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49
The Exact Computation Paradigm
, 1994
"... We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next ..."
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Cited by 95 (10 self)
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We describe a paradigm for numerical computing, based on exact computation. This emerging paradigm has many advantages compared to the standard paradigm which is based on fixedprecision. We first survey the literature on multiprecision number packages, a prerequisite for exact computation. Next we survey some recent applications of this paradigm. Finally, we outline some basic theory and techniques in this paradigm. 1 This paper will appear as a chapter in the 2nd edition of Computing in Euclidean Geometry, edited by D.Z. Du and F.K. Hwang, published by World Scientific Press, 1994. 1 1 Two Numerical Computing Paradigms Computation has always been intimately associated with numbers: computability theory was early on formulated as a theory of computable numbers, the first computers have been number crunchers and the original massproduced computers were pocket calculators. Although one's first exposure to computers today is likely to be some nonnumerical application, numeri...
Best choices for regularization parameters in learning theory: on the biasvariance problem
 Foundations of Computationals Mathematics
"... The goal of learning theory (and a goal in some other contexts as well) is to find an approximation of a function fρ: X → Y known only through a set of pairs z = (xi, yi) m i=1 drawn from an unknown probability measure ρ on X×Y ( fρ is the “regression function ” of ρ). ..."
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Cited by 40 (9 self)
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The goal of learning theory (and a goal in some other contexts as well) is to find an approximation of a function fρ: X → Y known only through a set of pairs z = (xi, yi) m i=1 drawn from an unknown probability measure ρ on X×Y ( fρ is the “regression function ” of ρ).
A Separation Bound for Real Algebraic Expressions
 In Lecture Notes in Computer Science
, 2001
"... Real algebraic expressions are expressions whose leaves are integers and whose internal nodes are additions, subtractions, multiplications, divisions, kth root operations for integral k, and taking roots of polynomials whose coefficients are given by the values of subexpressions. We consider the si ..."
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Cited by 38 (4 self)
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Real algebraic expressions are expressions whose leaves are integers and whose internal nodes are additions, subtractions, multiplications, divisions, kth root operations for integral k, and taking roots of polynomials whose coefficients are given by the values of subexpressions. We consider the sign computation of real algebraic expressions, a task vital for the implementation of geometric algorithms. We prove a new separation bound for real algebraic expressions and compare it analytically and experimentally with previous bounds. The bound is used in the sign test of the number type leda real. 1
Exact Geometric Computation in LEDA
 In Proc. 11th Annu. ACM Sympos. Comput. Geom
, 1995
"... real expressions with arbitrary precision. Figure 1 shows (part of) the LEDA manual page for reals. reals provide exact computation in a convenient way. In an implementation of a geometric algorithm in C++, reals can be used like doubles. The following example MaxPlanckInstitut fur Informatik, ..."
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Cited by 36 (5 self)
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real expressions with arbitrary precision. Figure 1 shows (part of) the LEDA manual page for reals. reals provide exact computation in a convenient way. In an implementation of a geometric algorithm in C++, reals can be used like doubles. The following example MaxPlanckInstitut fur Informatik, Im Stadtwald, 66123 Saarbrucken, Germany. Supported by the ESPRIT Basic Research Actions Program, under contract No. 7141 (project ALCOM II). y Fachbereich 14, Informatik, Universitat des Saarlandes, 66041 Saarbrucken, Germany. z MartinLutherUniversitat Halle, Fachbereich Mathematik und Informatik, 06099 Halle, Germany. 0 arises in the computation of Voronoi diagrams of line segments [2]. For i, 1 i 3, let l i : a i x + b i
An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems
, 1993
"... are addressed. Two algorithms are developed for certain feasibility versions of the SDP, and the rst of these is shown to have polynomial time complexity when the dimension of the matrix map involved is xed. The second algorithm is a globally convergent Newtonlike method applied to a leastsquares ..."
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Cited by 36 (4 self)
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are addressed. Two algorithms are developed for certain feasibility versions of the SDP, and the rst of these is shown to have polynomial time complexity when the dimension of the matrix map involved is xed. The second algorithm is a globally convergent Newtonlike method applied to a leastsquares penalty function. The problem of characterizing and identifying quadratic maps with convex images is analyzed from both structural and complexity theoretic points of view. Then a study is made of the geometry of a class of convex sets called spectrahedra, which are the feasible regions in semide nite programs. Finally, in Chapter 7, we develop some cutting plane techniques for MQP, based on eigenvalue inequalities. Acknowledgements I express my sincere gratitude to my thesis advisor Professor Alan Goldman for his support, ideas and encouragement. My special thanks to Professors Laszlo Lovasz and James Renegar for sparing their time generously and giving me very useful suggestions. I thank the warm and friendly Professors Dan Naiman and Ed Scheinerman for making my four year long stay at Johns Hopkins a very pleasant one. I am indebted to Prof. JongShi Pang and Prof. Roger Horn for giving a patient ear to many of my enthusiastic ideas and o ering suggestions. I also thank Prof. ShihPing Han for being a wonderful teacher, and Prof. Leslie Hall for being a patient second reader of my thesis. My interest in Multiquadratic Programming was initiated during my internship at AT&T Bell Laboratories in the summer of 1990, and I am obliged to Dr. Narendra Karmarkar for arranging this internship. I thank Dr. Farid Alizadeh, Dr. Florian Jarre, Profs. Raphael Loewy, Michael Overton and Stephen Vavasis for patiently answering my questions and sending me some literature. My heartfelt appreciation is due my parents, Lakshmi Pathy and Satyavathi, my sister,
Real Algebraic Numbers: Complexity Analysis and Experimentation
 RELIABLE IMPLEMENTATIONS OF REAL NUMBER ALGORITHMS: THEORY AND PRACTICE, LNCS (TO APPEAR
, 2006
"... We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both metho ..."
