Cylindrical algebraic decomposition I: the basic algorithm
SIAM Journal on Computing
Solving systems of polynomial inequalities in subexponential time
Journal of Symbolic Computation
Journal of Symbolic Computation
Some algebraic and geometric computations in PSPACE
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Arithmetic with real algebraic numbers is in NC
ISSAC '90 Proceedings of the international symposium on Symbolic and algebraic computation
Generic computation of the real closure of an ordered field
Selected papers presented at the international IMACS symposium on Symbolic computation, new trends and developments
Computing in the field of complex algebraic numbers
Journal of Symbolic Computation - Special issue: validated numerical methods and computer algebra
Journal of Automated Reasoning
CADE-15 Proceedings of the 15th International Conference on Automated Deduction: Automated Deduction
The complexity of elementary algebra and geometry
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
Algorithm 852: RealPaver: an interval solver using constraint satisfaction techniques
ACM Transactions on Mathematical Software (TOMS)
Efficient solving of quantified inequality constraints over the real numbers
ACM Transactions on Computational Logic (TOCL)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics)
Applications of MetiTarski in the Verification of Control and Hybrid Systems
HSCC '09 Proceedings of the 12th International Conference on Hybrid Systems: Computation and Control
Formal Verification of Curved Flight Collision Avoidance Maneuvers: A Case Study
FM '09 Proceedings of the 2nd World Congress on Formal Methods
δ-complete decision procedures for satisfiability over the reals
IJCAR'12 Proceedings of the 6th international joint conference on Automated Reasoning
IJCAR'12 Proceedings of the 6th international joint conference on Automated Reasoning
Univariate real root isolation in multiple extension fields
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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Recent applications of decision procedures for nonlinear real arithmetic (the theory of real closed fields, or RCF) have presented a need for reasoning not only with polynomials but also with transcendental constants and infinitesimals. In full generality, the algebraic setting for this reasoning consists of real closed transcendental and infinitesimal extensions of the rational numbers. We present a library for computing over these extensions. This library contains many contributions, including a novel combination of Thom's Lemma and interval arithmetic for representing roots, and provides all core machinery required for building RCF decision procedures. We describe the abstract algebraic setting for computing with such field extensions, present our concrete algorithms and optimizations, and illustrate the library on a collection of examples.