bakåtdifferentieringsformlerna

Bakåtdifferentiering, sometimes referred to as backward differentiation, is a numerical method used in mathematics and computer science to approximate the derivative of a function. It is a type of finite difference method. Unlike forward or central differencing, bakåtdifferentiering uses function values at previous points to estimate the derivative at a current point.

The basic formula for bakåtdifferentiering of a function f(x) at a point x_i using a step size h is given by:

f'(x_i) ≈ (f(x_i) - f(x_{i-1})) / h

where x_{i-1} is the point immediately preceding x_i, such that x_i - x_{i-1} = h. This formula represents a first-order approximation. Higher-order approximations can be derived using more preceding points.

Bakåtdifferentiering is particularly useful when dealing with differential equations, especially stiff ones, where forward differencing methods might become unstable. The stability of bakåtdifferentiering methods often makes them a preferred choice in numerical solvers for ordinary and partial differential equations. However, it requires that values at previous time steps or spatial locations are available, which can be a limitation in some scenarios. The accuracy of the approximation is dependent on the step size h; a smaller h generally leads to a more accurate derivative but can also increase computational cost and potential for round-off errors.