The stability region, or region of absolute stability, of a numerical method for ordinary differential equations
is the set of complex numbers 
for which the method does not
amplify the solution of the test equation 
. If a one-step method applied with step size 
has update 
, then its stability region is
 |
For the Euler forward method, 
, so the stability region is the closed
disk 
.
For the Euler backward method, 
, so it is the region 
. A method is A-stable
if its stability region contains the entire left half-plane.
