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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". It is used to write finite difference approximations to derivatives at grid points. It is an example for numerical differentiation.
In numerical analysis, given a square grid in two dimensions, the nine-point stencil of a point in the grid is a stencil made up of the point itself together with its eight "neighbors". It is used to write finite difference approximations to derivatives at grid points. It is an example for numerical differentiation.
The finite difference coefficients for a given stencil are fixed by the choice of node points. The coefficients may be calculated by taking the derivative of the Lagrange polynomial interpolating between the node points, [3] by computing the Taylor expansion around each node point and solving a linear system, [4] or by enforcing that the stencil is exact for monomials up to the degree of the ...
The above arithmetic can be generalized to calculate second order and higher derivatives of multivariate functions. However, the arithmetic rules quickly grow complicated: complexity is quadratic in the highest derivative degree. Instead, truncated Taylor polynomial algebra can be used. The resulting arithmetic, defined on generalized dual ...
A simple two-point estimation is to compute the slope of a nearby secant line through the points (x, f(x)) and (x + h, f(x + h)). [1] Choosing a small number h , h represents a small change in x , and it can be either positive or negative.
More generally though, simulation is employed for path dependent exotic derivatives, such as Asian options. In other cases, the source of uncertainty may be at a remove. For example, for bond options [3] the underlying is a bond, but the source of uncertainty is the annualized interest rate (i.e. the short rate).
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of () at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below.
In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as ...