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Cited by 30 (17 self)
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We present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size ≤ τ, using Sturm (Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both methods and we give an asymptotic complexity bound of eOB(d 4 τ 2). This matches the best known bounds for binary subdivision solvers. Moreover, we generalize this to cover the non squarefree polynomials and show that within the same complexity we can also compute the multiplicities of the roots. We also consider algorithms for sign evaluation, comparison of real algebraic numbers and simultaneous inequalities, and we improve the known bounds at least by a factor of d. Finally, we present our C++ implementation in synaps and some preliminary experiments on various data sets.
The LEDA class real number
 MaxPlanck Institut Inform
, 1996
"... We describe the implementation of the LEDA [MN95, Nah95] data type real. Every integer is a real and reals are closed under the operations addition, subtraction, multiplication, division and squareroot. The main features of the data type real are ffl The userinterface is similar to that of the bu ..."
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Cited by 16 (5 self)
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We describe the implementation of the LEDA [MN95, Nah95] data type real. Every integer is a real and reals are closed under the operations addition, subtraction, multiplication, division and squareroot. The main features of the data type real are ffl The userinterface is similar to that of the builtin data type double.
Computing monodromy groups defined by plane algebraic curves
 In: Proceedings of the 2007 International Workshop on Symbolicnumeric Computation. ACM, NewYork
, 2007
"... We present a symbolicnumeric method to compute the monodromy group of a plane algebraic curve viewed as a ramified covering space of the complex plane. Following the definition, our algorithm is based on analytic continuation of algebraic functions above paths in the complex plane. Our contribution ..."
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Cited by 13 (4 self)
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We present a symbolicnumeric method to compute the monodromy group of a plane algebraic curve viewed as a ramified covering space of the complex plane. Following the definition, our algorithm is based on analytic continuation of algebraic functions above paths in the complex plane. Our contribution is threefold: first of all, we show how to use a minimum spanning tree to minimize the length of paths; then, we propose a strategy that gives a good compromise between the number of steps and the truncation orders of Puiseux expansions, obtaining for the first time a complexity result about the number of steps; finally, we present an efficient numericalmodular algorithm to compute Puiseux expansions above critical points, which is a non trivial task.
On the complexity of real root isolation using Continued Fractions
 INRIA
, 2006
"... We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers. One motivation is to explain the method’s good performance in practice. We improve the previously known bou ..."
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Cited by 12 (6 self)
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We present algorithmic, complexity and implementation results concerning real root isolation of integer univariate polynomials using the continued fraction expansion of real algebraic numbers. One motivation is to explain the method’s good performance in practice. We improve the previously known bound by a factor of dτ, where d is the polynomial degree and τ bounds the coefficient bit size, thus matching the current record complexity for real root isolation by exact methods. Namely, the complexity bound is � OB(d 4 τ 2) using a standard bound on the expected bit size of the integers in the continued fraction expansion. Moreover, using a homothetic transformation we improve the expected complexity bound to � OB(d 3 τ) under the assumption that d = O(τ). We show how to compute the multiplicities within the same complexity and extend the algorithm to non squarefree polynomials. Finally, we present an efficient opensource C++ implementation in the algebraic library synaps, and illustrate its efficiency as compared to other available software. We use polynomials with coefficient bit size up to 8000 bits and degree up to 1000. 1
Polynomials with {0,+1,1} Coefficients and a Root Close to a Given Point.
 Canadian Journal of Math
, 1996
"... For a fixed ff we discuss how closely ff can be approximated by a root of a f0; +1; \Gamma1g polynomial of given degree. We show that the worst rate of approximation tends to occur for roots of unity, particularly those of small degree. For roots of unity these bounds depend on the order of vanishin ..."
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Cited by 11 (0 self)
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For a fixed ff we discuss how closely ff can be approximated by a root of a f0; +1; \Gamma1g polynomial of given degree. We show that the worst rate of approximation tends to occur for roots of unity, particularly those of small degree. For roots of unity these bounds depend on the order of vanishing, k, of the polynomial at ff. In particular we obtain the following. Let BN denote the set of roots of all f0; +1; \Gamma1g polynomials of degree at most N and BN (ff; k) the roots of those polynomials that have a root of order at most k at ff. For a Pisot number ff in (1; 2] we show that min fi2B N nfffg jff \Gamma fij i 1 ff N ; and for a root of unity ff that min fi2B N (ff;k)nfffg jff \Gamma fij i 1 N (k+1)d 1 2 OE(d)e+1 : We study in detail the case of ff = 1, where, by far, the best approximations are real. We give fairly precise bounds on the closest real root to 1. When k = 0 or 1 we can describe the extremal polynomials explicitly. Keywords: Mahler measure, zero one ...