[R] Solving Systems of Non-linear equations

[Gabor Grothendieck]

Sorry I seemed to have messed up the copying and pasting.
Here it is again.

---

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 simple
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.

This is what you enter:

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

eliminate b
a
simplify
code c

eliminate a
b
simplify
code c

and this is the entire session:


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-> code c
a = ((((mean * mean) - pow(mean, 3.0)) / variance) - mean);

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));




On 11/30/05, Gabor Grothendieck <ggrothendieck at gmail.com> wrote:
> 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
> >
> > ______________________________________________
> > R-help at stat.math.ethz.ch mailing list
> > https://stat.ethz.ch/mailman/listinfo/r-help
> > PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html
> >
>