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Julia set (in white) for the rational function associated to Newton's method for f : z → z3 −1. Coloring of Fatou set in red, green and blue tones according to the three attractors (the three roots of f). For some functions f (z) we can say beforehand that the Julia set is a fractal and not a simple curve.
Julia is a high-level, general-purpose [ 15 ] dynamic programming language, still designed to be fast and productive, [ 16 ] for e.g. data science, artificial intelligence, machine learning, modeling and simulation, most commonly used for numerical analysis and computational science. [ 17 ][ 18 ][ 19 ] Distinctive aspects of Julia's design ...
Complex dynamics. Complex dynamics, or holomorphic dynamics, is the study of dynamical systems obtained by iterating a complex analytic mapping. This article focuses on the case of algebraic dynamics, where a polynomial or rational function is iterated. In geometric terms, that amounts to iterating a mapping from some algebraic variety to itself.
A mathematical object X has the fixed-point property if every suitably well-behaved mapping from X to itself has a fixed point. The term is most commonly used to describe topological spaces on which every continuous mapping has a fixed point. But another use is in order theory, where a partially ordered set P is said to have the fixed point ...
A closed interval is an interval that includes all its endpoints and is denoted with square brackets. [2] For example, [0, 1] means greater than or equal to 0 and less than or equal to 1 . Closed intervals have one of the following forms in which a and b are real numbers such that a ≤ b : {\displaystyle a\leq b\colon }
Interval arithmetic (also known as interval mathematics;interval analysis or interval computation) is a mathematical technique used to mitigate rounding and measurement errors in mathematical computation by computing function bounds. Numerical methods involving interval arithmetic can guarantee relatively reliable and mathematically correct ...
A continuous function on the closed interval showing the absolute max (red) and the absolute min (blue). In calculus, the extreme value theorem states that if a real-valued function is continuous on the closed and bounded interval , then must attain a maximum and a minimum, each at least once.
A partition of an interval being used in a Riemann sum. The partition itself is shown in grey at the bottom, with the norm of the partition indicated in red. In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x0, x1, x2, …, xn of real numbers such that. a = x0 < x1 < x2 < … < xn = b.