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Taylor series

If we denote with $ y^{(n)}(t)$ the function value at point $ t$ of the $ n$'the derivative of a function $ y$, then the Taylor series of a continuous function $ y$ at point $ t$ is given by:

$\displaystyle y(t+h) = y(t) + hy^{(1)}(t) + \frac{1}{2!}h^2y^{(2)}(t) + \frac{1}{3!}h^3y^{(3)}(t) + \cdots + \frac{1}{n!}h^ny^{(n)}(t) + \cdots$ (1.1)

Some remarks about this expansion:

  1. $ h$ can be any element of $ \mathbb{R}$, such that a function is completely defined by its Taylor expansion at any single point of its domain. (Full knowledge of the function at a single point determines the full function at all points).
  2. The successive terms of the Taylor series are decreasing in magnitude in an exponential way.
  3. We can truncate a Taylor series to approximate the original function. This divides the Taylor series in two separate series: the numerical scheme and the error series. The error after truncation is mainly dependent on the first term of the error series.


next up previous contents
Next: Numerical Schemes Up: Numerical Preliminaries Previous: Numerical Preliminaries   Contents
2002-11-15