Algorithms trapezium

For example, the trapezium algorithm, also called the Crank-Nicolson or the second-order Adams-Moulton ... 

To give another example, the amplification factor for the trapezium algorithm applied to the ODE system (2.1) is given by... 

It seems that the trapezium algorithm is superior to the Euler backward method from all viewpoints Even though they are both A-stable and need to solve a nonlinear system in y +i, the trapezium algorithm is of second order, while Euler backward is of first order only. However, there is also an additional aspect to consider. 

For example, the Euler backward is strongly A-stable, whereas the trapeziimi algorithm is not. Actually, the trapezium algorithm has x - 1, while hX — oo. ... 

To understand the importance of this feature, let us consider the reference equation (2.38) with the initial condition y(0) =yQ.lit -> oo, the solution tends to zero with Re(X) < 0. If the initial condition is perturbed so as to have yg + e, the error with the trapezium algorithm propagates in line with the following relation ... 

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