Interval arithmetic block cyclic reduction on vector computers

We introduce several algorithms for interval arithmetic block cyclic reduction for efficient application to vector computers under the condition that interval arithmetic inclusion properties be preserved. Interval arithmetic block cyclic reduction is used as part of almost globally convergent Newton-like methods for some classes of large nonlinear systems of equations. We further introduce truncated variants of the above algorithms and discuss their integration into nonlinear methods. We report numerical results carried out on a CRAY-1/M.     

Solving nonlinear polynomial systems in the barycentric Bernstein basis

We present a method for solving arbitrary systems of N nonlinear polynomials in n variables over an n-dimensional simplicial domain based on polynomial representation in the barycentric Bernstein basis and subdivision. The roots are approximated to arbitrary precision by iteratively constructing a series of smaller bounding simplices. We use geometric subdivision to isolate multiple roots within a simplex. An algorithm implementing this method in rounded interval arithmetic is described and analyzed. We find that when the total order of polynomials is close to the maximum order of each variable, an iteration of this solver algorithm is asymptotically more efficient than the corresponding step in a similar algorithm which relies on polynomial representation in the tensor product Bernstein basis. We also discuss various implementation issues and identify topics for further study.     

An interval-based SAT modulo ODE solver for model checking nonlinear hybrid systems

This paper presents a bounded model checking tool called Hydlogic{\texttt{Hydlogic}} for hybrid systems. It translates a reachability problem of a nonlinear hybrid system into a predicate logic formula involving arithmetic constraints and checks the satisfiability of the formula based on a satisfiability modulo theories method. We tightly integrate (i) an incremental SAT solver to enumerate the possible sets of constraints and (ii) an interval-based solver for hybrid constraint systems (HCSs) to solve the constraints described in the formulas. The HCS solver verifies the occurrence of a discrete change by using a set of boxes to enclose continuous states that may cause the discrete change. We utilize the existence property of a unique solution in the boxes computed by the HCS solver as (i) a proof of the reachability of a model and (ii) a guide in the over-approximation refinement procedure. Our Hydlogic{\texttt{Hydlogic}} implementation successfully handled several examples including those with nonlinear constraints.     

Efficient Interval Linear Equality Solving in Constraint Logic Programming

Existing interval constraint logic programming languages, such as BNR Prolog, work under the framework of interval narrowing and are deficient in solving systems of linear constraints over real numbers, which constitute an important class of problems in engineering and other applications. In this paper, we suggest to separate linear equality constraint solving from inequality and non-linear constraint solving. The implementation of an efficient interval linear constraint solver, which is based on the preconditioned interval Gauss-Seidel method, is proposed. We show how the solver can be adapted to incremental execution and incorporated into a constraint logic programming language already equipped with a non-linear solver based on interval narrowing. The two solvers share common interval variables, interact and cooperate in a round-robin fashion during computation, resulting in an efficient interval constraint arithmetic language CIAL. The CIAL prototypes, based on CLP(R), are constructed and compared favorably against several major interval constraint logic programming languages.     

Synchronous and Asynchronous Interval Newton-Schwarz Methods for a Class of Large Systems of Nonlinear Equations

We introduce a class of parallel interval arithmetic iteration methods for nonlinear systems of equations, especially of the type Ax+(x) = f, diagonal, in R ^N . These methods combine enclosure and global convergence properties of Newton-like interval methods with the computational efficiency of parallel block iteration methods: algebraic forms of Schwarz-type methods which generalize both the well-known additive algebraic Schwarz Alternating Procedure and multisplitting methods. We discuss both synchronous and asynchronous variants. Besides enclosure and convergence properties, we present numerical results from a CRAY T3E.     

一种基于SAT的运算电路查错方法

基于SAT的运算电路查错方法将被验证系统中系统规范成立与否的问题转换为布尔公式和数学公式的混合形式E-CNF,通过采用了标志子句技术的E-SAT求解器进行求解.实验表明该方法自动化程度高,能处理大规模的运算电路,有较强的查找错误能力.     

The Symm-Wilkinson method for improving an approximate eigenvalue and its associated eigenvector

Here is discussed the Symm-Wilkinson method (called a relaxed algorithm in [4]) for improving an approximate simple eigenvalue of ann×n matrix and a corresponding approximate eigenvector which were obtained by some method. It is shown that their method is a Newton-like method applied to a system of nonlinear equations so that the process converges linearly under the usual assumptions. The Symm-Wilkinson method needs more multiplications than the standard Newton-like method applied to the same equations byn?1 at each step. Therefore, there does not seem to be any great advantage in using the former in place of the latter.     

