Purcell EM Problem 1.2: Theory Behind Numerical Solution?

[yucheng]

Homework Statement

(Purcell Electricity and Magnetism Problem 1.2)

Relevant Equations

N/A

The author start of with $\frac{1}{(y+\sqrt{3})^2} - 2 \cdot \frac{1}{1 + y^2} \left( \frac{y}{\sqrt{1+y^2}} \right) = 0$ and arrives at the equation $y = \frac{(1+y^2)^{3/2}}{2(y+\sqrt{3})^2}$ The solution is merely by iterating (use an initial guess value of y, calculate the RHS, then use this answer for the next calculation) Then the answer forms a converging sequence, with the limit $y \approx 0.1463$. Which method is this supposed to be? Newton's Method?

 

[fresh_42]

Homework Statement:: N/A
Relevant Equations:: N/A

The author start of with $$\frac{1}{(y+\sqrt{3})^2} - 2 \cdot \frac{1}{1 + y^2} \left( \frac{y}{\sqrt{1+y^2}} \right) = 0$$ and arrives at the equation $$y = \frac{(1+y^2)^{3/2}}{2(y+\sqrt{3})^2}$$ The solution is merely by iterating (use an initial guess value of y, calculate the RHS, then use this answer for the next calculation) Then the answer forms a converging sequence, with the limit $y \approx 0.1463$. Which method is this supposed to be? Newton's Method?

No, not really. It's the fixed-point-iteration because we are looking for a fixed point $y=f(y)$ where $f(y)$ is the function on the right.

Newton is looking for zeroes. Originally by approximating with $y_0+p$ and using only the constant and linear term for $p$, and $y_0$ the starting guess: $0=y-f(y)=y_0+p-f(y_0+p)=a_1+b_1p+ r(p)$ then solve for $0=a_1+b_1p_1$ and continue with $y_1 :=p_1+p_2$ and solve $0=a_2+b_2p_3$ etc.