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→Analysis: The fact that it "includes the whole left half of the complex plane" does not imply in A-stability. The backward Euler method may converge to an artificial solution when the dynamics of the real system is unstable (hence, not A-stable). |
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This differs from the (forward) Euler method in that the latter uses <math> f(t_k, y_k) </math> in place of <math>f(t_{k+1}, y_{k+1})</math>.
The backward Euler method is an implicit method: the new approximation <math> y_{k+1} </math> appears on both sides of the equation, and thus the method needs to solve an algebraic equation for the unknown <math> y_{k+1} </math>.
:<math> y_{k+1}^{[0]} = y_k, \quad y_{k+1}^{[i+1]} = y_k + h f(t_{k+1}, y_{k+1}^{[i]}). </math>
If this sequence converges (within a given tolerance), then the method takes its limit as the new approximation
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