An efficient method for unconstrained optimization problems ofnonlinear large mesh-interconnected systems

Presents an efficient method for solving unconstrained optimization problems for nonlinear large mesh-interconnected systems. This method combines an approximate scaled gradient method with a block Gauss-Seidel with line search method which is used to obtain an approximate solution of the unconstrained quadratic programming subproblem. The authors prove that their method is globally convergent and demonstrate by several numerical examples its superior efficiency compared to a sparse matrix technique based method. In an example of a system of more than 200 variables, the authors observe that their method is 3.45 times faster than the sparse matrix technique based Newton-like method and about 50 times faster than the Newton-like method without the sparse matrix technique     

Numerically generated tangent stiffness matrices using the complex variable derivative method for nonlinear structural analysis

Accurate tangent stiffness matrices are essential for solving nonlinear structural problems using Newton-like methods. This paper presents a numerical procedure for generating accurate tangent stiffness matrices from the derivative approximation of internal force using the complex variable derivative method (CVDM). The present procedure is suitable for problems that require complex constitutive relations and/or kinematic nonlinearities whose models are difficult to linearize, as well as for the incorporation of Newton-like solution strategies with nonlinear structural analyses.     

最优控制问题的Legendre 伪谱法求解及其应用

伪谱法通过全局插值多项式参数化状态和控制变量,将最优控制问题(OCP)转化为非线性规划问题(NLP)进行求解,是一类具有更高求解效率的直接法。总结Legendre伪谱法转化Bolza型最优控制问题的基本框架,推导OCP伴随变量与NLP问题KKT乘子的映射关系,建立基于拟牛顿法的LGL配点数值计算方法,并针对非光滑系统,进一步研究分段伪谱逼近策略。基于上述理论开发通用OCP求解器,并对3个典型最优控制问题进行求解,结果表明了所提出方法和求解器的有效性。     

Analysis of nonlinear electrical circuits using bernstein polynomials

In electrical circuit analysis, it is often necessary to find the set of all direct current (d.c.) operating points (either voltages or currents) of nonlinear circuits. In general, these nonlinear equations are often represented as polynomial systems. In this paper, we address the problem of finding the solutions of nonlinear electrical circuits, which are modeled as systems of n polynomial equations contained in an n-dimensional box. Branch and Bound algorithms based on interval methods can give guaranteed enclosures for the solution. However, because of repeated evaluations of the function values, these methods tend to become slower. Branch and Bound algorithm based on Bernstein coefficients can be used to solve the systems of polynomial equations. This avoids the repeated evaluation of function values, but maintains more or less the same number of iterations as that of interval branch and bound methods. We propose an algorithm for obtaining the solution of polynomial systems, which includes a pruning step using Bernstein Krawczyk operator and a Bernstein Coefficient Contraction algorithm to obtain Bernstein coefficients of the new domain. We solved three circuit analysis problems using our proposed algorithm. We compared the performance of our proposed algorithm with INTLAB based solver and found that our proposed algorithm is more efficient and fast.     

Numerically generated tangent stiffness matrices for nonlinear structural analysis

Accurate tangent stiffness matrices are essential for the solution of nonlinear structural problems via Newton-like methods. The present paper presents an accurate numerical procedure for the generation of tangent stiffness matrices from the internal force generation modules. The present procedure is attractive for problems that require complex constitutive relations and/or kinematic nonlinearities whose models are difficult to linearize and for incorporating Newton-like solution strategies into vectorial-based nonlinear structural analysis codes.     

A symmetric iterative interval method for systems of nonlinear equations

An iterative method for nonlinear systems of equations is presented that is based on the idea of symmetric methods known from linear systems. Due to the use of interval arithmetic the convergence to a solution can be proved under relatively weak conditions provided an initial inclusion of that solution is known. The concept of symmetry leads to a reduction of computation time compared to some well-known methods.     

