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As the backward difference operator, in the calculus of finite differences. As the widening operator, an operator that permits static analysis of programs to terminate in finite time, in the computer science field of abstract interpretation. As function definition marker and self-reference in the APL programming language
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:
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 backward differentiation formula (BDF) is a family of implicit methods for the numerical integration of ordinary differential equations.They are linear multistep methods that, for a given function and time, approximate the derivative of that function using information from already computed time points, thereby increasing the accuracy of the approximation.
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]
The "backward difference operator" is Δ, and the result when you apply this operator is the "backward difference". Similarly, Butzer & Berens say that Δ h r f {\displaystyle \Delta _{h}^{r}f} (the result of applying the operator to a function f ) is a "right difference".
In mathematics, an operator or transform is a function from one space of functions to another. Operators occur commonly in engineering, physics and mathematics. Many are integral operators and differential operators. In the following L is an operator :
In mathematics, time-scale calculus is a unification of the theory of difference equations with that of differential equations, unifying integral and differential calculus with the calculus of finite differences, offering a formalism for studying hybrid systems.