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In mathematics, the composition operator takes two functions, and , and returns a new function ():= () = (()). Thus, the function g is applied after applying f to x . Reverse composition , sometimes denoted f ↦ g {\displaystyle f\mapsto g} , applies the operation in the opposite order, applying f {\displaystyle f} first and g {\displaystyle g ...
In mathematics, an iterated function is a function that is obtained by composing another function with itself two or several times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial object, the result of applying a given function is fed again into the function as input, and ...
The name of the method comes from the fact that in the formula above, the function giving the slope of the solution is evaluated at = + / = + +, the midpoint between at which the value of () is known and + at which the value of () needs to be found.
The Gaussian quadrature chooses more suitable points instead, so even a linear function approximates the function better (the black dashed line). As the integrand is the third-degree polynomial y ( x ) = 7 x 3 – 8 x 2 – 3 x + 3 , the 2-point Gaussian quadrature rule even returns an exact result.
The Picard–Lindelöf theorem, which shows that ordinary differential equations have solutions, is essentially an application of the Banach fixed-point theorem to a special sequence of functions which forms a fixed-point iteration, constructing the solution to the equation. Solving an ODE in this way is called Picard iteration, Picard's method ...
Thus, the 3/8 rule is about twice as accurate as the standard method, but it uses one more function value. A composite 3/8 rule also exists, similarly as above. [6]
In mathematics, the method of matched asymptotic expansions [1] is a common approach to finding an accurate approximation to the solution to an equation, or system of equations. It is particularly used when solving singularly perturbed differential equations. It involves finding several different approximate solutions, each of which is valid (i ...
The n-th power of a solution of Schröder's equation provides a solution of Schröder's equation with eigenvalue s n, instead. In the same vein, for an invertible solution Ψ(x) of Schröder's equation, the (non-invertible) function Ψ(x) k(log Ψ(x)) is also a solution, for any periodic function k(x) with period log(s). All solutions of ...