Fixed Point for functions of several variables

Question

Can someone solve, or help me solve this problem in Numerical Analysis in chapter Fixed Point for functions of several variables. Chapter 10, exercise set 10.1 problem 4. The problem is explained in the picture. This is the book , page 637 is the problem, page 630 is Fixed Point for functions of several variables chapter. https://fac.ksu.edu.sa/sites/default/files/numerical_analysis_9th.pdf

The problem in photo

Graph

Formulas in x_i = g_i(x) form

Iterations for point (3,11)

Answer 1

The nonlinear system $$−x(x + 1) + 2y = 18, (x − 1)^2 + (x − 6)^2 = 25$$ has two solutions.

a. Approximate the solutions graphically. b. Use the approximations from part (a) as initial approximations for an appropriate function iteration, and determine the solutions to within $10^{−5}$ in the $l_\infty$ norm.

A Wolfram Alpha contour plot shows the two intersection points of $(-2, 10)$ and $(1.54695, 10.97)$.

After trial and error (there can be more than a single set of functions dervied), lets choose the first function as

$$\begin{align} g_1(x, y) &= \dfrac{2 y-18}{x+1} \\ g_2(x, y) &= \sqrt{25-(x-1)^2}+6 \end{align}$$

Starting at $x_0, y_0) = (3, 11)$, we have the iterates

$$\{\{3.00000000000000,11.0000000000000\},\{1.00000000000000,10.58257569495584\},\{1.58257569495584,11.00000000000000000000000000000\},\{1.54884133998961,10.965944578793316\},\{1.54261824614107,10.969786029953242\},\{1.54941547591172,10.970469337895043\},\{1.54582048827515,10.969722591335325\},\{1.54741671724848,10.970118710310557\},\{1.546758091811950,10.969943152358677\},\{1.547020236191428,10.9700156527960911\},\{1.546917943409719,10.9699868069439655\},\{1.546957421256077,10.9699980647055068\}$$

For the second point (this one has a very narrow area of convergence)

$$\begin{align} g_1(x, y) &= -\sqrt{2 y-17.75}-\dfrac{1}{2} \\ g_2(x, y) &= \sqrt{25-(x-1)^2}+6 \end{align}$$

Starting at $x_0, y_0) = (-3, 12)$, we have the iterates

$$\{\{-3.00000000000000,12.0000000000000\},\{-3.,9.00000000000000\},\{-1.,9.\},\{-1.,10.5826\},\{-2.34801,10.5826\},\{-2.34801,9.7136\},\{-1.79507,9.7136\},\{-1.79507,10.1458\},\{-2.09424,10.1458\},\{-2.09424,9.92756\},\{-1.9509,9.92756\},\{-1.9509,10.0364\},\{-2.02405,10.0364\},\{-2.02405,9.98185\},\{-1.98785,9.98185\},\{-1.98785,10.0091\},\{-2.00604,10.0091\},\{-2.00604,9.99546\},\{-1.99697,9.99546\},\{-1.99697,10.0023\},\{-2.00151,10.0023\},\{-2.00151,9.99887\},\{-1.99924,9.99887\},\{-1.99924,10.0006\},\{-2.00038,10.0006\},\{-2.00038,9.99972\},\{-1.99981,9.99972\},\{-1.99981,10.0001\},\{-2.00009,10.0001\},\{-2.00009,9.99993\},\{-1.99995,9.99993\},\{-1.99995,10.\},\{-2.00002,10.\},\{-2.00002,9.99998\},\{-1.99999,9.99998\},\{-1.99999,10.\},\{-2.00001,10.\},\{-2.00001,10.\},\{-2.,10.\}$$

You can verify the $l_\infty$ norm statement.