold then ICP can imply that x ∈ [1, 2] and y ∈ [1,2]. ...
We present the latest developments in SMT-RAT, a tool for the automated check of quantifier-free real and integer arithmetic formulas for satisfiability. As a distinguishing feature, \smtrat provides a set of solving modules and supports their strategic combination. We describe our CArL library for arithmetic computations, the available modules implemented on top of CArL, and how modules can be combined to satisfiability-modulo-theories (SMT) solvers. Besides the traditional SMT approach, some new modules support also the recently proposed and highly promising model-constructing @satisfiability calculus approach.
... (see, [3], [6]). Herbort and Ratz in [5] have been suggested a componentwise interval Newton method for finding enclosures of all solutions of a system of nonlinear equations. The proposed operator is applied on a temporarily univariate real-valued function of the system, which has interval coefficients. ...
... It adapts algebraic decision procedures to the needs of SMT solving and exploits powerful combinations of these procedures. Currently, it offers SMT-compliant implementations of the Fourier-Motzkin variable elimination, the simplex method [28], interval constraint propagation [36,39], methods based on Gröbner bases [68], the virtual substitution method [69], the cylindrical algebraic decomposition method [24], and a generalised branch-and-bound method. Additionally it provides a DPLL-style SAT solver as well as several preprocessing modules. ...
Satisfiability checking aims to develop algorithms and tools for checking the satisfiability of existentially quantified logical formulas. Besides powerful SAT solvers for solving propositional logic formulas, sophisticated SAT-modulo-theories (SMT) solvers are available for a wide range of theories, and are applied as black-box engines for many techniques in different areas. In this paper we give a short introduction to the theoretical foundations of satisfiability checking, mention some of the most popular tools, and discuss the successful embedding of SMT solvers in different technologies.
... Over the years many different methods for solving non-linear polynomial and non-polynomial systems of equations have been developed. The most common approaches for dealing with non-linear equations are either numerical or symbolic [18, 11, 12], continuation [16], reduction [19, 20] or iterative and interval methods [17, 14, 5, 9, 13], and sometimes even a combination of them, for example in most computer algebra tools [26] and [4]. But only one of these algorithms is based on the logic programming paradigm using the rule based programming language Constraint Handling Rules, namely INCLP(R) [6], which is also based on approximating results of a non-linear system. ...
Various methods for solving non-linear algebraic systems ex-ist, as this question is amongst the most popular in both the realm of mathematics and computation. As most of these methods use approx-imations, this work focuses on finding and directly solving a tractable subset. Bivariate binomial systems of non-linear polynomial equations were chosen and solved by simulating the by hand method, using the declarative logic programming language Constraint Handling Rules. Sub-stitution methods and different equation notations are used to extend the solvability of the subset.
... These constraints can be used to contract X j , the current search range for x j . There are various ways to do this, including univariate interval-Newton iteration [26] and methods [24] for direct calculation of new bounds for x j . This procedure can be repeated using a combination of any equation and any variable in the equation system. ...
Interval analysis provides techniques that make it possible to determine all solutions to a nonlinear algebraic equation system and to do so with mathematical and computational certainty. Such methods are based on the processing of granules in the form of intervals and thus can be regarded as one facet of granular computing. We review here some of the key concepts used in these methods and then focus on some specific application areas, namely ecological modeling, transition state analysis, and the modeling of phase equilibrium.
