Analysis

Newton-Iteration

Newton-Iteration is an approximation method for the calculation of a zero of f(x). If an initial value x0 is entered, which is close enough to the desired zero, then the next approximation calculated is the intersection of the tangent to the graph of f in the point P(x0|f(x0)) with the x-axis.

This leads to the recursion formula

    xn+1  =  xn  -  f(xn) / f'(xn)

The procedure converges, if   f(x0) · f"(x0) > 0   is valid for x0.

Example:

  f(x) = x-cos(x)

       x              f(x)           f´(x)
----------------------------------------------
 x0  = 1              
 x1  = 0,75036387     0,45969769     1,841471
 x2  = 0,73911289     0,018923074    1,681905
 x3  = 0,73908513     0,00004646     1,6736325
 x4  = 0,73908513     0,00000000     1,673612
 x5  = 0,73908513     0              1,673612
 x6  = 0,73908513     0              1,673612

See also:

Supported Functions
Wikipedia: Newton's method
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