[R] Solving Systems of Non-linear equations

[Gabor Grothendieck]

Just one addition to this.  I noticed that its not really true that
the output can be used in R verbatim since the C output uses
pow instead of ^; however, if one replaces the "code c" statement
with the statement "list export" then it is valid R.  That is the input
to mathomatic should be:

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

eliminate b
a
simplify
list export

eliminate a
b
simplify
list export


On 11/30/05, Gabor Grothendieck <ggrothendieck at gmail.com> wrote:
> 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
> > >
> >
>