2 x1+x2**3; x1**2*x2-x2**4; TITLE : decker2, with three regular root and one quadruple root at (0,0) For the quadruple root at the origin, three deflations are needed. Moreover, instead of the default tolerance of 1.0E-6 to decide the numerical rank, 1.0E-4 should be taken, as the initial roots in the list below are all quite far from (0,0). REFERENCES: D.W. Decker, H.B. Keller, C.T. Kelley: "Convergence rates for Newton's method at singular points." SIAM J. Numer. Anal. 20, 296-314, 1983 THE SOLUTIONS : 7 2 =========================================================== solution 1 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x2 : 1.73208017041855E-06 4.52248120137621E-05 x1 : 6.29486767127907E-15 5.45721660425622E-14 == err : 1.565E-05 = rco : 4.109E-13 = res : 4.185E-14 == solution 2 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x2 : -5.00000000000000E-01 -8.66025403784439E-01 x1 : -1.00000000000000E+00 8.62816615085482E-32 == err : 4.062E-16 = rco : 9.010E-02 = res : 5.863E-16 == solution 3 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x2 : 1.00000000000000E+00 0.00000000000000E+00 x1 : -1.00000000000000E+00 0.00000000000000E+00 == err : 0.000E+00 = rco : 1.098E-01 = res : 0.000E+00 == solution 4 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x2 : -5.00000000000000E-01 8.66025403784439E-01 x1 : -1.00000000000000E+00 -1.11703936774460E-31 == err : 3.772E-16 = rco : 9.010E-02 = res : 5.863E-16 == solution 5 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x2 : -3.91577428234014E-05 1.48946510335240E-06 x1 : 3.54258255519533E-14 -4.05820144229233E-15 == err : 1.355E-05 = rco : 2.665E-13 = res : 2.715E-14 == solution 6 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x2 : -1.77357885498062E-06 -4.67185535052115E-05 x1 : -6.87856850461687E-15 -6.01648175281317E-14 == err : 1.616E-05 = rco : 4.525E-13 = res : 4.610E-14 == solution 7 : t : 1.00000000000000E+00 0.00000000000000E+00 m : 1 the solution for t : x2 : 4.74973933678605E-05 -1.81900694130644E-06 x1 : -6.32194110579452E-14 7.29187155171923E-15 == err : 1.644E-05 = rco : 4.760E-13 = res : 4.848E-14 ==