RE: FindUnknowns[ ] (Mathematica wish list)

• To: mathgroup at smc.vnet.net
• Subject: [mg71313] RE: [mg71286] FindUnknowns[ ] (Mathematica wish list)
• From: "David Park" <djmp at earthlink.net>
• Date: Tue, 14 Nov 2006 05:06:48 -0500 (EST)

```I don't know how 'solvable' your system is for all cases, especially since
you seem to be using FindRoot, which requires approximate guesses.

But yes, assuming your system is well behaved and solvable you should be
able to write a routine that will do what you want. You can't expect
Mathematica to automatically have routines for doing this because there are
too many particular applications. I am always claiming that Mathematica is
not a 'tool for doing mathematics' but instead a metatool for constructing
the tools you need in your particular field or application. Mathematica, of
necessity, has many gaps and it is worthwhile filling them.

For example, in the free ConicSections package at my web site I have a
routine for solving for the various parameters of an ellipse, given two of
them. The usage message is as follows:

Needs["ConicSections`ConicSections`"]

?SolveStandardEllipse

SolveStandardEllipse[unknowns, knowns, vars:{a,b,c,d,e}] will solve for a
list of unknows among the variables and then substitute the known variables
given by a list of rules.
a = semimajor axis,
b = semiminor axis,
c = center-to-focus distance,
d = center-to-directrix distance,
e = eccentricity.
Alternative variables may be specified by giving the optional last argument.

As an example...

SolveStandardEllipse[{b, c, d}, {a -> 3, e -> 1/2}]

{{d -> a/e, b -> Sqrt[a^2 - a^2*e^2], c -> a*e},
{a -> 3, b -> (3*Sqrt[3])/2, c -> 3/2, d -> 6,
e -> 1/2}}

The first list gives the unknowns in terms of the knows, and the second list
gives all the values.

So one can do things like that, but unfortunately one has to actually write
the routine. The good news is that after you build up the routines you have
the makings of a good package.

David Park

From: AES [mailto:siegman at stanford.edu]

You have a set of E not too complicated equations which
contain/connect/constrain V variables/symbols.  You want
to assign values to V - E of those variables/symbols and
then solve for the remaining E values  -- but, making
*different* choices of which are the V - E "input" variables
and which are the E "output" variables on different runs
of your numerical routine. (This is a common situation in
design calculations, where you want to vary the values
of different variables of your problem at different times,
and have the remaining values be calculated).

So, you just put all your equations into a list; assign
values to any V - E of the variables for a particular
run; and call

{InputList, OutputList} := FindUnknowns[eqns]

which is a sort of FindRoot on steroids.  That is, when
called, FindUnknowns will:

*  Go thru eqns and identify all the symbols

*  Identify all of those symbols that presently have
a value, and put them into InList (more for user
convenience than anything else)

*  Put the remaining symbols into OutList

*  Run FindRoot[eqns, OutList]

Mathematica *could* do *all* these tasks, totally
automatically, not so?  (At least, for not too
pathological situations.)

```

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