Journal of Automated Reasoning
Journal of Automated Reasoning
Formal Verification of Square Root Algorithms
Formal Methods in System Design
Mechanical Verification of a Square Root Algorithm Using Taylor's Theorem
FMCAD '02 Proceedings of the 4th International Conference on Formal Methods in Computer-Aided Design
On the Mechanization of Real Analysis in Isabelle/HOL
TPHOLs '00 Proceedings of the 13th International Conference on Theorem Proving in Higher Order Logics
Codifying Guarded Definitions with Recursive Schemes
TYPES '94 Selected papers from the International Workshop on Types for Proofs and Programs
A Generic Root Operation for Exact Real Arithmetic
CCA '00 Selected Papers from the 4th International Workshop on Computability and Complexity in Analysis
Interactive Theorem Proving and Program Development
Interactive Theorem Proving and Program Development
Real Number Calculations and Theorem Proving
TPHOLs '08 Proceedings of the 21st International Conference on Theorem Proving in Higher Order Logics
Certified Exact Transcendental Real Number Computation in Coq
TPHOLs '08 Proceedings of the 21st International Conference on Theorem Proving in Higher Order Logics
Certified exact real arithmetic using co-induction in arbitrary integer base
FLOPS'08 Proceedings of the 9th international conference on Functional and logic programming
Formally verified conditions for regularity of interval matrices
AISC'10/MKM'10/Calculemus'10 Proceedings of the 10th ASIC and 9th MKM international conference, and 17th Calculemus conference on Intelligent computer mathematics
Computer certified efficient exact reals in Coq
MKM'11 Proceedings of the 18th Calculemus and 10th international conference on Intelligent computer mathematics
The picard algorithm for ordinary differential equations in coq
ITP'13 Proceedings of the 4th international conference on Interactive Theorem Proving
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We are interested in the verification of Newton's method. We use a formalization of the convergence and stability of the method done with the axiomatic real numbers of Coq's Standard Library in order to validate the computation with Newton's method done with a library of exact real arithmetic based on co-inductive streams. The contribution of this work is twofold. Firstly, based on Newton's method, we design and prove correct an algorithm on streams for computing the root of a real function in a lazy manner. Secondly, we prove that rounding at each step in Newton's method still yields a convergent process with an accurate correlation between the precision of the input and that of the result. An algorithm including rounding turns out to be much more efficient.