Linear multistep method: Difference between revisions

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Multistep methods use information from the previous <math> s </math> steps to calculate the next value. In particular, a ''linear'' multistep method uses a linear combination of <math> y_i </math> and <math> f(t_i,y_i) </math> to calculate the value of <math> y </math> for the desired current step. Thus, a linear multistep method is a method of the form

: <math> \begin{align}

& y_{n+s} =+ a_{s-1} \cdot y_{n+s-1} + a_{s-2} \cdot y_{n+s-2} + \cdots + a_0 \cdot y_n \\

& \qquad {} += h\cdot\left( b_s \cdot f(t_{n+s},y_{n+s}) + b_{s-1} \cdot f(t_{n+s-1},y_{n+s-1}) + \cdots + b_0 \cdot f(t_n,y_n) \right),

\end{align} </math>

The coefficients <math> a_0, \dotsc, a_{s-1} </math> and <math> b_0, \dotsc, b_s </math> determine the method. The designer of the method chooses the coefficients, balancing the need to get a good approximation to the true solution against the desire to get a method that is easy to apply. Often, many coefficients are zero to simplify the method.