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CXSC 2.0  A C++ library for extended scientific computing
 Numerical Software with Result Verification: International Dagstuhl Seminar, Dagstuhl
, 2003
"... Abstract. In this note the main features and newer developments of the C++ class library for extended scientific computing CXSC 2.0 will be discussed. The original version of the CXSC library is about ten years old. But in the last decade the underlying programming language C++ has been developed ..."
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Cited by 16 (2 self)
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Abstract. In this note the main features and newer developments of the C++ class library for extended scientific computing CXSC 2.0 will be discussed. The original version of the CXSC library is about ten years old. But in the last decade the underlying programming language C++ has been developed significantly. Since November 1998 the C++ standard is available and more and more compilers support (most of) the features of this standard. The new version CXSC 2.0 conforms to this standard. Application programs written for older CXSC versions have to be modified to run with CXSC 2.0. Several examples will help the user to see which changes have to be done. Note, that all sample codes given in [6] have to be modified to work properly with CXSC 2.0. All sample codes listed in this note will be made available on the web page
Decision Making Under Interval Probabilities
, 1998
"... If we know the probabilities p1 ; : : : ; pn of different situations s1 ; : : : ; sn , then we can choose a decision A i for which the expected benefit C i = p1 \Delta c i1 + : : : + pn \Delta cin takes the largest possible value, where c ij denotes the benefit of decision A i in situation s j . In ..."
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Cited by 9 (3 self)
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If we know the probabilities p1 ; : : : ; pn of different situations s1 ; : : : ; sn , then we can choose a decision A i for which the expected benefit C i = p1 \Delta c i1 + : : : + pn \Delta cin takes the largest possible value, where c ij denotes the benefit of decision A i in situation s j . In many real life situations, however, we do not know the exact values of the probabilities p j ; we only know the intervals p j = [p \Gamma j ; p + j ] of possible values of these probabilities. In order to make decisions under such interval probabilities, we would like to generalize the notion of expected benefits to interval probabilities. In this paper, we show that natural requirements lead to a unique (and easily computable) generalization. Thus, we have a natural way of decision making under interval probabilities. 1 Introduction to the Problem Decision making: case of exactly known consequences. One of the main problems in decision making is the problem of choosing one of (finitel...
Theoretical Justification of a Heuristic Subbox Selection Criterion for Interval Global Optimization
, 2001
"... The most widely used guaranteed methods for global optimization are probably the intervalbased branchandbound techniques. In these techniques, we start with a single box  the entire function domain  as a possible location of the global minimizer points, and then, in each step, subdivide some ..."
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Cited by 8 (4 self)
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The most widely used guaranteed methods for global optimization are probably the intervalbased branchandbound techniques. In these techniques, we start with a single box  the entire function domain  as a possible location of the global minimizer points, and then, in each step, subdivide some of the boxes, use interval computations to compute the enclosure [F (X); F (X)] f(X) of the range f(X) of the objective function f(x) on each new subbox X , and, based on these computations, eliminate the boxes which cannot contain a global minimizer point. The computational eciency of these methods strongly depends on which boxes we select for subdivision. Traditionally, for subdivision, the algorithms selected a box with the smallest value of F (X). Recently, it was shown that the algorithm converges much faster if we select, instead, a box with the largest possible value of the ratio e f F (X) F (X) F (X) ; where e f is a current upper bound on the actual global minimum. In this paper, we give a theoretical justication for this empirical criterion. Namely, we show that:  first, this criterion is the only one that is invariant w.r.t. some reasonable symmetries; and  second, that this criterion is optimal in some reasonable sense.
Improving the Efficiency of a NonlinearSystemSolver Using a Componentwise Newton Method
 Institut für Angewandte Mathematik, Universität Karlsruhe (TH
, 1997
"... A Componentwise Interval Newton Method: We give an efficient branchandprune 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 onedimensi ..."
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Cited by 7 (0 self)
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A Componentwise Interval Newton Method: We give an efficient branchandprune 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 realvalued 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 "realworld" 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 ...
Interval rational = algebraic
 ACM SIGNUM Newsletter
, 1995
"... Abstract. Rational functions can be defined as compositions of arithmetic operations (+, −, ·,:). What class of functions will be get if we add to this list the “interval ” operation (that transforms a function f of n variables and given intervals X1,..., Xn into the bounds for the range f(X1,..., X ..."
