# Example 1 in Nagasaka's ISSAC'2002 paper:
F := expand( (x^2 + u*x + 2*u - 1)*(x^3 + u^2*x - u + 7) + 0.2*x);

NF := (0.0032 * u*x^2 + 0.9360*x^2 + 0.9407 * u*x + 0.0283*x + 1.8791 * u
- 0.9396) * (-0.0020*u^2*x^3 + 0.0033*u*x^3 + 1.0766*x^3
+7.4501 - 1.0651*u - 0.0015*u^2 + 0.0126*x + 0.0025*u*x
+ 1.0666*u^2*x - 0.0276*x^2 - 0.0136*u*x^2 - 0.0011*u^2*x^2);
# Tolerance 0.000753 in Nagasaka's paper.

read NearestNpoly:

NF := NearestNpoly(F, 5, x, u);
#
# tolerance 0.000953 with Version 0.01.
#
# tolerance 0.001107 with Version 0.02
#   (error got larger because in this version we only allow non-zero
#    coefficients where F itself has a non-zero coefficient).
#                  
# This tolerance 0.000953 in Version 0.01 is significantly larger than
# the tolerance  0.000753 mentioned in Nagasaka's paper, so our program is
# not close to optimal in this example, which probably means that O(epsilon^2)
# is not small compared to O(epsilon) in this example.  This example shows
# then that we should still implement the Newton iteration on the factors
# of NF to bring NF closer to F.