Abstract
This chapter is devoted to describing basic tools and related topics necessary to numerical verification and computer-assisted proofs. We introduce fixed-point approaches, some surprising pitfalls in numerical computation, interval arithmetic, and verification methods for finite-dimensional problems.
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Notes
- 1.
Note that, in order to compute with rational number arithmetic, input data in Mathematica should be rational numbers such as 418695205/10 (not 41869520.5).
- 2.
MATLAB R2019a returns different results.
- 3.
In Part I we define ρ and \(\hat \rho \) (in the next subsection) as an upper bound corresponding to the 2-norm.
- 4.
In some cases, when intvalinit( ’SharpIVmult’) is active, isequal judges that the given interval matrix is not symmetric.
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Nakao, M.T., Plum, M., Watanabe, Y. (2019). Basic Tools. In: Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations. Springer Series in Computational Mathematics, vol 53. Springer, Singapore. https://doi.org/10.1007/978-981-13-7669-6_12
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