Principles of Linear Algebra With Maple The Newton–Raphson ...

26 Chapter 2. Newton-Raphson Method for Square SystemsThe real solutions are the real intersection points of these two curves.> with(linalg): with(plots):> f:= (x,y) -> 3*xˆ2*y-yˆ3+5*x-8:> g:= (x,y) -> 3*x*yˆ2-xˆ3-4*y+2:> plotf:= implicitplot(f(x,y) = 0, x = -10..10, y = -10..10, color = blue, grid= [50,50]):> plotg:= implicitplot(g(x,y) = 0, x = -10..10, y = -10..10, color = red, grid= [50,50]):> display({plotf, plotg});10y5–10 –5 5 10x–5–10Figure 2.3: Intersection of the level curves f(x,y) = 0 and g(x,y) = 0It is clear from Figure 2.3 that this system of equations has exactly threereal solutions which are the intersection points of these two curves. How manytotal real and complex solutions are there to this system?> F:= (x,y) -> vector([f(x,y), g(x,y)]):> F(-2.+I, 5-3*I);[ 1.+116.I −22.+229.I]> jacobian(F(x,y), [x,y]);[6xy +5 3x 2 −3y 2]3y 2 −3x 26xy −4>G:=unapply(convert(simplify(evalm(vector([x,y])-jacobian(F(x,y),[x,y])ˆ(-