运用区间算法的约束非线性系统鲁棒模型预测控制

针对一类具有输入和状态约束的干扰有界非线性系统,提出了基于区间分析的约束非线性鲁棒模型预测控制,以降低计算量并扩大系统吸引域.首先,在集合运算的基础上,利用区间运算和函数区间扩展,给出了一种计算效能更好、保守性更低的非线性系统鲁棒一步集计算方法;其次,构造重叠的多面体控制不变集序列并以此计算约束非线性系统的鲁棒多步集,并通过设计基于集合的在线优化策略,提出了基于鲁棒一步集的单步优化非线性模型预测控制,有效降低了非线性优化的在线计算量;最后,仿真实例验证了算法的有效性.     

Computing simple bifurcation points using a minimally extended system of nonlinear equations

The present paper deals with the computation of simple bifurcation points of nonlinear parameter dependent equations. At first, a minimally extended system of nonlinear equations is constructed by addition of one parameter and two equations. This augmented system has an isolated solution which yields to the simple bifurcation point directly. Using the structural properties of this auxiliary system an adapted Newton-like method is developed not requiring evaluations of second derivatives. Finally, the results of some computer experiments show the efficiency of theR-quadratically convergent method.     

A special extended system and a Newton-like method for simple singular nonlinear equations

A special extended system and a locally quadratically convergent Newton-like method are discussed for approximations of simple singular solutions of nonlinear equations. A relation between the extended system for singular problems and the modified Newton's methods is established.     

Very fast and exact accumulation of products

The IFIP Working Group on Numerical Software and other scientists repeatedly requested that a future arithmetic standard should consider and specify an exact dot product (EDP) [The IFIP WG—IEEE 754R letter, dated September 4 (2007), The IFIP WG—IEEE P1788 letter, dated September 9 (2009)]. On 18 November 2009 the IEEE standards committee P1788 on interval arithmetic accepted a motion [Kulisch and Snyder (The exact dot product as basic tool for long interval arithmetic, passed on Nov 18, 2009 as official IEEE P1788 document)] for including the EDP into a future interval arithmetic standard. Actually the simplest and fastest way for computing a dot product is to compute it exactly. By pipelining, it can be computed in the time the processor needs to read the data, i.e., it comes with utmost speed. A hardware implementation of the EDP exceeds any approximate computation of the dot product in software by several orders of magnitude. By a sample illustration the paper informally specifies the implementation of the EDP on computers. While [Kulisch and Snyder (The exact dot product as basic tool for long interval arithmetic, passed on Nov 18, 2009 as official IEEE P1788 document)] defines what has to be provided, how to embed the EDP into the new standard IEEE 754, [IEEE Floating-Point Arithmetic Standard 754 (2008)] and how exceptions like NaN are to be dealt with, this article illustrates how the EDP can be implemented on computers. There is indeed no simpler way of accumulating a dot product. Any method that just computes an approximation also has to consider the relative values of the summands. This results in a more complicated method. The hardware needed for the EDP is comparable to that for a fast multiplier by an adder tree, accepted years ago and now standard technology in every modern processor. The EDP brings the same speedup for accumulations at comparable costs. In Numerical Analysis the dot product is ubiquitous. It is not merely a fundamental operation in all vector and matrix spaces. It is the EDP which makes residual correction effective. This has a direct and positive influence on all iterative solvers of systems of equations. The EDP is essential for fast long real and long interval arithmetic, as well as for assessing and managing uncertainty in computing. By operator overloading variable precision interval arithmetic is very easy to use. With it the result of every arithmetic expression can be guaranteed to a number of correct digits.     

On the Combination of Interval Constraint Solvers

This paper tackles the combination of interval methods for solving nonlinear systems. A cooperative strategy of application of elementary solvers is designed in order to accelerate the whole computation while weakening the local domain contractions. It is implemented in a prototype solver which efficiently combines interval-based local consistencies and the multidimensional interval Newton method. A set of experiments shows a gain of one order of magnitude on average with respect to Numerica.     

A Fast Domain Decomposition High Order Poisson Solver

We present a fast high-order Poisson solver for implementation on parallel computers. The method uses deferred correction, such that high-order accuracy is obtained by solving a sequence of systems with a narrow stencil on the left-hand side. These systems are solved by a domain decomposition method. The method is direct in the sense that for any given order of accuracy, the number of arithmetic operations is fixed. Numerical experiments show that these high-order solvers easily outperform standard second-order ones. The very fast algorithm in combination with the coarser grid allowed for by the high-order method, also makes it quite possible to compete with adaptive methods and irregular grids for problems with solutions containing widely different scales.