Error of truncated Chebyshev series and other near minimax polynomial approximations
Journal of Approximation Theory
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
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.)
On Taylor Model Based Integration of ODEs
SIAM Journal on Numerical Analysis
Chebyshev expansions for solutions of linear differential equations
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
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
Polynomial evaluation and interpolation on special sets of points
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
Standardized interval arithmetic and interval arithmetic used in libraries
ICMS'10 Proceedings of the Third international congress conference on Mathematical software
Certification of bounds of non-linear functions: the templates method
CICM'13 Proceedings of the 2013 international conference on Intelligent Computer Mathematics
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Performing numerical computations, yet being able to provide rigorous mathematical statements about the obtained result, is required in many domains like global optimization, ODE solving or integration. Taylor models, which associate to a function a pair made of a Taylor approximation polynomial and a rigorous remainder bound, are a widely used rigorous computation tool. This approach benefits from the advantages of numerical methods, but also gives the ability to make reliable statements about the approximated function. Despite the fact that approximation polynomials based on interpolation at Chebyshev nodes offer a quasi-optimal approximation to a function, together with several other useful features, an analogous to Taylor models, based on such polynomials, has not been yet well-established in the field of validated numerics. This paper presents a preliminary work for obtaining such interpolation polynomials together with validated interval bounds for approximating univariate functions. We propose two methods that make practical the use of this: one is based on a representation in Newton basis and the other uses Chebyshev polynomial basis. We compare the quality of the obtained remainders and the performance of the approaches to the ones provided by Taylor models.