[R] Two or more dimensional root (Zero) finding

[Ben Bolker]

Ravi Varadhan wrote:

> That is not a good approach, i.e. finding the zero, x*, of F(x), such that
> F(x*) = 0, as a minimum of ||F(x)|| is NOT a good approach.  Any root of
> F(x) is indeed a global minimum of ||F(x)||, or for that matter, the global
> minimum of any f(F(x)), where f(.) is a mapping from R^p to R, such that it
> has a uniques global minimizer x=0.  However, the converse does not
> generally hold, i.e. a (local) minimizer of f(F(x)) is not necessarily a
> root of F(x).  See Ortega and Rheinboldt (p. 97, 1970) for theorem on this.
>
> There are better approaches that directly solve the non-linear system (e.g.
> Newton's method and spectral appproaches).  There are 2 packages in R that
> are quite useful for finding roots of nonlinear systems of equations:  "BB"
> and "nleqslv".  For more information, You can try, for example:
> 
> 	library(BB)
> 	?dfsane
> 
> Hope this helps,
> Ravi.

  Thanks for the correction!  I should also clarify that I misremembered
what Numerical Recipes said -- it explains (as you did) why collapsing
does _not_ work well.

  cheers
     Ben


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