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Variations of Newtons method

  1. Jun 11, 2010 #1
    Are there any variations of newtons method, say where you use higher derivatives?
     
  2. jcsd
  3. Jun 11, 2010 #2

    HallsofIvy

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    I don't know of any that are commonly used but it's not hard to extend the basic idea.

    In "Newton' method", we approximate the function by its tangent line at a given point, and determine where that line crosses y= 0.

    We could, as well, approximate the function by a parabola that (1) passes through the given point, (2) has the same slope their, and (3) has the same second derivative.

    That is, to solve f(x)= 0, start with some given point [itex](x_0, f(x_0))[/itex], at which f(x) has derivative [itex]f'(x_0)[/itex] and second derivative [itex]f''(x_0)[/itex].

    We can write a parabola through [itex](x_0, f(x_0))[/itex] as [itex]y= a(x- x_0)^2+ b(x- x_0)+ f(x_0)[/itex]. That has derivative, at [itex]x= x_0[/itex], [itex]y'= b[/itex] and second dervative [itex]y''= 2a[/itex] so that the "approximating parabola" is [itex]f''(x_0)(x- x_0)^2/2+ f'(x_0)(x- x_0)+ f(x_0)[/itex]. Set that equal to 0 and solve for x to get the next x at which to approximate.

    But the fact is that there are just some "approximating" methods that are just so good, any improvement by a "better" approximation just isn't worth the additional work. Newton's method is one of those!
     
  4. Jun 11, 2010 #3
    As usually presented Newton's method is intended for a single unknown variable.
    Hamming gives an improved method, sensitive to the difference between the current approximation and the last.

    There is an equivalent method when you have a system of equations usng the Jacobian in place of the first derivative.
     
    Last edited: Jun 11, 2010
  5. Jun 11, 2010 #4
    Isnt this method the basic idea behind Taylor Polynomials?
     
  6. Jun 11, 2010 #5

    hotvette

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    Yep, Halley's method uses 2nd derivatives in addition to first derivatives. If Newton's method converges quickly, Halley's method is turbo charged.

    http://en.wikipedia.org/wiki/Halley's_method
     
  7. Jun 13, 2010 #6
    The following method is often used when the evaluating the function is very time consuming. You first find a few starting values. E.g. if y(x) is your finction, you find two points x1 and x2 such that y changes siign and then you do, say, two bisection steps, ending up with four points. Then you do an interpolation using the four points, but not of y asa function of x, but instead of x as a function of y. In that interpolating polynoimial, x(y) you simply put y equal to zero and you obtain a very accurate estimate of the zero of y.

    You then throw away the first point and construct a new interpolating polynomial using the last four points, insert y = 0 in there and then iterate this process.
     
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