... For improving the efficiency of IN/GB methods, there are various approaches, including: 1. Methods for dealing with the " dependency " issue in interval arithmetic, which may prevent the computation of interval extensions that tightly bound the function range (e.g. Ratscheck and Rokne, 1984; Makino and Berz, 1999; Jansson, 2000); 2. Techniques that involve changes to the methodology at the level of the nonlinear equation system (e.g., Alefeld et al., 1998; Granvilliers and Hains, 2000; van Hentenryck et al., 1997; Ratz, 1994; Herbort and Ratz, 1997; Yamamura et al., 1998; Yamamura and Nishizawa, 1999); 3. Methods that seek to make improvements in solving the linear interval system defined by Eq. (1), the interval-Newton equation (e.g., Kearfott, 1990; Hu, 1990; Kearfott et al., 1991; Gan et al., 1994; Hansen, 1997); or some combination of the above (e.g., Madan, 1990; Dinkel et al., 1991; Kearfott, 1991; Kearfott, 1997 ). A comprehensive review or these and other techniques is beyond the scope of this paper. ...
The use of interval methods, in particular interval-Newton/generalized-bisection (IN/GB) techniques, provides an approach that is mathematically and computationally guaranteed to reliably solve difficult nonlinear equation solving and global optimization problems, such as those that arise in chemical process modeling. The most significant drawback of the currently used interval methods is the potentially high computational cost that must be paid to obtain the mathematical and computational guarantees of certainty. New methodologies are described here for improving the efficiency of the interval approach. In particular, a new hybrid preconditioning strategy, in which a simple pivoting preconditioner is used in combination with the standard inverse-midpoint method, is presented, as is a new scheme for selection of the real point used in formulating the interval-Newton equation. These techniques can be implemented with relatively little computational overhead, and lead to a large reduction in the number of subintervals that must be tested during the interval-Newton procedure. Tests on a variety of problems arising in chemical process modeling have shown that the new methodologies lead to substantial reductions in computation time requirements, in many cases by multiple orders of magnitude.
The European Erasmus+ project ARC – Automated Reasoning in the Class aims at improving the academic education in disciplines related to Computational Logic by using Automated Reasoning tools. We present the technical aspects of the tools as well as our education experiments, which took place mostly in virtual lectures due to the COVID pandemics. Our education goals are: to support the virtual interaction between teacher and students in the absence of the blackboard, to explain the basic Computational Logic algorithms, to study their implementation in certain programming environments, to reveal the main relationships between logic and programming, and to develop the proof skills of the students. For the introductory lectures we use some programs in C and in Mathematica in order to illustrate normal forms, resolution, and DPLL (Davis-Putnam-Logemann-Loveland) with its Chaff version, as well as an implementation of sequent calculus in the Theorema system. Furthermore we developed special tools for SAT (propositional satisfiability), some based on the original methods from the partners, including complex tools for SMT (Satisfiability Modulo Theories) that allow the illustration of various solving approaches. An SMT related approach is natural-style proving in Elementary Analysis, for which we developed and interesting set of practical heuristics. For more advanced lectures on rewrite systems we use the Coq programming and proving environment, in order on one hand to demonstrate programming in functional style and on the other hand to prove properties of programs. Other advanced approaches used in some lectures are the deduction based synthesis of algorithms and the techniques for program transformation.
In this chapter the author presents algorithms for the simplest kinds of problems that can be solved using interval branch-and-bound-type methods. These are the non-quantified problems: systems of inequalities and systems of equations. Specific tools, policies and heuristics are presented. The HIBA_USNE solver, developed by the author, is presented.
The satisfiability problem is the problem of deciding whether a logical formula is satisfiable. For first-order arithmetic theories, in the early 20th century some novel solutions in form of decision procedures were developed in the area of mathematical logic. With the advent of powerful computer architectures, a new research line started to develop practically feasible implementations of such decision procedures. Since then, symbolic computation has grown to an extremely successful scientific area, supporting all kinds of scientific computing by efficient computer algebra systems.
Independently, around 1960 a new technology called SAT solving started its career. Restricted to propositional logic, SAT solvers showed to be very efficient when employed by formal methods for verification. It did not take long till the power of SAT solving for Boolean problems had been extended to cover also different theories. Nowadays, fast SAT-modulo-theories (SMT) solvers are available also for arithmetic problems.
