TY - JOUR

T1 - Global solutions of well-constrained transcendental systems using expression trees and a single solution test

AU - Aizenshtein, Maxim

AU - Bartoň, Michael

AU - Elber, Gershon

N1 - Funding Information:
This research was partly supported by the Israel Science Foundation (grant No. 346/07), in part by the Israeli Ministry of Science Grant No. 3-8273, and in part by the New York metropolitan research fund, Technion.

PY - 2012/6

Y1 - 2012/6

N2 - We present an algorithm which is capable of globally solving a well-constrained transcendental system over some sub-domain D⊂ℝ n, isolating all roots. Such a system consists of n unknowns and n regular functions, where each may contain non-algebraic (transcendental) functions like sin, exp or log. Every equation is considered as a hyper-surface in ℝ n and thus a bounding cone of its normal (gradient) field can be defined over a small enough sub-domain of D. A simple test that checks the mutual configuration of these bounding cones is used that, if satisfied, guarantees at most one zero exists within the given domain. Numerical methods are then used to trace the zero. If the test fails, the domain is subdivided. Every equation is handled as an expression tree, with polynomial functions at the leaves, prescribing the domain. The tree is processed from its leaves, for which simple bounding cones are constructed, to its root, which allows to efficiently build a final bounding cone of the normal field of the whole expression. The algorithm is demonstrated on curve-curve intersection, curve-surface intersection, ray-trap and geometric constraint problems and is compared to interval arithmetic.

AB - We present an algorithm which is capable of globally solving a well-constrained transcendental system over some sub-domain D⊂ℝ n, isolating all roots. Such a system consists of n unknowns and n regular functions, where each may contain non-algebraic (transcendental) functions like sin, exp or log. Every equation is considered as a hyper-surface in ℝ n and thus a bounding cone of its normal (gradient) field can be defined over a small enough sub-domain of D. A simple test that checks the mutual configuration of these bounding cones is used that, if satisfied, guarantees at most one zero exists within the given domain. Numerical methods are then used to trace the zero. If the test fails, the domain is subdivided. Every equation is handled as an expression tree, with polynomial functions at the leaves, prescribing the domain. The tree is processed from its leaves, for which simple bounding cones are constructed, to its root, which allows to efficiently build a final bounding cone of the normal field of the whole expression. The algorithm is demonstrated on curve-curve intersection, curve-surface intersection, ray-trap and geometric constraint problems and is compared to interval arithmetic.

KW - Bounding cone

KW - Expression tree

KW - Non-algebraic equation system

KW - Root solver

KW - Single root criteria

UR - http://www.scopus.com/inward/record.url?scp=84860493845&partnerID=8YFLogxK

U2 - 10.1016/j.cagd.2011.07.002

DO - 10.1016/j.cagd.2011.07.002

M3 - Article

AN - SCOPUS:84860493845

SN - 0167-8396

VL - 29

SP - 265

EP - 279

JO - Computer Aided Geometric Design

JF - Computer Aided Geometric Design

IS - 5

ER -