Linear multistep method: Difference between revisions

Revision as of 18:40, 31 March 2011

Revision as of 06:20, 12 May 2011

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

\end{align} </math>

where ''h'' denotes the step size and ''f'' the right-hand side of the differential equation. The coefficentscoefficients <math> a_0, \ldots, a_{s-1} </math> and <math> b_0, \ldots, b_s </math> determine the method. The designer of the method chooses the coefficients; often, many coefficients are zero. Typically, the designer chooses the coefficients so they will exactly interpolate <math>y(t)</math> when it is an ''n''th order polynomial.

If the value of <math>b_s</math> is nonzero, then the value of <math>y_{n+s}</math> depends on the value of <math> f(t_{n+s}, y_{n+s}) </math>. Consequently, the method is explicit if <math> b_s = 0 </math>. In that case, the formula can directly compute <math> y_{n+s} </math>. If <math> b_s \ne 0 </math> then the method is implicit and the equation for <math> y_{n+s} </math> must be solved. [[Iterative methods]] such as [[Newton's method]] are often used to solve the implicit formula.