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Samrat Mukhopadhyay
Samrat Mukhopadhyay
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Let $\{x_n\}_{n\ge 0}$ be a sequence of reals such that $x_{n+1}=g(x_n)$, where $g:\mathbb{R}\to \mathbb{R}$ is a continuous function such that $0$ is a fixed point of $g$. My question is the following:

If for an initial condition $x_0$, we have $x_n\to 0$ as $n\to \infty$, how should one proceed to find the rate of convergence of the sequence $x_n$?

I know that if $g$ is a contraction, then this rate of convergence is linear and the rate is just the contraction factor of $g$. However,

how should one proceed if $g$ is not a contraction?

For example, let $g(x) = x^2$. If $x_0<1$, then we have $x_n = x_0^{2^n}\stackrel{n\to \infty}{\to} 0$, and the rate of convergence is quadratic.

Note that if $g$ is Lipschitz, then I can always find an initial condition suitably so that the sequence converges to $0$ linearly. However, this estimated rate would only be an upper bound on the actual rate, as it was seen for the $x^2$ example, the actual rate is quadratic and is thus very fast.

Please provide some references to relevant literature. Thanks in advance.

Let $\{x_n\}_{n\ge 0}$ be a sequence of reals such that $x_{n+1}=g(x_n)$, where $g:\mathbb{R}\to \mathbb{R}$ is a continuous function such that $0$ is a fixed point of $g$. My question is the following:

If for an initial condition $x_0$, we have $x_n\to 0$ as $n\to \infty$, how should one proceed to find the rate of convergence of the sequence $x_n$?

I know that if $g$ is a contraction, then this rate of convergence is linear and the rate is just the contraction factor of $g$. However,

how should one proceed if $g$ is not a contraction?

For example, let $g(x) = x^2$. If $x_0<1$, then we have $x_n = x_0^{2^n}\stackrel{n\to \infty}{\to} 0$, and the rate of convergence is quadratic.

Note that if $g$ is Lipschitz, then I can always find an initial condition suitably so that the sequence converges to $0$ linearly. However, this estimated rate would only be an upper bound on the actual rate, as it was seen for the $x^2$ example, the actual rate is quadratic and is thus very fast.

Please provide some references to relevant literature. Thanks in advance.

Let $\{x_n\}_{n\ge 0}$ be a sequence of reals such that $x_{n+1}=g(x_n)$, where $g:\mathbb{R}\to \mathbb{R}$ is a continuous function such that $0$ is a fixed point of $g$. My question is the following:

If for an initial condition $x_0$, we have $x_n\to 0$ as $n\to \infty$, how should one proceed to find the rate of convergence of the sequence $x_n$?

I know that if $g$ is a contraction, then this rate of convergence is linear. However,

how should one proceed if $g$ is not a contraction?

For example, let $g(x) = x^2$. If $x_0<1$, then we have $x_n = x_0^{2^n}\stackrel{n\to \infty}{\to} 0$, and the rate of convergence is quadratic.

Note that if $g$ is Lipschitz, then I can always find an initial condition suitably so that the sequence converges to $0$ linearly. However, this estimated rate would only be an upper bound on the actual rate, as it was seen for the $x^2$ example, the actual rate is quadratic and is thus very fast.

Please provide some references to relevant literature. Thanks in advance.

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Samrat Mukhopadhyay
Samrat Mukhopadhyay
  • 554
  • 3
  • 13

How to find the best convergence rate of a dynamical system $x_{n+1} = g(x_n),\ n\ge 0,\ x_0\in \mathbb{R}$?

Let $\{x_n\}_{n\ge 0}$ be a sequence of reals such that $x_{n+1}=g(x_n)$, where $g:\mathbb{R}\to \mathbb{R}$ is a continuous function such that $0$ is a fixed point of $g$. My question is the following:

If for an initial condition $x_0$, we have $x_n\to 0$ as $n\to \infty$, how should one proceed to find the rate of convergence of the sequence $x_n$?

I know that if $g$ is a contraction, then this rate of convergence is linear and the rate is just the contraction factor of $g$. However,

how should one proceed if $g$ is not a contraction?

For example, let $g(x) = x^2$. If $x_0<1$, then we have $x_n = x_0^{2^n}\stackrel{n\to \infty}{\to} 0$, and the rate of convergence is quadratic.

Note that if $g$ is Lipschitz, then I can always find an initial condition suitably so that the sequence converges to $0$ linearly. However, this estimated rate would only be an upper bound on the actual rate, as it was seen for the $x^2$ example, the actual rate is quadratic and is thus very fast.

Please provide some references to relevant literature. Thanks in advance.