Sorry for late reply.

An Image with A0 and A4 is just the scheme how the Euler method works. I
took it from wikipedia. There is explanation under that image. It says:"
Illustration of the Euler method. The unknown curve is in blue, and its
polygonal approximation is in red."
http://en.wikipedia.org/wiki/Euler_method

I wrote in blog post, that if we take the point y(A0) - 1 as the initial
one, than we will get a value less than A4 and closer to the function which
is drawn with blue colour.

It is clear if you will draw the same red line, but from the point (A0 -
1),  then we will get a new one point instead the A4, that will be less
than A4 and closer to the function which is drawn with blue colour.
It has this result:
1. If we take the actual initial point minus some value, the result will be
equal to the real value and blue line at point y=0 ( blue one, because the
deviation is negative - initial point minus some value);
2. If we take the actual initial point (A0), the result will be equal to
(A4) the real plus some value and black line at some value ( black one,
because the deviation is zero);

The algorithms of other methods are different. For example for Midpoint
method we look for midpoint at each step, for multistep methods there are
refining and because of that the bars are different.

Best regards,
Natalia

2013/9/24 Ben Coman <btc@openinworld.com>