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Cited by 6 (6 self)
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Abstract. Rational functions can be defined as compositions of arithmetic operations (+, −, ·,:). What class of functions will be get if we add to this list the “interval ” operation (that transforms a function f of n variables and given intervals X1,..., Xn into the bounds for the range f(X1,..., Xn))? In this paper, we prove that adding this “interval ” operation to rational functions leads exactly to the set of all (locally) algebraic functions. In other words, algebraic functions can be described as compositions of arithmetic operations and the “interval ” operation. This result provides an additional explanation of why naive interval computations sometimes overshoot: • the desired dependency is (locally) a general algebraic function; • naive interval methods results in a (locally) rational function; • not all algebraic functions are rational. 1. FORMULATION OF THE PROBLEM 1.1. Estimating accuracy of the results of data processing, and interval mathematics
On Hardware Support For Interval Computations And For Soft Computing: Theorems
, 1994
"... This paper provides a rationale for providing hardware supported functions of more than two variables for processing incomplete knowledge and fuzzy knowledge. The result is in contrast to Kolmogorov's theorem in numerical (nonfuzzy) case. ..."
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Cited by 6 (3 self)
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This paper provides a rationale for providing hardware supported functions of more than two variables for processing incomplete knowledge and fuzzy knowledge. The result is in contrast to Kolmogorov's theorem in numerical (nonfuzzy) case.
On Branching Rules in SecondOrder BranchandBound Methods for Global Optimization
 Scienti Computing and Validated Numerics, AkademieVerlag
, 1996
"... This paper investigates different branching rules, i.e. rules for selecting the subdivision direction, in interval branchandbound algorithms for global optimization. Earlier studies ([2],[9]) dealt with the subdivision direction selection in methods which do not use secondorder information about ..."
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Cited by 5 (2 self)
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This paper investigates different branching rules, i.e. rules for selecting the subdivision direction, in interval branchandbound algorithms for global optimization. Earlier studies ([2],[9]) dealt with the subdivision direction selection in methods which do not use secondorder information about the objective function. The investigated model algorithm (similar to that in [3]) now uses the enclosure of the Hessian matrix to incorporate a concavity test for boxdiscarding and an interval Newton GaussSeidel step to reduce the widths of the boxes resulting from the underlying generalized bisection method. Four different branching rules are investigated, and a wide spectrum of test problems is used for numerical tests. The results indicate that there are substantial differences between the rules with respect to the performance of the model algorithm. This is clarified by comparing the required CPU times, the numbers of function and derivative evaluations, and the necessary amounts of st...
Enclosing the maximum likelihood of the simplest DNA model evolving on fixed topologies: towards a rigorous framework for phylogenetic inference. BSCB Dept
, 2004
"... Summary. An interval extension of the recursive formulation for the likelihood function of the simplest Markov model of DNA evolution on unrooted phylogenetic trees with a fixed topology is used to obtain rigorous enclosure(s) of the global maximum likelihood. Validated global maximizer(s) inside an ..."
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Cited by 5 (3 self)
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Summary. An interval extension of the recursive formulation for the likelihood function of the simplest Markov model of DNA evolution on unrooted phylogenetic trees with a fixed topology is used to obtain rigorous enclosure(s) of the global maximum likelihood. Validated global maximizer(s) inside any compact set of the parameter space which is the set of all branch lengths of the tree are thus obtained. The algorithm is an adaptation of a widely applied global optimization method using interval analysis for the phylogenetic context. The method is applied to enclose the maximizer(s) and the global maximum for the simplest DNA model evolving on trees with 2, 3, and 4 taxa. The method is also applicable to a wide class of inclusion isotonic likelihood functions. 1.
W.: The interval library filib++ 2.0  design, features and sample programs. Preprint 2001/4, Universität
, 2001
"... InternetZugriff Die Berichte sind in elektronischer Form erhältlich über die World Wide Web Seiten ..."
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Cited by 5 (0 self)
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InternetZugriff Die Berichte sind in elektronischer Form erhältlich über die World Wide Web Seiten
A BranchandPrune Method for Global Optimization: the Univariate Case
 Scientific Computing, Validated Numerics, Interval Methods
, 2000
"... We present a branchandprune algorithm for univariate optimization. Pruning is achieved by using first order information of the objective function by means of an interval evaluation of the derivative over the current interval. First order information aids fourfold. Firstly, to check monotonicity. S ..."
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Cited by 4 (2 self)
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We present a branchandprune algorithm for univariate optimization. Pruning is achieved by using first order information of the objective function by means of an interval evaluation of the derivative over the current interval. First order information aids fourfold. Firstly, to check monotonicity. Secondly, to determine optimal centers which, along with the mean value form, are used to improve the enclosure of the function range. Thirdly, to prune the search interval using the current upper bound of the global minimum, and finally, to apply a more sophisticated splitting strategy. Results of numerical experiments are also presented. 1.