Fixed Point Iterations

\(g1[x_] := x^2 - 2; 
g2[x_] := Sin[x]; 
g3[x_] := x - Tan[x]; \)

Plot[{g3[x], x}, {x, -2, 3}];

x = 1;

\(\(For[i = 1, i <= 10, \(i++\), 
	x = g3[x] // N; 
	
      Print[i, "    ", x]; 
	]; \)\)

g1 has repelling fixed points
g2 has attracting fixed point, but very slow.
g3 has attracting fixed point, quadratic convergence.

x=x-Tan[x]=0  ⇒  Tan[x]=0 ⇒ Sin[x]=0.

g2 and g3 are both trying to find zeroes of Sin[x].  One works well, the other not so good.