Bakåtdifferens

Bakåtdifferens, also known as backward difference, is a method for approximating the derivative of a function at a point using function values at that point and previous points. It is a fundamental concept in numerical analysis, particularly in the study of finite differences.

The simplest form of the backward difference is denoted by $\nabla y_i$ and is defined as $y_i - y_{i-1}$, where $y_i$ is the value of a function at point $x_i$ and $y_{i-1}$ is the value at the preceding point $x_{i-1}$. This represents the change in the function value over the interval $[x_{i-1}, x_i]$.

The backward difference can be used to approximate the first derivative of a function $f(x)$ at a point $x_i$. Using a step size $h = x_i - x_{i-1}$, the approximation is given by $\frac{y_i - y_{i-1}}{h}$ or $\frac{f(x_i) - f(x_i-h)}{h}$. This approximation is generally less accurate than the central difference but is often easier to implement, especially when dealing with boundary conditions or when only past data is available.

Higher-order backward differences can also be defined recursively. For example, the second backward difference is $\nabla^2 y_i = \nabla y_i - \nabla y_{i-1} = (y_i - y_{i-1}) - (y_{i-1} - y_{i-2}) = y_i - 2y_{i-1} + y_{i-2}$. These higher-order differences are useful in approximating higher-order derivatives and in constructing finite difference methods for solving differential equations. The accuracy of the backward difference approximation depends on the function's smoothness and the step size $h$. As $h$ approaches zero, the backward difference approximation converges to the true derivative.