Talk:Backward Euler method

Mistake in formula for fixed-point iteration
$$ y_{k+1}^{[0]} = y_k, \quad y_{k+1}^{[i+1]} = y_k + h f(t_{k+1}^{[i]}, y_{k+1}^{[i]}). $$

in this formula, i guess it should read:

$$ y_{k+1}^{[0]} = y_k, \quad y_{k+1}^{[i+1]} = y_k + h f(t_{k+1}, y_{k+1}^{[i]}). $$

as we don't improve the value of $$t_{k+1}$$ during iterations. wrong?


 * Thanks for the correction. I changed the article accordingly. -- Jitse Niesen (talk) 15:10, 24 April 2012 (UTC)

thanks — Preceding unsigned comment added by 182.73.186.2 (talk) 05:22, 28 March 2013 (UTC)

Local truncation error
I'm fairly certain that the 'Local truncation error' is of size $$ - \frac{1}{2}h^2y''(\xi) $$, thus $$ O(h^2)$$. And, assuming the function and method don't 'amplify' the errors, the error at time $$ t_n$$ is given by $$ n \times \tau_n = \frac{1}{h} \times O(h^2) = O(h) $$. So, it is only the 'global truncation error' which is of order $$O(h)$$. — Preceding unsigned comment added by 213.124.211.182 (talk) 11:35, 18 January 2014 (UTC)


 * This does depend on the definition of 'local truncation error'. But I agree, using the definition in our article (which I think is the most common one), it is $$ O(h^2) $$, so I'll change it. -- Jitse Niesen (talk) 16:56, 18 January 2014 (UTC)

Region of absolute stability
The article says: "The region of absolute stability for the backward Euler method is the complement in the complex plane of the disk with radius 1 centered at 1, depicted in the figure.", and then provides a nice figure of a circle on a plane. It's probably evident to whomever constructed this figure, but it isn't to the reader: What are the axes of this plane? Could the figure provide a title for the axes? — Preceding unsigned comment added by 2001:480:20:22:0:0:0:10 (talk) 14:35, 3 June 2014 (UTC)