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An illustration of the five-point stencil in one and two dimensions (top, and bottom, respectively). In numerical analysis, given a square grid in one or two dimensions, the five-point stencil of a point in the grid is a stencil made up of the point itself together with its four "neighbors".
With n = 1, the slopes or first derivatives of the smoothstep are equal to zero at the left and right edge (x = 0 and x = 1), where the curve is appended to the constant or saturated levels. With higher integer n , the second and higher derivatives are zero at the edges, making the polynomial functions as flat as possible and the splice to the ...
Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function ...
The classical finite-difference approximations for numerical differentiation are ill-conditioned. However, if is a holomorphic function, real-valued on the real line, which can be evaluated at points in the complex plane near , then there are stable methods.
To get the coefficients of the backward approximations from those of the forward ones, give all odd derivatives listed in the table in the previous section the opposite sign, whereas for even derivatives the signs stay the same. The following table illustrates this: [5]
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
Characteristics may fail to cover part of the domain of the PDE. This is called a rarefaction, and indicates the solution typically exists only in a weak, i.e. integral equation, sense. The direction of the characteristic lines indicates the flow of values through the solution, as the example above demonstrates.
It allows the translation of various partial differential equations, e.g., the heat equation, to the graph setting. Based on the first-order partial difference operators on graphs one can formally derive a family of weighted graph -Laplace operators,: () for < by minimization of the discrete -Dirichlet energy functional