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For r < 1, exists outside [0, 1] as an unstable fixed point, but for r = 1, the two fixed points collide, and for r > 1, appears between [0, 1] as a stable fixed point. When the parameter r = 1, the trajectory of the logistic map converges to 0 as before, but the convergence speed is slower at r = 1.
If μ is greater than 1 the system has two fixed points, one at 0, and the other at μ/(μ + 1). Both fixed points are unstable, i.e. a value of x close to either fixed point will move away from it, rather than towards it. For example, when μ is 1.5 there is a fixed point at x = 0.6 (since 1.5(1 − 0.6) = 0.6) but starting at x = 0.61 we get
A common algorithm design tactic is to divide a problem into sub-problems of the same type as the original, solve those sub-problems, and combine the results. This is often referred to as the divide-and-conquer method; when combined with a lookup table that stores the results of previously solved sub-problems (to avoid solving them repeatedly and incurring extra computation time), it can be ...
Map is sometimes generalized to accept dyadic (2-argument) functions that can apply a user-supplied function to corresponding elements from two lists. Some languages use special names for this, such as map2 or zipWith. Languages using explicit variadic functions may have versions of map with variable arity to support variable-arity functions ...
The shortest path between two intersections on a city map can be found by this algorithm using pencil and paper. Every intersection is listed on a separate line: one is the starting point and is labeled (given a distance of) 0. Every other intersection is initially labeled with a distance of infinity.
The type of the fixed point is the return type of the function being fixed. This may be a real or a function or any other type. In the untyped lambda calculus, the function to apply the fixed-point combinator to may be expressed using an encoding, like Church encoding. In this case particular lambda terms (which define functions) are considered ...
Use the line formed by the two points to divide the set into two subsets of points, which will be processed recursively. We next describe how to determine the part of the hull above the line; the part of the hull below the line can be determined similarly. Determine the point above the line with the maximum distance from the line.
The value of the line function at this midpoint is the sole determinant of which point should be chosen. The adjacent image shows the blue point (2,2) chosen to be on the line with two candidate points in green (3,2) and (3,3). The black point (3, 2.5) is the midpoint between the two candidate points.