Iteration: (not defined if x=0 or y=0)

F(x,y) = (y, a+b+c - (ab+bc+ca - abc/x) /y )

Fixed points: (a,a), (b,b), (c,c)

Invariant curves:

y = a+b - ab/x (magenta/green)

y = b+c - bc/x, y = c+a - ca/x (red)

The red curves are attracting for (c,c)

and repelling for (a,a), (b,b)

the magenta/green curve is attracting

for (b,b) and repelling für (a,a).

Because of rounding errors in floating point

computations, iterated points cannot stay on

the magenta/green curve and converge to (b,b).

Eventually all points are attracted by (c,c),

just wait until Start/Stop stops!

For rational points on y = a+b - ab/x one can

write code which uses arbitrarily large integers

and allows the iteration to converge to (b,b).

No program iterates (pi, a+b-ab/pi) correctly.

Berkeley University used this iteration to allert students to the consequences
of rounding errors in floating point computations. It is easy to check that the
Muller iteration can be expressed on the invariant curves by far simpler formulas,
for example on the magenta/green curve by (x,y) --> (y, a+b - ab/x). Of course,
if one computes with these formulas, iterated points stay on the invariant curves,
without rounding errors showing up. But Muller's original iteration iterates ** all **
points - except the fixed points (a,a) und (b,b) - eventually to (c,c).

in most computers real numbers are represented as binary numbers with finitely many digits (e.g. 64 bit). This is not even exact for all rational numbers and has rounding errors for all irrational numbers. Most real numbers are in fact transcendental, which implies that there is no way to**exactly compute** with them.

in most computers real numbers are represented as binary numbers with finitely many digits (e.g. 64 bit). This is not even exact for all rational numbers and has rounding errors for all irrational numbers. Most real numbers are in fact transcendental, which implies that there is no way to