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The Newton-Raphson Method

Introduction The Newton-Raphson method, or Newton  Method,  is a powerful technique for solving equations numerically. Like so much of the differential calculus, it is based on the simple idea of linear approximation. The Newton Method, properly used, usually homes in on a root with devastating efficiency. The essential part of these notes is Section 2.1, where the basic formula is derived, Section  2.2,  where  the procedure  is interpreted geometrically, and—of course—Section  6, where the problems are. Peripheral but perhaps interesting is Section 3, where the birth of the Newton Method is described.

Using Linear Approximations  to  Solve Equa- tions Let f (x) be a well-behaved  function,  and  let  r  be a root  of the equation f (x) = 0. We start with  an  estimate  x0  of r.   From  x0, we produce  an improved—we  hope—estimate x1. From x1,  we produce  a  new estimate x2. From x2, we produce  a new estimate  x3. We go on until we are ‘close enough’ to r—or until it becomes clear that we are getting nowhere. The above general style of proceeding is called iterative. Of the many it- erative root-finding procedures, the Newton-Raphson method, with its com- bination  of simplicity  and  power,  is the  most widely used. Section 2.4 de- scribes another iterative root-finding procedure,  the Secant  Method. Comment. The initial  estimate  is sometimes  called  x1, but  most  mathe- maticians prefer to start counting  at 0. Sometimes the initial  estimate is called a “guess.”  The Newton Method is usually very  very  good  if x0  is  close  to  r,  and  can The “guess” x0 should be chosen with care.

The Newton-Raphson Iteration Let x0 be a good estimate of r and let r = x0 + h.  Since the true root is r, and h = r − x0, the number  h measures  how far the estimate x0 is from the truth. Since h is ‘small,’ we can use the linear (tangent line) approximation to conclude that