Dan Bron wrote:
> What I don't buy is this philosophy that it's better to have no inverse
> than an imperfect inverse. We've had this discussion numerous times. No
> one complains about (e.g.) ;:^:_1 even though ;: is not 1-to-1.
>
Once again, the domain has been restricted in a useful way.
> If we restrict the range of p.^:_1 it may not systematic useful, but an
> error is systematically useless. Further, the cost is not so great: the
> change (like all new inverses) is completely backwards compatible.
I am generally in support of Dan's suggestions for new inverses (for
example, for I. ) but I disagree with the proposal for p. .
You can certainly define any function f:R->R to have inverse g defined
by g(y)=min{x : f(x)=y}. If f is a polynomial, we can effectively
calculate g(y).
However, this is going to give arbitrary discontinuities.
require 'plot'
x=:steps _3 3 100
c=:0 _3 0 1
f=:c&p.
isreal=:1e_6 > |@{:@+.
g=:3 : 0"0
roots=.1&{:: p. c-y*0=i.#c
[EMAIL PROTECTED] <./ (#~ isreal) roots
)
plot x;f x
plot g;g x
For functions which are not globally invertible but may be locally
invertible, one can turn to the inverse and implicit function theorems
at a point. You can have a map of part of the Earth's surface that is
accurate in a small area, you can have a map with almost all the
Earth's surface with arbitrarily inaccurate distances, but you can't
have a map of all of the Earth's surface.
Best wishes,
John
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