[R] Solving Systems of Non-linear equations

[Gabor Grothendieck]

Go to http://mathomatic.orgserve.de/math/ and install mathomatic
(its free) or just connect to the online server and do this.

The C output, i.e the result of the two code c commands,
can be used verbatim in R.

Note that mathomatic does not support logs but for simply
problems like this its very useful.

Note that 1-> and 2-> are the mathomatic prompts and what
comes after them are what I typed in.  The entry goes into
the corresponding equation space, i.e. equation 1 or equation 2.

1-> mean = a/(a+b)

              a
#1: mean = -------
           (a + b)

1-> variance = (a*b)/(((a+b)^2) * (a+b+1))

                          a*b
#2: variance = -------------------------
               (((a + b)^2)*(a + b + 1))

2-> eliminate b
Solving equation #1 for (b)...

                                        1
                                (a^2)*(---- - 1)
                                       mean
#2: variance = ---------------------------------------------------
                           1                       1
               (((a + (a*(---- - 1)))^2)*(a + (a*(---- - 1)) + 1))
                          mean                    mean

2-> a

              mean*(1 - mean)
#2: a = mean*(--------------- - 1)
                 variance

2-> simplify

        ((mean^2) - (mean^3))
#2: a = --------------------- - mean
              variance

2-> eliminate a
Solving equation #1 for (a)...

      b*mean     ((mean^2) - (mean^3))
#2: ---------- = --------------------- - mean
    (1 - mean)         variance

2-> b

         mean*(1 - mean)
#2: b = (--------------- - 1)*(1 - mean)
            variance
2-> simplify

             ((mean^2) - mean)
#2: b = (1 + -----------------)*(mean - 1)
                 variance


2-> code c
b = ((1.0 + (((mean * mean) - mean) / variance)) * (mean - 1.0));

2-> #1

          b*mean
#1: a = ----------
        (1 - mean)

1-> code c
a = (b * mean / (1.0 - mean));



On 11/30/05, Scott Story <sstory at montana.edu> wrote:
> I am trying to write a function that will solve a simple system of
> nonlinear equations for the parameters that describe the beta
> distribution (a,b) given the mean and variance.
>
>
> mean = a/(a+b)
> variance = (a*b)/(((a+b)^2) * (a+b+1))
>
> Any help as to where to start would be welcome.
>
>
>
> --
> Scott Story
> Graduate Student
> MSU Ecology Department
> 319 Lewis Hall
> Bozeman, Mt 59717
> 406.994.2670
> sstory at montana.edu
>
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