Numerical analysis: 4th ed
Combinatorial complexity bounds for arrangements of curves and spheres
Discrete & Computational Geometry - Special issue on the complexity of arrangements
Symbolic treatment of geometric degeneracies
Journal of Symbolic Computation
Complexity theory of real functions
Complexity theory of real functions
Arrangements of curves in the plane—topology, combinatorics, and algorithms
Theoretical Computer Science
Cutting hyperplanes for divide-and-conquer
Discrete & Computational Geometry
Static analysis yields efficient exact integer arithmetic for computational geometry
ACM Transactions on Graphics (TOG)
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Computational geometry: algorithms and applications
Computational geometry: algorithms and applications
Feasible real random access machines
Journal of Complexity
Computable analysis: an introduction
Computable analysis: an introduction
Topological complexity of zero finding with algebraic operations
Journal of Complexity
Computability of Linear Equations
TCS '02 Proceedings of the IFIP 17th World Computer Congress - TC1 Stream / 2nd IFIP International Conference on Theoretical Computer Science: Foundations of Information Technology in the Era of Networking and Mobile Computing
CCA '00 Selected Papers from the 4th International Workshop on Computability and Complexity in Analysis
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algebraic Complexity Theory
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Algebraic models of real computation and their induced notions of time complexity neglect stability issues of numerical algorithms. Recursive Analysis on the other hand appropriately describes stable numerical computations while, based on Turing Machines, usually lacks significant lower complexity bounds.We propose a synthesis of the two models, namely a restriction of algebraic algorithms to computable primitives. These are thus inherently stable and allow for nontrivial complexity considerations. In this model, one can prove on a sound mathematical foundation the empirically well-known observation that stability and speed may be contradictory goals in algorithm design.More precisely we show that solving the geometric point location problem among hyperplanes by means of a total computable decision tree (i.e., one behaving numerically stable for all possible input points) has in general complexity exponentially larger than when permitting the tree to be partial, that is, to diverge (behave in an instable way) on a 'small' set of arguments.The trade-off between the extremes is investigated quantitatively for the planar case. Proofs involve both topological and combinatorial arguments.