the region of absolute stability
When the true solution of an equation is quietly DECAYING, you expect your numerical solution to decay too, not to blow up. Whether it does depends on the step size: too big a step can make a stable method produce growing garbage. The region of absolute stability is the map that tells you, for a given method, exactly which step sizes keep the numerical solution bounded — it is the safe operating zone of the method.
We test every method on the same simple yardstick, the linear test equation y' = lambda*y, whose true solution e^(lambda*t) decays when the real part of lambda is negative. Apply a method with step h and it produces y_{n+1} = R(z)*y_n, where z = lambda*h and R is the method's STABILITY FUNCTION (a polynomial for explicit methods, a rational function for implicit ones). The numerical solution stays bounded exactly when |R(z)| <= 1. The set of complex values z = lambda*h for which |R(z)| <= 1 is the region of absolute stability, drawn in the complex plane. For forward Euler, R(z) = 1 + z, so the region is the disk |1 + z| <= 1, a circle of radius 1 centered at -1 — meaning for real negative lambda you must keep h <= 2/|lambda|, or it explodes. For backward Euler, R(z) = 1/(1 - z), whose region is the ENTIRE left half-plane and more.
This picture is the single most useful tool for understanding stability. To use it: find your problem's fast decaying modes (the eigenvalues lambda of the Jacobian with large negative real part), and you must choose h so that every lambda*h lands inside the region. The honesty here: explicit methods have SMALL, bounded stability regions, so a large negative lambda (a stiff mode) forces h tiny — this is the mechanism of stiffness. Implicit methods can have HUGE regions covering the whole left half-plane (A-stability), letting them take big steps. Absolute stability is about not BLOWING UP; it is separate from accuracy — a step can be stable yet inaccurate.
For y' = -50y, forward Euler is stable only when |1 + h*(-50)| <= 1, i.e. h <= 2/50 = 0.04; at h = 0.05 the factor is 1 + 0.05*(-50) = -1.5, |R| = 1.5 > 1, so y_n = (-1.5)^n oscillates and grows without bound even though the true solution decays. Backward Euler's factor 1/(1 + 50h) stays below 1 for every h > 0, so it is stable at any step.
Plot |R(lambda*h)| <= 1 in the complex plane — the safe zone of step sizes for each method.
Absolute stability concerns boundedness, NOT accuracy: a step inside the region can still be inaccurate, and a step that is accurate enough can still lie outside the region and explode. They are different requirements, and you must satisfy both.