lagrange.c running on 0<x<1 for N=6 
 L_order,point  order is highest power of x, point 0..n 
      L_n,j(x) = product i=0,n i!=j (x-x_i)/(x_j-x_i)  
      let 0 < x < 1 for points x_0, x_1, ... , x_n 
      let h = x_i - x_i-1,  x_i = i * h,  h = 1/n 
      L_n(x) = sum j=0,n f(x_j) L_n,j(x) 
      This L_n(x) fits f(x) with a n th order polynomial 
L_1,0(x) = 1 x^0 +
           -1 x^1 + 

L_1,1(x) = -0 x^0 +
           1 x^1 + 


L_2,0(x) = 1 x^0 +
           -3 x^1 + 
           2 x^2 + 

L_2,1(x) = -0 x^0 +
           4 x^1 + 
           -4 x^2 + 

L_2,2(x) = 0 x^0 +
           -1 x^1 + 
           2 x^2 + 


L_3,0(x) = 1 x^0 +
           -5.5 x^1 + 
           9 x^2 + 
           -4.5 x^3 + 

L_3,1(x) = -0 x^0 +
           9 x^1 + 
           -22.5 x^2 + 
           13.5 x^3 + 

L_3,2(x) = 0 x^0 +
           -4.5 x^1 + 
           18 x^2 + 
           -13.5 x^3 + 

L_3,3(x) = -0 x^0 +
           1 x^1 + 
           -4.5 x^2 + 
           4.5 x^3 + 


L_4,0(x) = 1 x^0 +
           -8.33333 x^1 + 
           23.3333 x^2 + 
           -26.6667 x^3 + 
           10.6667 x^4 + 

L_4,1(x) = -0 x^0 +
           16 x^1 + 
           -69.3333 x^2 + 
           96 x^3 + 
           -42.6667 x^4 + 

L_4,2(x) = 0 x^0 +
           -12 x^1 + 
           76 x^2 + 
           -128 x^3 + 
           64 x^4 + 

L_4,3(x) = -0 x^0 +
           5.33333 x^1 + 
           -37.3333 x^2 + 
           74.6667 x^3 + 
           -42.6667 x^4 + 

L_4,4(x) = 0 x^0 +
           -1 x^1 + 
           7.33333 x^2 + 
           -16 x^3 + 
           10.6667 x^4 + 


L_5,0(x) = 1 x^0 +
           -11.4167 x^1 + 
           46.875 x^2 + 
           -88.5417 x^3 + 
           78.125 x^4 + 
           -26.0417 x^5 + 

L_5,1(x) = -0 x^0 +
           25 x^1 + 
           -160.417 x^2 + 
           369.792 x^3 + 
           -364.583 x^4 + 
           130.208 x^5 + 

L_5,2(x) = 0 x^0 +
           -25 x^1 + 
           222.917 x^2 + 
           -614.583 x^3 + 
           677.083 x^4 + 
           -260.417 x^5 + 

L_5,3(x) = -0 x^0 +
           16.6667 x^1 + 
           -162.5 x^2 + 
           510.417 x^3 + 
           -625 x^4 + 
           260.417 x^5 + 

L_5,4(x) = 0 x^0 +
           -6.25 x^1 + 
           63.5417 x^2 + 
           -213.542 x^3 + 
           286.458 x^4 + 
           -130.208 x^5 + 

L_5,5(x) = -0 x^0 +
           1 x^1 + 
           -10.4167 x^2 + 
           36.4583 x^3 + 
           -52.0833 x^4 + 
           26.0417 x^5 + 


L_6,0(x) = 1 x^0 +
           -14.7 x^1 + 
           81.2 x^2 + 
           -220.5 x^3 + 
           315 x^4 + 
           -226.8 x^5 + 
           64.8 x^6 + 

L_6,1(x) = -0 x^0 +
           36 x^1 + 
           -313.2 x^2 + 
           1044 x^3 + 
           -1674 x^4 + 
           1296 x^5 + 
           -388.8 x^6 + 

