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

Chapter 2The Newton-RaphsonMethod for Square Systemsof Equations2.1 Newton-Raphson for TwoEquations in TwoUnknownsIn this section we will discuss the Newton-Raphson method for solving square(as many equations as variables) systems of two non-linear equationsf(x,y) = 0, g(x,y) = 0in the two variables x and y. In order to do this we must combine these twoequations into a single equation of the form F(x,y) = 0 where F must give usa two component column vector and 0 is also the two component zero columnvector, that is,[ 0and 0 =0F(x,y) =[ f(x,y)g(x,y)]. Then clearly the equation F(x,y) = 0 is the same as](2.1)[ f(x,y)g(x,y)] [ 0=0](2.2)which is equivalent to the system of two equations f(x,y) = 0 and g(x,y) = 0.Now let F : R 2 → R 2 be a continuous function which has continuous firstpartial derivatives where F is defined as in equation (2.1) for variables x, y andcomponent functions f(x,y), g(x,y). We wish to solve the equation F(x,y) =19