D.7.4.2 triMNewton

Procedure from library ntsolve.lib (see ntsolve_lib).

Usage:

triMNewton(G,a[,ipar]); G,a= ideals, ipar=list/intvec

Assume:

G: g1,..,gn, a triangular system of n equations in n vars, i.e. gi=gi(var(n-i+1),..,var(n)),
a: ideal of numbers, coordinates of an approximation of a common zero of G to start with (with a[i] to be substituted in var(i)),
ipar: control integer vector (default: ipar = 100,10)

ipar[1]: max. number of iterations ipar[2]: accuracy (we have as norm |.| absolute value ): accept solution sol if |G(sol)| < |G(a)|*(0.1^ipar[2]).

Return:

an ideal, coordinates of a better approximation of a zero of G

Example:

LIB "ntsolve.lib";
ring r = (real,30),(z,y,x),(lp);
ideal i = x^2-1,y^2+x4-3,z2-y4+x-1;
ideal a = 2,3,4;
intvec e = 20,10;
ideal l = triMNewton(i,a,e);
l;
→ l[1]=-2.000000000042265738880279143423
→ l[2]=1.41421356237309504880168872421
→ l[3]=1