be easily avoided if compensated summation is used in the formula for the Euler method. Modifications and extensions A simple modification of the Euler method which eliminates the stability problems noted above is the backward Euler method: This differs from the (standard, or forward) Euler method in that the function is evaluated at the end point of the step, instead of the starting point. The backward Euler method is an implicit method, meaning that the formula for the backward Euler method has on both sides, so when applying the backward Euler method we have to solve an equation. This makes the implementation more costly. Other modifications of the Euler method that help with stability yield the exponential Euler method or the semi-implicit Euler method. More complicated methods can achieve a higher order (and more accuracy). One possibility is to use more function evaluations. This is illustrated by the midpoint method which is already mentioned in this article: . This leads to the family of Runge–Kutta methods. The other possibility is to use more past values, as illustrated by the two-step Adams–Bashforth method: This leads to the family of linear multistep methods. There are other modifications which uses techniques from compressive sensing to minimize memory usage In popular culture In the film Hidden Figures, Katherine Goble resorts to the Euler method in calculating the re-entry of astronaut John Glenn from Earth orbit. See also Crank–Nicolson method Gradient descent similarly uses finite steps, here to find minima of functions List of Runge-Kutta methods Linear multistep method Numerical integration (for calculating definite integrals) Numerical methods for ordinary differential equations Notes References External links Euler method implementations in different languages by Rosetta Code Numerical differential equations Runge–Kutta methods First order methods Leonhard Euler Articles with example R code Articles with example MATLAB/Octave code