How to prove newton's method?

As above. Thanks

Answer 1

# x_(n+1) = x_n - f(x_n)/(f'(x_n)) #

Suppose we seek a solution to the equation:

# f(x) = 0 #

And that we have an initial estimate #x_0# of the solution. We seek a second (hopefully more accurate) solutions, #x_1#. If #f(x)# is a smooth well behaved continuous function, then it is reasonable to assume that the point where the tangent to the curve at the point #x_0# crosses the #x#-axis is a better approximation.

The tangent to the curve at he point #(x_0,f(x_0))# has slope #f'(x_0)#, thus the equation of the tangent line is given by:

# y - f(x_0) = f'(x_0)(x-x_0) #

The point #(x_1,0)# lies on this tangent, and so it satisfies:

# 0 - f(x_0) = f'(x_0)(x_1-x_0) #

And if we rearrange for #x_1#:

# :. f'(x_0)(x_1-x_0) = -f(x_0) #

# :. x_1-x_0 = -f(x_0)/(f'(x_0)) #

# :. x_1 = x_0 - f(x_0)/(f'(x_0)) #

Leading to the general Newton's Method iterative method:

# x_(n+1) = x_n - f(x_n)/(f'(x_n)) #

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Answer 2

Newton's method, also known as Newton-Raphson method, is a numerical method used to find the roots of a real-valued function. The method relies on iterative approximation to converge towards the roots of the function. Here's a brief outline of how Newton's method is proven:

  1. Understanding Newton's Method: Newton's method starts with an initial guess ( x_0 ) for the root of the function ( f(x) ). It then iteratively improves this guess using the formula: [ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} ] Where:

    • ( x_{n+1} ) is the next approximation of the root.
    • ( x_n ) is the current approximation of the root.
    • ( f(x_n) ) is the value of the function at ( x_n ).
    • ( f'(x_n) ) is the derivative of the function evaluated at ( x_n ).
  2. Proving Convergence: Newton's method is proven to converge under certain conditions. One key condition is that the initial guess ( x_0 ) must be sufficiently close to the actual root of the function. Additionally, the function ( f(x) ) must be continuously differentiable and the derivative ( f'(x) ) should not be zero near the root. Under these conditions, Newton's method is guaranteed to converge to a root.

  3. Using Taylor Series Expansion: The proof of Newton's method often involves using Taylor series expansion of the function ( f(x) ) around the root. By truncating the Taylor series to the first-order term, we can derive the formula for ( x_{n+1} ) as given above.

  4. Convergence Analysis: The proof also involves analyzing the convergence rate of Newton's method. It is typically quadratic, meaning that with each iteration, the number of correct digits roughly doubles. This rapid convergence is a key advantage of Newton's method over other root-finding algorithms.

In summary, the proof of Newton's method involves understanding its iterative formula, ensuring the conditions for convergence are met, utilizing Taylor series expansion, and analyzing its convergence rate.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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