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Unformatted text preview: The Newton Method James Keesling 1 The Newton Formula Suppose that we are trying to find a solution to the equation f ( x ) = 0 and we have an approximation, x , near a solution, z . The Newton Method finds a new approximation by following the tangent line from f ( x ) to where it hits the x- axis. The new approximation is given by x 1 = x- f ( x ) f ( x ) . Figure 1: Geometry of the Newton Method 2 Fixed Point Iteration Consider the following equation. 1 h ( x ) = x To solve the equation we are looking for a fixed point of h . In addition suppose that h is a continuous function. Let x be an arbitrary point and suppose that we define x n +1 = h ( x n ) for n = 0 , 1 , 2 ,... . If x n so defined converges to z , then, by the continuity of h , h ( z ) = z . Hence the limit point z is a fixed point of h and thus a solution of our equation. The above method of finding a fixed point is called Fixed Point Iteration . When the iteration converges, we have located a fixed point for the function h . Of course, the iteration may not converge. We need a criterion that will guarantee convergence. We will use the Mean Value Theorem to come up with such a criterion. Theorem 2.1 (Mean Value Theorem) . Let f be a differential equation on the interval [ a,b ] . Then there is a c with a < c < b such that f ( c ) = f ( b )- f ( a ) b- a . Now suppose that h has a fixed point z and that | h ( x ) | < 1 for all x in the interval [ z- ,z + ] for some > 0. Consider any x 6 = z in the interval [ z- ,z + ]. The Mean Value Theorem guarantees that there will be a point c between x and z such that h ( c ) = h ( x )- h ( z ) x- z = h ( x )- z x- z . Since | h ( c ) | < 1, we must have that | h ( x )- z | < | x- z | . Thus h ( x ) is closer to z than x . Under these circumstances, this guarantees that Fixed Point Iteration will converge to z for any starting point x [ z- ,z + ]. On the other hand, suppose that | h ( x ) | > 1 for all x [ z- ,z + ]. Then if x 6 = z in [ z- ,z + ], then h ( x ) is further away from z than x ....
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