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D.12.6.1 nrRootsProbab
Procedure from library rootsmr.lib (see rootsmr_lib).
- Return:
- int: the number of real roots of the ideal I by a probabilistic
algorithm
- Assume:
- If I is not a Groebner basis, then a Groebner basis will be computed
by using std. If I is already a Groebner basis (i.e. if
attrib(I,"isSB"); returns 1) then this Groebner basis will be
used, hence it must be one w.r.t. (any) global ordering. This may
be useful if the ideal is known to be a Groebner basis or if it
can be computed faster by a different method.
- Note:
- If n<10 is given, n is the number of digits being used for
constructing a random characteristic polynomial, a bigger n is
more safe but slower (default: n=5).
If printlevel>0 the number of complex solutions is displayed
(default: printlevel=0).
Example:
| | LIB "rootsmr.lib";
ring r = 0,(x,y,z),lp;
ideal i = (x-1)*(x-2),(y-1)^3*(x-y),(z-1)*(z-2)*(z-3)^2;
nrRootsProbab(i); //no of real roots (using internally std)
==> 9
i = groebner(i); //using the hilbert driven GB computation
int pr = printlevel;
printlevel = 2;
nrRootsProbab(i);
==> //ideal has 32 complex solutions, counted with multiplicity
==> *********************************************************************
==> * WARNING: This polynomial was obtained using pseudorandom numbers.*
==> * If you want to verify the result, please use the command *
==> * *
==> * verify(p,b,i) *
==> * *
==> * where p is the polynomial I returned, b is the monomial basis *
==> * used, and i the Groebner basis of the ideal *
==> *********************************************************************
==> 9
printlevel = pr;
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See also:
nrRootsDeterm;
nrroots;
randcharpoly;
solve.
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