#678 nroots(sqrt(3),minf,inf) & ....

[Barton Willis]

nroots claims that 1 isn't a univariate polynomial,
and that sqrt(3) is. This is chaotic. Further,

nroots(%pi,minf,inf) --> 1,
nroots(sqrt(3),minf,inf) --> 1.

These are both wrong. It's not clear from the
documentation, but it seems that nroots does
not allow algebraic coefficients; if that's the
case, the documentation should say so. Finally,
nroots counts multiplicities, but the user
documentation doesn't say that it does.

(%i1) nroots(1,minf,inf);
Argument must be a univariate polynomial
-- an error. Quitting. To debug this try DEBUGMODE
(TRUE);
(%i2) nroots(sqrt(3),minf,inf);
(%o2) 1
(%i3) nroots(%pi,minf,inf);
(%o3) 1
(%i4) nroots(sqrt(3)*x+sqrt(7));
Argument must be a univariate polynomial
-- an error. Quitting. To debug this try DEBUGMODE
(TRUE);

(%i14) nroots(x^2,minf,inf);
(%o14) 2

One more comment about the documentation--it
doesn't mention what happends when low > high.

(%i18) nroots(x^2,1,-1);
(%o18) - 2

Maxima version: 5.9.1
Maxima build date: 7:34 9/24/2004
host type: i686-pc-mingw32
lisp-implementation-type: Kyoto Common Lisp
lisp-implementation-version: GCL 2.6.5

Barton

[Stavros Macrakis]

Logged In: YES
user_id=588346

Agreed that the univariate polynomial stuff is random.

As for excluding algebraic coefficients.... I agree that
this should be documented. The method (Sturm sequences) is
exact for exact evaluation of polynomials. Exact evaluation
is easy for rational coefficients. The result is
approximate for approximate coefficients (floats), which is
also fine. However, though algebraic numbers are exact,
calculating with them exactly is a problem. If this matters
to you, I suppose you can use increasingly precise intervals
until the signs are unambiguous (though checking for 0 is
harder...), but Maxima doesn't currently support interval
arithmetic.

Anyway, what I would suggest is not changing the
documentation, but instead converting algebraic numbers to
rationals (at what precision?) and giving the warning:

Warning: nroots will not always give exact results with
non-rational coefficients

Unfortunately, nroots does not currently support bfloats....