Due to their different roots, symbolic computation and SMT solving tackle the satisfiability problem differently. We discuss differences and similarities in their approaches, highlight potentials of combining their strengths, and discuss the challenges that come with this task.
In this paper, we address the problem of finding all solutions to polynomial systems, a fundamental and important problem in the research of real algebra from the viewpoint of algorithm research. An efficient intelligent hybrid algorithm for finding real solutions of non-linear systems is presented here. It is based on a branch-and-prune algorithm, combination with classical numerical methods, symbolic methods and interval methods. Besides these, there are some intelligent judgments which can improve the system's efficiency significantly for some kinds of applications. KEYWORDS hybrid method, interval arithmetic, symbolic methods, interval Newton methods, Wu's method.
Each Sard kernels theorem supplies multiple error estimates for a bounded linear functional, provided the underlying multivariate function is sufficiently smooth. The error estimation due to Sard generally concerns the
-norms of Sard kernels and the supremum norms of different partial derivatives of a given function. This article presents a verified method for computing Sard error constants. To assist the verified computation, we examine relevant properties of Sard kernels for general bounded linear functionals of bivariate functions. The derived
-norms, together with interval Taylor arithmetic, make the multiple error estimates possible. We demonstrate the flexibility and superiority of Sard kernels method by different numerical examples that concern non-product cubature for bivariate functions.
This paper gives an efficient hybrid algorithm for real root isolation. It is based on an branch-and-prune algorithm, combination with classical numerical methods, symbolic methods and interval methods. Besides these, there are some intelligent judgments which can improve the system's efficiency signifficantly for some kinds of applications. The algorithm presented here works rather efficiently.
We give a hybrid algorithm for solving non-linear polynomial systems. It is based on a branch-and-prune algorithm, combined
with classical numerical methods, symbolic methods and interval methods. For some kinds of problems, Gather-and-Sift method,
a symbolic method proposed by L. Yang, was used to reduce the dependency of variables or occurrences of the same variable,
then interval methods were used to isolate the real roots. Besides these, there are some intelligent judgments which can improve
the system’s efficiency significantly. The algorithm presented here works rather efficiently for some kinds of tests.
Interval methods are known to be a precise and robust tool of global optimization. Several interval algorithms have been developed
to deal with various kinds of this problem. Far less has been written about the use of interval methods in multicriterial
optimization. The paper surveys two methods presented by other researchers and proposes a modified approach, combining PICPA
algorithm with the use of derivative information. Preliminary numerical results are presented.
A Survey of PASCAL--XSC and a Language Reference Supplement: PASCAL--XSC is a general purpose programming language which provides special support for the implementation of sophisticated numerical algorithms. The new PASCAL--XSC system has the advantage of being portable across many platforms and is available for personal computers, workstations, mainframes and supercomputers by means of a portable compiler which translates to ANSI-C language. By using the mathematical modules of PASCAL--XSC, numerical algorithms which deliver highly accurate and automatically verified results can be programmed easily. PASCAL--XSC simplifies the design of programs in engineering and scientific computation by modular program structure, userdefined operators, overloading of functions, procedures, and operators, functions and operators with arbitrary result type and dynamic arrays. Arithmetic standard modules for additional numerical data types including operators and standard functions of high accuracy an...
An Optimized Interval Slope Arithmetic and its Application: We describe an interval slope arithmetic which is optimized in view of the possibility to compute sharper enclosures of the actual slope values. For this purpose, special formulas are used for convex/concave or locally convex/concave elementary functions to replace the normally used derivative values. We treat the details of a practical realization and we give an implementation of the slope arithmetic. Moreover, we show how our interval slope arithmetic can succesfully be applied in the field of one-dimensional global optimization. We describe the practical aspects of a new pruning technique based on slopes in the context of interval branch-and-bound methods. We show, that it is possible to replace the frequently used monotonicity test by a new pruning step which provides considerable improvement in efficiency of the global optimization method. 1 Introduction and Notation Many interval methods use interval slopes (together wi...
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