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see Newton's Method As An Optimization Algorithm . DESCRIPTION OF THE METHOD The idea of the method is as follows: one starts with a value which is reasonably close to the true zero, then replaces the function by its Tangent (which can be computed using the tools of Calculus ) and computes the zero of this tangent (which is easily done with elementary algebra). This zero of the tangent will typically be a better approximation to the function's zero, and the method can be Iterated . Suppose ''f'' : ''b'' -> R is a Differentiable function defined on the Interval ''b'' with values in the Real Numbers R. We start with an arbitrary value ''x''0 (the closer to the zero the better) and then define for each Natural Number ''n'': : Here, ''f'' ' denotes the Derivative of the function ''f''. Newton's method will usually converge provided the initial guess is close enough to the unknown zero. Furthermore, for a zero of Multiplicity 1, the Convergence Is Quadratic , which intuitively means that the number of correct digits roughly doubles in every step. More details can be found in the "Analysis" Section . ::
EXAMPLE Consider the problem of finding the positive number ''x'' with cos(''x'') = ''x''3. We can rephrase that as finding the zero of ''f''(''x'') = cos(''x'') - ''x''3. We have ''f'' '(''x'') = -sin(''x'') - 3''x''2. Since cos(''x'') ≤ 1 for all ''x'' and ''x''3 > 1 for ''x''>1, we know that our zero lies between 0 and 1. We try a starting value of ''x''0 = 0.5. : The correct digits are underlined in the above example. In particular, all digits of ''x''6 are correct. We see that the number of correct digits after the decimal point increases from 2 (for ''x''3) to 5 and 10, illustrating the quadratic convergence. Consider the following example in pseudocode. function newtonIterationFunction(x) {
} var x := 0.5 for i '''from''' 0 '''to''' 99 { print "Iteration: " + i print "Guess: " + x xold := x x := newtonIterationFunction(x) if x = xold { print "Solution found!" break } } Here is the same using a calculator. HISTORY Newton's method was described by . The essence of Viète's method can be found in the work of the Persian Mathematician Sharaf Al-Din Al-Tusi . Heron Of Alexandria used essentially the same method in book 1, chapter 8, of his ''Metrica'' to determine the square root of 720. Newton's method was first published in 1685 in ''A Treatise of Algebra both Historical and Practical'' by John Wallis . In 1690 , Joseph Raphson published a simplified description in '' Analysis Aequationum Universalis ''. Raphson again viewed Newton's method purely as an algebraic method and restricted its use to polynomials, but he describes the method in terms of the successive approximations ''x''''n'' instead of the more complicated sequence of polynomials used by Newton. Finally, in 1740 , Thomas Simpson described Newton's method as an iterative method for solving general nonlinear equations using fluxional calculus, essentially giving the description above. In the same publication, Simpson also gives the generalization to systems of two equations and notes that Newton's method can be used for solving optimization problems by setting the gradient to zero. PRACTICAL CONSIDERATIONS In general the of a line through two points on the function, the Secant Method results; this can be more efficient depending on how one measures computational effort.) Second, if the initial value is too far from the true zero, Newton's method can fail to converge. Because of this, most practical implementations of Newton's method put an upper limit on the number of iterations and perhaps on the size of the iterates. Third, if the Root being sought has Multiplicity greater than one, the convergence rate is merely linear (errors reduced by a constant factor at each step) unless special steps are taken. ANALYSIS Suppose that the function ''f'' has a zero at α, i.e., ''f''(α) = 0. If ''f'' is continuously differentiable and its derivative does not vanish at α, then there exists a Neighborhood of α such that for all starting values ''x''0 in that neighborhood, the Sequence {''x''''n''} will Converge quadratically towards α. If the derivative does vanish at α, then the convergence is usually only linear. Specifically, if ''f'' is twice continuously differentiable, ''f'' GENERALIZATIONS One may use Newton's method also to solve systems of ''k'' (non-linear) equations, which amounts to finding the zeros of continuously differentiable functions ''F'' : R''k'' -> R''k''. In the formulation given above, one then has to multiply with the inverse of the ''k''-by-''k'' Jacobian Matrix ''J''''F''(''x''''n'') instead of dividing by ''f'' '(''x''''n''). Rather than actually computing the inverse of this matrix, one can save time by solving the System Of Linear Equations : for the unknown ''x''''n''+1 - ''x''''n''. Again, this method only works if the initial value ''x''''0'' is close enough to the true zero. Typically, a region which is Well-behaved is located first with some other method and Newton's method is then used to "polish" a root which is already known approximately. Complex functions See Also: Newton fractal When dealing with Complex Functions , however, Newton's method can be directly applied to find their zeros. For many complex functions, the boundary of the set (also known as the Basin Of Attraction ) of all starting values that cause the method to converge to the true zero is a Fractal . REFERENCES
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