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I am using an Adam Bashforth 4 method to solve an IVP problem so I need other numerical method to estimate the first 3 values. I am very much interested in finding a way to estimate the first 3 values and that particular method should have order of accuracy of 4. Which method would you suggest?

From a theoretical point of view, is it even possible to estimate the first 3 values with order of accuracy at 4?

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What you are looking for is called "bootstrapping". It is a common problem of all multistep ODE integrators and is discussed in many books on the topic.

Among your options are to use a lower-order method with smaller time step, or to use a one-step method of higher order for the first few steps (e.g., a Runge-Kutta method).

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