I have a computer that I don?t use much working on the difference. I'll let
you know when it comes up with the answer. I could take a long time, maybe
a day or more. FWIW.
Rich
-----Original Message-----
From: Richard Hennessy
Sent: Saturday, January 05, 2013 3:09 PM
To: Raymond Toy ; maxima at math.utexas.edu
Cc: Aleksas Domarkas
Subject: Re: [Maxima] Find roots
Ray,
One of the roots is approximately 8.185668853475937, the value of
exp(-1000*x) at that point is 1.0213774096019683602413700960[611
digits]449347701342852478546561b-3555 so the difference in this root of f
and the cubic are, at this point, probably in the 3556th digit. Maybe it's
not worth the effort to try to compute that many digits....
Rich
-----Original Message-----
From: Richard Hennessy
Sent: Saturday, January 05, 2013 2:36 PM
To: Raymond Toy ; maxima at math.utexas.edu
Cc: Aleksas Domarkas
Subject: Re: [Maxima] Find roots
>>>>> "Aleksas" == Aleksas Domarkas <aleksasd873 at gmail.com> writes:
Aleksas> see
http://www.math.utexas.edu/pipermail/maxima/2013/031439.html
Aleksas> Find roots of f=log(x^3-x*78+90)+1000*x
Aleksas> (%i1) f:log(x^3-x*78+90)+1000*x$
Aleksas> (%i2) allroots(x^3-x*78+90); map(rhs,%)$
Aleksas> (%o2)
[x=1.174624074847539,x=8.185668853475939,x=-9.360292928323476]
Aleksas> (%i4) [x1,x2,x3]:sort(%);
Aleksas> (%o4) [-9.360292928323476,1.174624074847539,8.185668853475939]
Aleksas> x^3-x*78+90<0 if x in (-infinity, x1)U (x2, x3). Then
Aleksas> function f is defined for all x in (x1, x2)U(x3, infinity)
Aleksas> (%i5) limit(f,x,x1,plus);
Aleksas> (%o5) -inf
Aleksas> (%i6) limit(f,x,x2,minus);
Aleksas> (%o6) 1140.659863000102
Aleksas> (%i7) limit(f,x,x3,plus);
Aleksas> (%o7) 8155.863524711861
What is the intent of computing the limits above with approximate
roots of the cubic?
--> wxplot2d([f], [x,-20,20])$
Aleksas> Function f has only one root:
"No, the function has 3 roots. You've found the negative root.
The other two roots, which don't show up in the graph, are very close
the to positive roots of the cubic. As mentioned before, this is
easier to see if you transform log(x^3-78*x+90)+1000*x = 0 to
(x^3-78*x+90)=exp(-1000*x). Since exp(-1000*x2) and exp(-1000*x3) are
zero (for floats), x2 and x3 are very close to the roots."
I agree with Ray on this. The function f definitely has three real roots.
But two of them are very close to causing the error "log(0) encountered"
which means the function has infinities in very close proximity to the
roots. I guess they are called poles. Anyway, I am not sure if there is an
easy way in Maxima to find the roots since find_root() fails on complex
values. Finding the three roots to high precision is hard. Ray has found
two of them to low precision by stating they are more or less the same as
the roots of the cubic, which is true but the roots of f differ slightly
from the roots of the cubic, but not by very much. The challenge is maybe
to figure out the difference between the cubic's roots and f's roots.
Rich
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