2. Error Estimates and convergence

Recall that
1. $a=x_0 < x_1 < x_2 ... < x_a =b , \,\, \, x_i - x_{i-1}=h$
2.$y_i$ is the approximate solution of the initial value problem

defined by the Euler's methods namely

$$y_i = y_{i-1}+f(x_i,y_{i-1}),\,\,\, 1\leq i \leq n$$ (2.2)

Note here $y_i$ is the approximate value of $y(x_i)\,\,(1\leq i \leq n)$.The quantity $e_i$, defined by

$$e_i = y(x_i)-y_i,\,\,\, (1\leq i \leq n)$$

is therefore the deviation of $y_i$ from $y(x_i)$ or the error committed at the $i^{th}$ step. It is also called the truncation or discretigation error.
In this section, we examine the nature of $e_i$. It is desirable that $e_i$ is "small" when the step size h is small. In this connection we have the following result.