least_square_fit.c
terms used to find fit 
a^0 b^0 c^0 

a^1 b^0 c^0 
a^0 b^1 c^0 
a^0 b^0 c^1 

the function generating the data set is 
u = 1.0 + 2.0*a + 3.0*b + 4.0*c + 5.0*a*a + 6.0*b*b +
    7.0*c*c +8.0*a*a*a + 9.0*b*b*b + 10.0*c*c*c 
fit with 3 variables to 1 power using 4 terms 
check_3d k=63, amin=0, amax=3, bmin=0, bmax=3 
               cmin=0, cmax=3, umin=16, umax=919 

Least Square Fit is u = 
   -145 * a^0 * b^0 * c^0 
   90.1375 * a^1 * b^0 * c^0 
   102.938 * a^0 * b^1 * c^0 
   115.738 * a^0 * b^0 * c^1 
rms_err=80.3658, avg_err=68.5397, max_err=165.625 

terms used to find fit 
a^0 b^0 c^0 

a^1 b^0 c^0 
a^0 b^1 c^0 
a^0 b^0 c^1 

a^2 b^0 c^0 
a^1 b^1 c^0 
a^1 b^0 c^1 
a^0 b^2 c^0 
a^0 b^1 c^1 
a^0 b^0 c^2 

fit with 3 variables to 2 power using 10 terms 
check_3d k=63, amin=0, amax=3, bmin=0, bmax=3 
               cmin=0, cmax=3, umin=16, umax=919 

Least Square Fit is u = 
   12.5508 * a^0 * b^0 * c^0 
   -37.1377 * a^1 * b^0 * c^0 
   -40.8377 * a^0 * b^1 * c^0 
   -44.5377 * a^0 * b^0 * c^1 
   41.1805 * a^2 * b^0 * c^0 
   0.259893 * a^1 * b^1 * c^0 
   0.259893 * a^1 * b^0 * c^1 
   46.6805 * a^0 * b^2 * c^0 
   0.259893 * a^0 * b^1 * c^1 
   52.1805 * a^0 * b^0 * c^2 
rms_err=10.5126, avg_err=8.59898, max_err=24.1412 

terms used to find fit 
a^0 b^0 c^0 

a^1 b^0 c^0 
a^0 b^1 c^0 
a^0 b^0 c^1 

a^2 b^0 c^0 
a^1 b^1 c^0 
a^1 b^0 c^1 
a^0 b^2 c^0 
a^0 b^1 c^1 
a^0 b^0 c^2 

a^3 b^0 c^0 
a^2 b^1 c^0 
a^2 b^0 c^1 
a^1 b^2 c^0 
a^1 b^1 c^1 
a^1 b^0 c^2 
a^0 b^3 c^0 
a^0 b^2 c^1 
a^0 b^1 c^2 
a^0 b^0 c^3 

fit with 3 variables to 3 power using 20 terms 
check_3d k=63, amin=0, amax=3, bmin=0, bmax=3 
               cmin=0, cmax=3, umin=16, umax=919 

Least Square Fit is u = 
   1 * a^0 * b^0 * c^0 
   2 * a^1 * b^0 * c^0 
   3 * a^0 * b^1 * c^0 
   4 * a^0 * b^0 * c^1 
   5 * a^2 * b^0 * c^0 
   1.43503e-12 * a^1 * b^1 * c^0 
   7.56093e-13 * a^1 * b^0 * c^1 
   6 * a^0 * b^2 * c^0 
   1.0936e-12 * a^0 * b^1 * c^1 
   7 * a^0 * b^0 * c^2 
   8 * a^3 * b^0 * c^0 
   -1.62588e-13 * a^2 * b^1 * c^0 
   -9.89597e-14 * a^2 * b^0 * c^1 
   -1.87207e-13 * a^1 * b^2 * c^0 
   -1.56922e-13 * a^1 * b^1 * c^1 
   -3.36059e-14 * a^1 * b^0 * c^2 
   9 * a^0 * b^3 * c^0 
   -1.64241e-13 * a^0 * b^2 * c^1 
   -7.66826e-14 * a^0 * b^1 * c^2 
   10 * a^0 * b^0 * c^3 
rms_err=6.1088e-13, avg_err=4.55142e-13, max_err=2.40163e-12