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D.7.2.3 ures_solve

Procedure from library solve.lib (see solve_lib).

Usage:
ures_solve(i [, k, p] ); i = ideal, k, p = integers
k=0: use sparse resultant matrix of Gelfand, Kapranov and Zelevinsky,
k=1: use resultant matrix of Macaulay which works only for homogeneous ideals,
p>0: defines precision of the long floats for internal computation if the basering is not complex (in decimal digits),
(default: k=0, p=30)

Assume:
i is a zerodimensional ideal given by a quadratic system, that is,
nvars(basering) = ncols(i) = number of vars actually occurring in i,

Return:
If the ground field is the field of complex numbers: list of numbers (the complex roots of the polynomial system i=0).
Otherwise: ring R with the same number of variables but with complex coefficients (and precision p). R comes with a list SOL of numbers, in which complex roots of the polynomial system i are stored:

Example:
 
LIB "solve.lib";
// compute the intersection points of two curves
ring rsq = 0,(x,y),lp;
ideal gls=  x2 + y2 - 10, x2 + xy + 2y2 - 16;
def R=ures_solve(gls,0,16);
==> 
==> // 'ures_solve' created a ring, in which a list SOL of numbers (the compl\
   ex
==> // solutions) is stored.
==> // To access the list of complex solutions, type (if the name R was assig\
   ned
==> // to the return value):
==>         setring R; SOL; 
setring R; SOL;
==> [1]:
==>    [1]:
==>       -2.82842712474619
==>    [2]:
==>       -1.414213562373095
==> [2]:
==>    [1]:
==>       -1
==>    [2]:
==>       3
==> [3]:
==>    [1]:
==>       1
==>    [2]:
==>       -3
==> [4]:
==>    [1]:
==>       2.82842712474619
==>    [2]:
==>       1.414213562373095
// result is a list (x,y)-coordinates as strings
// now with complex coefficient field, precision is 20 digits
ring rsc= (real,20,I),(x,y),lp;
ideal i = (2+3*I)*x2 + (0.35+I*45.0e-2)*y2 - 8, x2 + xy + (42.7)*y2;
list l= ures_solve(i,0,10);
// result is a list of (x,y)-coordinates of complex numbers
l;
==> [1]:
==>    [1]:
==>       (-1.31584587549391830705+I*0.70396753310002259573)
==>    [2]:
==>       (-0.092006639590217681983-I*0.20902112035965287775)
==> [2]:
==>    [1]:
==>       (-1.315392899374542198+I*0.70468233142752928117)
==>    [2]:
==>       (0.12292646536251281054+I*0.19245727404407015049)
==> [3]:
==>    [1]:
==>       (1.315392899374542198-I*0.70468233142752928117)
==>    [2]:
==>       (-0.12292646536251281054-I*0.19245727404407015049)
==> [4]:
==>    [1]:
==>       (1.31584587549391830705-I*0.70396753310002259573)
==>    [2]:
==>       (0.092006639590217681983+I*0.20902112035965287775)
// check the result
subst(subst(i[1],x,l[1][1]),y,l[1][2]);
==> 0


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