[R] solving system of nonlinear equations

[Spencer Graves]

      Have you considered "nls"?  If you read the help file and work 
through the examples, there is a good chance you can make it work, I 
think.  I think I would start trying "plinear" in "nls", parameterizing 
the problem in terms of alpha, beta, ln.sigma, and ln.tau, unless you 
think a solution might require sigma < 0 or tau < 0.  Using logarithms 
will get rid of the constraint and may make the problem numerically 
easier.  Using alpha and beta rather than lambda and delta transforms 
the problem into an ordinary least squares problem for alpha and beta 
given any two numbers for sigma and tau (or ln.sigma and ln.tau).  

      If I had trouble with this, I might try two other things: 
     
      (a) The "solver" in Excel. 

      (b) I might generate a grid in ln.sigma and ln.tau using 
expand.grid.  For each combination of levels, I'd set up the linear 
regression problem and use "lm" to estimate alpha and beta and compute 
and store the sum of squares of residuals.  Then I'd use "contour" to 
visualize the sum of squares surface. 

      I've done all these things with crudely similar problems in the 
past and been happy with the results.  If I only had this one problem, 
I'd be surprised if it would require more than a few hours.  If I wanted 
a general algorithm for other purposes, I might do it two or three 
different ways both to help select a good algorithm and to build 
confidence in the results. 

      hope this helps. 
      spencer graves
p.s.  Some of these techniques are discussed in Venables and Ripley 
(2002) Modern Applied Statistics with S, 4th ed. (Springer).  If you 
don't have this, I'd encourage you to consider spending some time with it. 

Enayetur RAHEEM wrote:

>Hello there
>
>Can anybody please tell me if there is any package in R to solve the
>following 4 nonlinear equations with 4 unknowns:
>
>alpha*exp(20/sigma)+ beta*exp(21/tau) = 2
>alpha*exp(22/sigma)+ beta*exp(9/tau) = 4
>alpha*exp(10/sigma)+ beta*exp(30/tau) = 6
>alpha*exp(40/sigma)+ beta*exp(39/tau) = 5
>
>where 
>
>alpha = exp(lambda/sigma)
>beta= exp(delta/tau)
>
>I need to estimate lambda, sigma, delta, tau
>
>Thanks.
>E Raheem
>
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-- 
Spencer Graves, PhD, Senior Development Engineer
O:  (408)938-4420;  mobile:  (408)655-4567