# Re: [R] Solving Systems of Non-linear equations

From: Gabor Grothendieck <ggrothendieck_at_gmail.com>
Date: Thu 01 Dec 2005 - 03:37:42 EST

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@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
> MSU Ecology Department
> 319 Lewis Hall
> Bozeman, Mt 59717
> 406.994.2670
> sstory@montana.edu
>
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