[Maxima-discuss] putting a symbol into a numeric formula, a use for a NaN?

[Richard F. <fa...@gm...>]

In the context of taking a numerical program and reusing it
with partial symbolic input, here's another example..

Here is one step of a newton iteration:  x[n+1] = x[n] - f(x[n])/df(x[n])...

newtstep(f,x,val):= block([d:diff(f,x)], val-  subst(val, x,f/d));

how does this work?  val is a guess... here, a guess the root of x^2-9...

newtstep(x^2-9,x, 5.0);  --> returns 3.4
newtstep(x^2-9,x, 3.4);  --> returns 3.023529411764706
 newtstep(x^2-9,x, 3.023529411764706); --> returns 3.00009155413138
   you get the idea.

Now try putting an expression in there for the guess.. [ could use a NaN
perhaps]
newtstep(x^2-9,x,3-eps)$ /* could put 5-eps or anything else...
  newtstep(x^2-9,x,%)$
  newtstep(x^2-9,x,%)$    -->

  -((eps^8-24*eps^7+504*eps^6-6048*eps^5+45360*eps^4-217728*eps^3+653184*eps^2-1119744*eps+839808)/(8*eps^7-168*eps^6+2016*eps^5-15120*eps^4+72576*eps^3-217728*eps^2+373248*eps-279936))

whew!   try to understand this by using Taylor expansion..

  taylor(%,eps,0, 9) -->  3+eps^8/279936+eps^9/209952

The newtstep program could instead be some large mysterious
 iterative numerical program in a language that normally does not
allow symbols. But we might fool it by using NaNs as substitutes for
symbolic
expressions.
If you had this all hooked up via trap handling
you might get some insight as to how a program works..

RJF