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Root of Non-Linear Eqn (Numerical Method)

  1. Dec 22, 2008 #1

    I have a non-linear function [tex]F: \Re^{3}\rightarrow\Re[/tex]. I would like to find the roots of this equation numerically, since an explicit formula cannot be derived.

    As far as I am aware Newton's method can only be utilized when the domain and the range of the function are of the same degree. (i.e. [tex]F: \Re^{n}\rightarrow\Re^{n}[/tex])

    Is there a method that can used for the case[tex]F: \Re^{n}\rightarrow\Re^{m}[/tex] with [tex]m \neq n[/tex]?

    Thanks in advance,
  2. jcsd
  3. Dec 23, 2008 #2


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    Science Advisor

    So you have something like f(x,y,z)= 0 to solve? I see no reason why Newton's method could not be generalized to this. Choose some starting point [itex](x_0, y_0, z_0). The tangent (hyper-)plane to u= f(x,y,z) at that point is [itex]u= \nabla f(x_0,y_0,z_0)\cdot(x- x_0,y-y_0,z- z_0)+ f(x_0, y_0, z_0)[/itex]. Set that equal to 0 and solve for (x, y, z) as the next approximation.
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