L_6,2(x) = 0 x^0 +
           -45 x^1 + 
           526.5 x^2 + 
           -2074.5 x^3 + 
           3699 x^4 + 
           -3078 x^5 + 
           972 x^6 + 

L_6,3(x) = -0 x^0 +
           40 x^1 + 
           -508 x^2 + 
           2232 x^3 + 
           -4356 x^4 + 
           3888 x^5 + 
           -1296 x^6 + 

L_6,4(x) = 0 x^0 +
           -22.5 x^1 + 
           297 x^2 + 
           -1381.5 x^3 + 
           2889 x^4 + 
           -2754 x^5 + 
           972 x^6 + 

L_6,5(x) = -0 x^0 +
           7.2 x^1 + 
           -97.2 x^2 + 
           468 x^3 + 
           -1026 x^4 + 
           1036.8 x^5 + 
           -388.8 x^6 + 

L_6,6(x) = 0 x^0 +
           -1 x^1 + 
           13.7 x^2 + 
           -67.5 x^3 + 
           153 x^4 + 
           -162 x^5 + 
           64.8 x^6 + 



check Lagrange 5 point fit of exp(x) in 0 <= x <=1 
L_5(x) = 1 x^0 +
        1.00008 x^1 + 
        0.499069 x^2 + 
        0.17041 x^3 + 
        0.0348664 x^4 + 
        0.0138543 x^5 + 

check exp(0)=1, fit=1, err=0 
check exp(0.1)=1.10517, fit=1.10517, err=2.0592e-06 
check exp(0.2)=1.2214, fit=1.2214, err=8.88178e-16 
check exp(0.3)=1.34986, fit=1.34986, err=-7.05634e-07 
check exp(0.4)=1.49182, fit=1.49182, err=5.9952e-15 
check exp(0.5)=1.64872, fit=1.64872, err=5.18455e-07 
check exp(0.6)=1.82212, fit=1.82212, err=1.88738e-14 
check exp(0.7)=2.01375, fit=2.01375, err=-7.47076e-07 
check exp(0.8)=2.22554, fit=2.22554, err=4.21885e-14 
check exp(0.9)=2.4596, fit=2.45961, err=2.30826e-06 
check exp(1)=2.71828, fit=2.71828, err=7.77156e-14 
maxerr=2.30826e-06, rmserr=9.95134e-07 

check Lagrange 6 point fit of exp(x) in 0 <= x <=1 
L_6(x) = 1 x^0 +
        0.999995 x^1 + 
        0.50007 x^2 + 
        0.166287 x^3 + 
        0.0426744 x^4 + 
        0.00694954 x^5 + 
        0.00230584 x^6 + 

check exp(0)=1, fit=1, err=0 
check exp(0.0833333)=1.0869, fit=1.0869, err=-9.06716e-08 
check exp(0.166667)=1.18136, fit=1.18136, err=1.33227e-15 
check exp(0.25)=1.28403, fit=1.28403, err=2.5232e-08 
check exp(0.333333)=1.39561, fit=1.39561, err=0 
check exp(0.416667)=1.5169, fit=1.5169, err=-1.43078e-08 
check exp(0.5)=1.64872, fit=1.64872, err=4.21885e-15 
check exp(0.583333)=1.792, fit=1.792, err=1.46087e-08 
check exp(0.666667)=1.94773, fit=1.94773, err=2.95319e-14 
check exp(0.75)=2.117, fit=2.117, err=-2.68577e-08 
check exp(0.833333)=2.30098, fit=2.30098, err=1.01252e-13 
check exp(0.916667)=2.50094, fit=2.50094, err=1.00619e-07 
check exp(1)=2.71828, fit=2.71828, err=2.51799e-13 
maxerr=1.00619e-07, rmserr=3.93424e-08 

 end lagrange.c