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Thread: Finding square root. Newton's method

  1. #1 Finding square root. Newton's method 
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    Hi The Science Forum members!

    Could somebody tell me how Newton came to an idea for:

    New (x) = 0.5 * (x + b/x)

    where b is the number that we are looking square root for?

    I understand (x^2+b)/2 but (x^2+b)/2x do not understand.

    Please help.


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  3. #2 Re: Finding square root. Newton's method 
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    Quote Originally Posted by scientist91
    Hi The Science Forum members!

    Could somebody tell me how Newton came to an idea for:

    New (x) = 0.5 * (x + b/x)

    where b is the number that we are looking square root for?

    I understand (x^2+b)/2 but (x^2+b)/2x do not understand.

    Please help.
    Using use notation (replace new(x) by y for notation simplification)

    y=x-(b-x^2)/2x = (b+x^2)/2x = (b/x +x)/2.

    The first "equality" is obtained from the first two terms of the Taylor series for
    f(u) around u0, where u corresponds to b and u0 corresponds to x^2, with f(u) being the square root.


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  4. #3  
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    Newton often thought geometrically - geometric reasoning, involving pictures and triangles and straightedge constructions was a much more important part of a mathematical education then than now.

    So if you want to follow Newton's probable approach, draw a picture of the function and make lines and intersections with a ruler as you proceed with the iteration. Then you can see how the tangent line via the derivative hands you the b/x, how x and b/x bracket the real root, , and how averaging them tightens the bracket at each step (with a suitable choice of x and function).

    With a little thought, you don't need the tangent (or the derivative, or any calculus): if x is less than the square root, b/x will be greater, and the average brings you in. If x is greater, b/x will be smaller, once again a bracket and tightening.
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