Ratio of variables (recurrence relation)
I have the following recurrence relation.
Time is divided into synchronous steps (t is an integer).
There are n variables and initially all variables are the same: $x_1(0) = x_2(0) = dots = x_n(0) = c$.
begin{align*}
x_1(t+1) &= (frac{1}{2-frac{2}{n}}) x_1(t) + frac{1}{2}x_2(t) \
x_i(t+1) &= frac{1}{2} x_{i-1}(t) + x_{i+1}(t) \
x_n(t+1) &= frac{1}{2}x_{n-1}(t) + frac{1}{2}x_n(t)
end{align*}
I would like to prove that $x_1(t) leq c' x_n(t)$ for a constant $c'$ (this seems to be true in simulations).
Does anyone know how to prove this?
One way could be to derive the general formula for the variables, but currently I do not really know how to do this.
analysis recurrence-relations relations
asked Nov 28 '18 at 13:21
Jannik
587
add a comment |
I have the following recurrence relation.
Time is divided into synchronous steps (t is an integer).
There are n variables and initially all variables are the same: $x_1(0) = x_2(0) = dots = x_n(0) = c$.
begin{align*}
x_1(t+1) &= (frac{1}{2-frac{2}{n}}) x_1(t) + frac{1}{2}x_2(t) \
x_i(t+1) &= frac{1}{2} x_{i-1}(t) + x_{i+1}(t) \
x_n(t+1) &= frac{1}{2}x_{n-1}(t) + frac{1}{2}x_n(t)
end{align*}
I would like to prove that $x_1(t) leq c' x_n(t)$ for a constant $c'$ (this seems to be true in simulations).
Does anyone know how to prove this?
One way could be to derive the general formula for the variables, but currently I do not really know how to do this.
analysis recurrence-relations relations
asked Nov 28 '18 at 13:21
Jannik
587
add a comment |
I have the following recurrence relation.
Time is divided into synchronous steps (t is an integer).
There are n variables and initially all variables are the same: $x_1(0) = x_2(0) = dots = x_n(0) = c$.
begin{align*}
x_1(t+1) &= (frac{1}{2-frac{2}{n}}) x_1(t) + frac{1}{2}x_2(t) \
x_i(t+1) &= frac{1}{2} x_{i-1}(t) + x_{i+1}(t) \
x_n(t+1) &= frac{1}{2}x_{n-1}(t) + frac{1}{2}x_n(t)
end{align*}
I would like to prove that $x_1(t) leq c' x_n(t)$ for a constant $c'$ (this seems to be true in simulations).
Does anyone know how to prove this?
One way could be to derive the general formula for the variables, but currently I do not really know how to do this.
analysis recurrence-relations relations
asked Nov 28 '18 at 13:21
Jannik
587
I have the following recurrence relation.
Time is divided into synchronous steps (t is an integer).
There are n variables and initially all variables are the same: $x_1(0) = x_2(0) = dots = x_n(0) = c$.
begin{align*}
x_1(t+1) &= (frac{1}{2-frac{2}{n}}) x_1(t) + frac{1}{2}x_2(t) \
x_i(t+1) &= frac{1}{2} x_{i-1}(t) + x_{i+1}(t) \
x_n(t+1) &= frac{1}{2}x_{n-1}(t) + frac{1}{2}x_n(t)
end{align*}
I would like to prove that $x_1(t) leq c' x_n(t)$ for a constant $c'$ (this seems to be true in simulations).
Does anyone know how to prove this?
One way could be to derive the general formula for the variables, but currently I do not really know how to do this.
analysis recurrence-relations relations
analysis recurrence-relations relations
asked Nov 28 '18 at 13:21
Jannik
587
asked Nov 28 '18 at 13:21
Jannik
587
asked Nov 28 '18 at 13:21
Jannik
587
asked Nov 28 '18 at 13:21
Jannik
587
asked Nov 28 '18 at 13:21
Jannik
587
587
add a comment |
add a comment |
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