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The Newton-Raphson method is an iterative method in the sense that it generates a sequence of successive approximations by repeating, or iterating, the same formula. However, the term iterative method as commonly used refers to a particular class of algorithms which might more descriptively be called implicit iterative methods. Such algorithms occur in many numerical contexts as we’ll see in subsequent sections of this course. At this point, we apply the approach to the system of simultaneous nonlinear equations.
a. General form
Let  be the solution matrix to the equation  . I.e.,  . Now, solve algebraically each  for xi. This creates a new set of equations,  , where  refers to the set of unknowns {xj} excluding xi. Algebraically, this looks funny, because each unknown is expressed in terms of all the other unknowns, hence the term implicit. Of course, what we really mean is
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