Adams-Moulton formula

One of the most popular predictor-corrector methods is the fourth order Adams-Bashford and Adams-Moulton formula. 

The fully implicit scheme is based on the second-order accurate Adams-Moulton formulae, that is, at each time step, n + 1, all terms (both linear and nonlinear) in Eqs. (1.11)-(1.13) are integrated implicitly as follows ... 

The value of is then substituted in the implicit Adams-Moulton Formula to correct the value of yu+i-Correct ... 

The Adam-Bashforth methods are frequently used as predictors and the Adam-Moulton methods are often used as correctors. The combination of the two formulas results in predictor-corrector schemes. 

The primary advantage of the single-step methods is that they are self starting. We can also vary the step sizes. In contrast, the multistep methods require a single-step formula to start the calculations. Step size variation is difficult. However, the efficiency of both the Milne s and Adams-Moulton methods is about twice that of the single-steps methods. We need two function evaluations per step in the former while four or five are required with the single step. 

Ccxnputation times also follow the expected evolution, increasing with the niunber of function evaluations per step required by the method. The Adams-Moulton scheme, however, which requires only two e uations per step, like the modified Euler scheme, was only slightly faster than the fourth-order Runge-Kutta due to the amount of computation involved in the predictor and corrector formulas. The former also provided extremely low errors, when applicable, but showed a tendency to become unstable at higher step sizes. 

Example 4.1.2 For equal (constant) step sizes the Adams-Moulton methods are given by the following formulas... 

The matrixes Y, C, D, and v are the output states, the output matrix, the feed forward control force matrix, and the noise matrix, respectively. In the case where the output variables are the same with the states of the system and there is no application of the control forces to the output variables, the matrixes C, D are the identity and zero matrix, respectively. The noise matrix depends oti the sensor that is used to measure the response of the system. The above equation can be solved by any numerical technique for differential equations, like an explicit Rtmge-Kutta formula, the Dormand-Prince pair, Bogacki-Shampine, and Adams-Bashforth-Moulton PECE solver. 

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