Convergence Interval for Newton's Method

[Scootertaj]

1. The problem statement:

In what region can we choose x₀ and get convergence to the root x = 0 for f(x) = e^{-1/x^2}

Homework Equations

x_n+1 = x_n - f(x_n) / f'(x_n)

The Attempt at a Solution

The only thing I've come across is a formula that says |root - initial point| < 1/M where M = max|f''(x)|/(2min|f'(x)| where x belongs to a "sufficiently small interval"

My thought: [-1,1]

 

[Ray Vickson]

1. The problem statement:

In what region can we choose x₀ and get convergence to the root x = 0 for f(x) = e^{-1/x^2}

Homework Equations

x_n+1 = x_n - f(x_n) / f'(x_n)

The Attempt at a Solution

The only thing I've come across is a formula that says |root - initial point| < 1/M where M = max|f''(x)|/(2min|f'(x)| where x belongs to a "sufficiently small interval"

My thought: [-1,1]

The exponential function does not equal zero for any real argument, so there is NO root. (I suppose you could regard x = +-infinity as "roots", but things like |root - x_n| are then not real numbers, either.) If you regard the question as: "for what x_0 does x_n --> + infinity (or -infinity)?", then you might have a sensible question.

RGV