Implicit backward Euler approximation

However, the implicit, backward Euler approximation gives... 

Because Xk+i appears on both sides of this equation, additional steps are required to solve for x +i before the approximation can be used to calculate it. (This can be done via iteration, such as through the Newton-Raphson method.) Hence, the backward Euler method is also referred to as an implicit method. The trapezoidal algorithm averages the information from the forward and backward Euler algorithms such that the iteration equation to be used is... 

Another possibility is to let the same derivative approximation pertain to the next time this is the backward implicit (BI) method, also called backward Euler [8] ... 

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