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The order of differencing can be reversed for the time step (i.e., forward/backward followed by backward/forward). For nonlinear equations, this procedure provides the best results. For linear equations, the MacCormack scheme is equivalent to the Lax–Wendroff method. [4]
Backward finite difference [ edit ] 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 forward jump, backward jump, and graininess operators on a discrete time scale The forward jump and backward jump operators represent the closest point in the time scale on the right and left of a given point t {\displaystyle t} , respectively.
In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for f ′(x + h / 2 ) and f ′(x − h / 2 ) and applying a central difference formula for the derivative of f ′ at x, we obtain the central difference approximation of the second derivative of f:
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
Data-flow analysis is a technique for gathering information about the possible set of values calculated at various points in a computer program.A program's control-flow graph (CFG) is used to determine those parts of a program to which a particular value assigned to a variable might propagate.
Reverse accumulation is more efficient than forward accumulation for functions f : R n → R m with n ≫ m as only m sweeps are necessary, compared to n sweeps for forward accumulation. Backpropagation of errors in multilayer perceptrons, a technique used in machine learning , is a special case of reverse accumulation.
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.