Reapply Functions

Question

Reading this question made me think: for a given function f, how can we know that the cycle of this form:

while (x > 2)
   x = f(x)

will stop for any value x? Is there a simple criterion?

(The fact that f(x) < xfor x > 2does not seem to help, as the series may converge).

In particular, can we prove this for sqrtand for log?

Answer 1

For these functions, it suffices to prove that ceil(f(x))<xfor x > 2. You can do one iteration - to get an integer, and then continue using simple induction.

, , , . , , , .

Amnon:

, , . : x << y , ceil(x) < ceil(y), << . , , 2, sqrt, log , .

, . , , " " , .

Answer 2

, . ( , . , , .)

x, f (x), f (f (x)),... , f . k < 1 , x y, | f (x) - f (y) | <= k | x-y |.

Answer 3

( , f (x) < x x > 2, -, , ).

, . x > n f(x) , x, n - ( ).

, , , f(x) x (.. , , x, , f(x) = x - ).

Answer 4

, f x . .

sqrt log , . , sqrt 1, log . , x < 2 .

, .

Answer 5

, , , , x _i & le; 2. , , , 2. , , 2.

, , , , ( ) . x _{+ 1}= sqrt (x _i), x 1. y _{+ 1}= log (y _i), 2, undefined R ( , C *, , , , (.. z = log (z)). , , .

x _i z , & epsilon; > 0, n, i > n, | x _i - z | < & ;.

Mandelbrot Set, M. c C M , z _{+ 1}= z _i ² + c , , | z _i | > 2. M (, 0), (, -1).

Answer 6

. x :

 log(x) <= x - 1

( , , log x, x-1 x = 1). , , , while ceil(x) - 2 - , , .

f(x) = sqrt(x); , , :

sqrt(x) <= x/(2 sqrt(2)) + 1/sqrt(2)

x.

, , , , . , log, , , . , , ; log.

Answer 7

, . f .