How to solve a really nasty system of nonlinear equations?

Looking to solve a system of nonlinear equations using the newton-raphson method.?

• The two equations are as follows: f(x) = x^2 + 3xy - 5 g(x) = 3x - y + 3 There are two roots and must show the work necessary to derive the iteration equations. Basically just need to get the equations in a form that can be evaluated and then a good initial guess for the determination of the roots (x_1 , y_1) and (x_2 , y_2).

• Answer:

  Since you looking for points where y = 3x + 3, can't you just seek the roots of f1(x) = x^2 - 3x(3x+3) - 5 ? Thus, f1(x) = -2x^2 - 9x - 5. I can make a really good guess at the roots by observing that the quadratic formula gives (-18 +/- sqrt(81-40))/2 and just guess that the square root of 41 is about 6.4 (because 64*64=4096). Hence x_1 = -12.2, y_1 = -33.6 and the convergence should be pretty rapid since f1'(-12.2) is large, around -200 Obviously if you accept this way of doing it, x_2 is -5.8, etc.