Checking geometric programs or verification of geometric structures
Proceedings of the twelfth annual symposium on Computational geometry
A Mechanically Checked Proof of the AMD5K86TM Floating-Point Division Program
IEEE Transactions on Computers
Fast Algorithms for Manipulating Formal Power Series
Journal of the ACM (JACM)
Evaluating derivatives: principles and techniques of algorithmic differentiation
Evaluating derivatives: principles and techniques of algorithmic differentiation
Formal Verification of Floating Point Trigonometric Functions
FMCAD '00 Proceedings of the Third International Conference on Formal Methods in Computer-Aided Design
Program Result Checking: A New Approach to Making Programs More Reliable
ICALP '93 Proceedings of the 20th International Colloquium on Automata, Languages and Programming
Divider Circuit Verification with Model Checking and Theorem Proving
TPHOLs '00 Proceedings of the 13th International Conference on Theorem Proving in Higher Order Logics
ACM Transactions on Mathematical Software (TOMS)
Efficient isolation of polynomial's real roots
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
Methods and Applications of Interval Analysis (SIAM Studies in Applied and Numerical Mathematics) (Siam Studies in Applied Mathematics, 2.)
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Assisted verification of elementary functions using Gappa
Proceedings of the 2006 ACM symposium on Applied computing
A Certified Infinite Norm for the Implementation of Elementary Functions
QSIC '07 Proceedings of the Seventh International Conference on Quality Software
Rigorous global search using taylor models
Proceedings of the 2009 conference on Symbolic numeric computation
Certified and Fast Computation of Supremum Norms of Approximation Errors
ARITH '09 Proceedings of the 2009 19th IEEE Symposium on Computer Arithmetic
Verifying nonlinear real formulas via sums of squares
TPHOLs'07 Proceedings of the 20th international conference on Theorem proving in higher order logics
Handbook of Floating-Point Arithmetic
Handbook of Floating-Point Arithmetic
Formal global optimisation with taylor models
IJCAR'06 Proceedings of the Third international joint conference on Automated Reasoning
Rigorous polynomial approximation using taylor models in Coq
NFM'12 Proceedings of the 4th international conference on NASA Formal Methods
Estimation of error propagation in multiprocessor systems
Advances in Engineering Software
| Hi-index | 5.23 |
For purposes of actual evaluation, mathematical functions f are commonly replaced by approximation polynomials p. Examples include floating-point implementations of elementary functions, quadrature or more theoretical proof work involving transcendental functions. Replacing f by p induces a relative error @e=p/f-1. In order to ensure the validity of the use of p instead of f, the maximum error, i.e. the supremum norm @?@e@?"~^I must be safely bounded above over an interval I, whose width is typically of order 1. Numerical algorithms for supremum norms are efficient, but they cannot offer the required safety. Previous validated approaches often require tedious manual intervention. If they are automated, they have several drawbacks, such as the lack of quality guarantees. In this article, a novel, automated supremum norm algorithm on univariate approximation errors @e is proposed, achieving an a priori quality on the result. It focuses on the validation step and paves the way for formally certified supremum norms. Key elements are the use of intermediate approximation polynomials with bounded approximation error and a non-negativity test based on a sum-of-squares expression of polynomials. The new algorithm was implemented in the Sollya tool. The article includes experimental results on real-life examples.