Editing Rate of convergence (section)

=== Example ===
Consider the ordinary differential equation
:<math> \frac{dy}{dx} = -\kappa y </math>

with initial condition <math>y(0) = y_0</math>. We can approximate a solution to this one-dimensional equation using a sequence <math>(y_n)</math> applying the [[forward Euler method]] for numerical discretization using any regular grid spacing <math>h</math> and grid points indexed by <math>n </math> as follows:
:<math> \frac{y_{n+1} - y_n}{h} = -\kappa y_{n}, </math>

which implies the first-order [[linear recurrence with constant coefficients]]
:<math> y_{n+1} = y_n(1 - h\kappa). </math>

Given <math>y(0) = y_0</math>, the sequence satisfying that recurrence is the [[geometric progression]]

<math display="block"> y_{n} = y_0(1 - h\kappa)^n = y_0\left(1 - nh\kappa + \frac{n(n-1)}{2}h^2\kappa^2 + ....\right). </math>

The exact analytical solution to the differential equation is <math>y = f(x) = y_0\exp(-\kappa x)</math>, corresponding to the following [[Taylor expansion]] in <math>nh\kappa </math>:
<math display="block">f(x_n) = f(nh) = y_0\exp(-\kappa nh)
 = y_0\left(1 - nh\kappa + \frac{n^2 h^2\kappa^2}{2} + ...\right).</math>

Therefore the error of the discrete approximation at each discrete point is
:<math display="block">|y_n - f(x_n)| = \frac{nh^2\kappa^2}{2} + \ldots</math>
For any specific <math>x = p</math>, given a sequence of forward Euler approximations <math>((y_n)_k)</math>, each using grid spacings <math>h_k</math> that divide <math>p</math> so that <math>n_{p,k} = p/h_k</math>, one has

<math display="block">\lim_{h_k \rightarrow 0} \frac{|y_k(p) - f(p)|}{h_k} = \lim_{h_k \rightarrow 0} \frac{|y_{k, n_{p,k}} - f(h_k n_{p,k})|}{h_k} = \frac{h_k n_{p,k} \kappa^2}{2} = \frac{p \kappa^2}{2}</math>

for any sequence of grids with successively smaller grid spacings <math>h_k</math>. Thus <math>((y_n)_k)</math> converges to <math>f(x)</math> [[Pointwise convergence|pointwise]] with a convergence order <math>q = 1</math> and asymptotic error constant <math>p \kappa^2 / 2</math> at each point <math>p > 0.</math> Similarly, the sequence converges [[Uniform convergence|uniformly]] with the same order and with rate <math>L \kappa^2 / 2</math> on any bounded interval of <math>p \leq L</math>, but it does not converge uniformly on the unbounded set of all positive real values, <math>[0, \infty).</math>