Backward induction is the process of reasoning backwards in time from the end of a problem or situation to determine a sequence of optimal actions. It proceeds by examining the last point at which a decision is to be made and then identifying what action would be most optimal at that moment.
The backward Euler method is an implicit method: the new approximation + appears on both sides of the equation and thus the method needs to solve an algebraic equation for the unknown +. For non- stiff problems this can be done with fixed-point iteration :
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.
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If instead of (2) we use the approximation ′ ≈ − (−) we get the backward Euler method: + = + (+ +). ()The backward Euler method is an implicit method meaning that we have to solve an equation to find y n+1.One often uses fixed-point iteration or (some modification of) the Newton–Raphson method to achieve this.. It costs more time to solve this equation than explicit methods …
In numerical analysis the Runge–Kutta methods are a family of implicit and explicit iterative methods which include the well-known routine called the Euler Method used in temporal discretization for the approximate solutions of ordinary differential equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm…