Calling
f(x,y) = ((f_1),(f_2))=((x y -a),(y log_e x+x log_e y - b))
with a = 64, b = 2.5
Expanding near the point p_0 = {x_0,y_0} in Taylor series for the linear term
f(x,y) = f(x_0,y_0) + grad f_0 cdot(p - p_0) +O^2(p-p_0) where
grad f = (((df_1)/dx,(df_1)/dy),((df_2)/dx,(df_2)/dy))
p-p_0 = ((x-x_0),(y-y_0))
O^2(p-p_0)->{0,0} as abs(p-p_0)->0
For abs(p-p_0) small then
f(x,y) approx f(x_0,y_0) + grad f_0 cdot(p - p_0)
If abs(p-p_0) is small then f(x,y) approx {0,0}
and
p = p_0 - (grad f_0)^{-1}p_0
or
p_{k+1} = p_k - (grad f_k)^{-1}p_k
Here
grad f = ((y, x),(y/x + Log(y), x/y + Log(x)))
(grad f_0)^{-1} = (((x/y + Log(x))/(x - y + y Log(x) - x Log(y)),
x/(-x + y - y Log(x) + x Log(y))),((y + x Log(y))/(
x (-x + y - y Log(x) + x Log(y))), y/(x - y + y Log(x) - x Log(y))))
Begining with
p_0 = {1,20} we obtain
p_1 = {0.877443,66.4511}
p_2 = {0.970524,65.8899}
p_3 = {0.9761,65.5652}
p_4 = {0.976141,65.5643}
p_5 = {0.976141,65.5643}
