Validated Numerics for Dynamical Systems and Related Topics in Rio de Janeiro

School @UFRJ 18-22 February 2019

Workshop @IMPA 25 February-1 March 2019

What is Validated Numerics?

Validated Numerics is a set of mathematical and numerical tools that allows us to prove mathematical results with the aid of a computer.

One of the most successful tecniques in the area is Interval Arithmetic, a computational framework that, in its most basic use, allows us to enclose the range of functions, i.e., given an interval I and a function f which takes values in the real numbers, we can find an interval that bounds the range of f over I automatically with the use of a computer.

While, in principle, this may seem quite limited, the applications of this simple process are striking: from bounding tails of series, root finding, enclosure of orbits of ODEs and dynamical systems, enclosures for values of integrals, and many others.

The goals of this summer program are to give an access point to researchers and students into the topic of Validated Numerics and to disseminate research results obtained with interval tecniques to the wider mathematical community.

Workshop in Validated Numerics @ IMPA (25 February - 1 March 2019)

This research workshop aims to disseminate recent results in Validated Numerics to a wider mathematical audience.

Some of the topics to be discussed are

  • Ergodic and statistical properties of deterministic and random dynamics
  • Renormalization
  • Parameter exclusion

We plan to have a social dinner on Thursday night, more information will be given on Monday.

School in Validated Numerics @UFRJ (18- 22 February 2019)

The school in Validated Numerics at UFRJ aims to introduce students and researchers to the basic tools of Validated Numerics, with a view towards Computer-Assisted Proofs in Dynamical Systems.

We are going to follow the book:

W. Tucker

Validated Numerics: A Short Introduction to Rigorous Computations, Princeton University Press

The school will have morning lectures introducing the concepts of Validated Numerics, Computer Assisted Proofs and some basic notions of Dynamical Systems and afternoon exercise sessions in which we will replicate O.E. Lanford III enclosure of the value of the Feigenbaum constant (article).

We are currently checking if we can give some sort of financial support to participants.

(Image from Wikimedia

School in Validated Numerics: Schedule

The school is going to be held at IM-UFRJ, Block B and C of the Institute of Mathematics of UFRJ, localized at Ilha do Fundão, Technology Center (CT).

Location of the Institue

Public bus services take to Ilha do Fundão:

- the 321-323-325-327 lines that start at the town center take you to the Fundão transport hub and in campus shuttle take you to the technology center

- the 485 bus, that starts at Copacabana, takes you directly to the technology center.

The lectures are going to be held from 8:30-12, with practice session in the afternoon.

If you have the possibility, bring your own laptop: the platform we are going to use is cloud-based (, so there is no need to install any software, but we have access to only a limited number of computers in the laboratory.

Monday: Introduction to Validated Numerics, Range Inclusion

Tuesday: Automatic Differentiation, Interval Newton Method

Wednesday: Taylor series

Thursday: Inclusion of integrals

Friday: Ordinary Differential Equations

Material for the school

The following is the material used during the school.

The material from the lecture of Luiz Henrique de Figueiredo:

  1. Slides from Interval methods for computer graphics and geometric modeling, which was an invited plenary talk at SIBGRAPI 2017
  2. Demos from mini-course on Introdução aos Métodos Numéricos Intervalares:,,
  3. Files with the scripts will be made accessible on Google Drive

The notebooks for the exercise sessions are going to be made accessible on Google Drive.


To register to participate in the school or the workshop, fill this form; moreover, to have access to the facilities at IMPA, we ask you to register with their system.

The events are free and open to everyone but registration is necessary since space is limited and to allow access to the premises.

Organizing commitee: