plotting complicated function
I would like to approximate the sum h(a,x)=−2nn−1∑n=0log|Tna(x)| where n is large like n=1000−5000 and for a fixed a Ta(x)=|1x|−⌊|1x|−1+a⌋ where x∈(0,1).
By fixing x to be a value x0∈(0,1), e.g. x0=1/π, h(a,x0)=h(a,1/π) a function of one variable, and I want to plot a 2D graph of point (a,h(a,1/π)), by fixing n=2000, for a∈[0,1].
I figure how to calculate the value at one given a using SageMath, for example, when a=1,
T(x) = 1/x - floor(1/x)
s = 0
for k in xrange(0, 1000):
a = 0
a = nest(T, k, 0.79)
b = abs(a)
c = log(b)
s = s + cThen −21000s give the approximation for the sum when x=0.79, n=1000, a=1.
But for plotting, I think I need to define the function h(a,x) which is a summation over composition of functions. I tried to use sum and symbolic_sum but failed.
Any help how to achieve this please?
I suspect that you want to write |Ta(x)|k instead of Tna(x), right? Otherwsie, what means Tn here?
@Masacroso: it means the function Ta is applied n times. For example, T2a(x)=Ta(Ta(x)).
well, in first place note that Ta(x)=x−1−⌊⌊x−1⌋+{x−1}−r⌋={x−1}−⌊{x−1}−r⌋={x−1}+[r>{x−1}] for r∈[0,1] and where [⋅] is an Iverson bracket and {x−1} is the fractional part of x−1. But then note that Ta(Ta((x)) is not defined when {x−1}=0 and r≤{x−1}.
In that branch of mathematics, it's usual to study functions that are defined only almost everywhere. The set of measure zero where they are not defined is just disregarded and "does not matter". Of course, when experimenting with a computer, it can turn out to matter.