Linear multistep method: Difference between revisions

Revision as of 07:38, 19 June 2010

Revision as of 19:21, 19 June 2010

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The coefficients <math>b_j</math> can be determined as follows. Use [[polynomial interpolation]] to find the polynomial ''p'' of degree <math>s-1</math> such that

:<math> p'(t_{n+i}) = f(t_{n+i}, y_{n+i}), \qquad \text{for } i=0,\ldots,s-1. </math>

The [[Lagrange polynomial|Lagrange formula]] for polynomial interpolation yields

:<math> p(t) = \sum_{j=0}^{s-1} \frac{(-1)^{s-j-1}f(t_{n+j}, y_{n+j})}{j!(s-j-1)!h^{s-1}} \prod_{i=0 \atop i\ne j}^{s-1} (t-t_{n+i}). </math>