lagrange.f90 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) =  1.00000000000000  x^0 +
             -1.00000000000000  x^ 1  + 
  
 L_ 1 , 1 (x) =  -0.000000000000000  x^0 +
             1.00000000000000  x^ 1  + 
  
  
 L_ 2 , 0 (x) =  1.00000000000000  x^0 +
             -3.00000000000000  x^ 1  + 
             2.00000000000000  x^ 2  + 
  
 L_ 2 , 1 (x) =  -0.000000000000000  x^0 +
             4.00000000000000  x^ 1  + 
             -4.00000000000000  x^ 2  + 
  
 L_ 2 , 2 (x) =  0.000000000000000  x^0 +
             -1.00000000000000  x^ 1  + 
             2.00000000000000  x^ 2  + 
  
  
 L_ 3 , 0 (x) =  1.00000000000000  x^0 +
             -5.50000000000000  x^ 1  + 
             9.00000000000000  x^ 2  + 
             -4.50000000000000  x^ 3  + 
  
 L_ 3 , 1 (x) =  -0.000000000000000  x^0 +
             9.00000000000000  x^ 1  + 
             -22.5000000000000  x^ 2  + 
             13.5000000000000  x^ 3  + 
  
 L_ 3 , 2 (x) =  0.000000000000000  x^0 +
             -4.50000000000000  x^ 1  + 
             18.0000000000000  x^ 2  + 
             -13.5000000000000  x^ 3  + 
  
 L_ 3 , 3 (x) =  -0.000000000000000  x^0 +
             1.00000000000000  x^ 1  + 
             -4.50000000000000  x^ 2  + 
             4.50000000000000  x^ 3  + 
  
  
 L_ 4 , 0 (x) =  1.00000000000000  x^0 +
             -8.33333333333333  x^ 1  + 
             23.3333333333333  x^ 2  + 
             -26.6666666666667  x^ 3  + 
             10.6666666666667  x^ 4  + 
  
 L_ 4 , 1 (x) =  -0.000000000000000  x^0 +
             16.0000000000000  x^ 1  + 
             -69.3333333333333  x^ 2  + 
             96.0000000000000  x^ 3  + 
             -42.6666666666667  x^ 4  + 
  
 L_ 4 , 2 (x) =  0.000000000000000  x^0 +
             -12.0000000000000  x^ 1  + 
             76.0000000000000  x^ 2  + 
             -128.000000000000  x^ 3  + 
             64.0000000000000  x^ 4  + 
  
 L_ 4 , 3 (x) =  -0.000000000000000  x^0 +
             5.33333333333333  x^ 1  + 
             -37.3333333333333  x^ 2  + 
             74.6666666666667  x^ 3  + 
             -42.6666666666667  x^ 4  + 
  
 L_ 4 , 4 (x) =  0.000000000000000  x^0 +
             -1.00000000000000  x^ 1  + 
             7.33333333333333  x^ 2  + 
             -16.0000000000000  x^ 3  + 
             10.6666666666667  x^ 4  + 
  
  
 L_ 5 , 0 (x) =  1.00000000000000  x^0 +
             -11.4166666666667  x^ 1  + 
             46.8750000000000  x^ 2  + 
             -88.5416666666667  x^ 3  + 
             78.1250000000000  x^ 4  + 
             -26.0416666666667  x^ 5  + 
  
 L_ 5 , 1 (x) =  -0.000000000000000  x^0 +
             25.0000000000000  x^ 1  + 
             -160.416666666667  x^ 2  + 
             369.791666666667  x^ 3  + 
             -364.583333333333  x^ 4  + 
             130.208333333333  x^ 5  + 
  
 L_ 5 , 2 (x) =  0.000000000000000  x^0 +
             -25.0000000000000  x^ 1  + 
             222.916666666667  x^ 2  + 
             -614.583333333333  x^ 3  + 
             677.083333333333  x^ 4  + 
             -260.416666666667  x^ 5  + 
  
 L_ 5 , 3 (x) =  -0.000000000000000  x^0 +
             16.6666666666667  x^ 1  + 
             -162.500000000000  x^ 2  + 
             510.416666666667  x^ 3  + 
             -625.000000000000  x^ 4  + 
             260.416666666667  x^ 5  + 
  
 L_ 5 , 4 (x) =  0.000000000000000  x^0 +
             -6.25000000000000  x^ 1  + 
             63.5416666666667  x^ 2  + 
             -213.541666666667  x^ 3  + 
             286.458333333333  x^ 4  + 
             -130.208333333333  x^ 5  + 
  
 L_ 5 , 5 (x) =  -0.000000000000000  x^0 +
             1.00000000000000  x^ 1  + 
             -10.4166666666667  x^ 2  + 
             36.4583333333333  x^ 3  + 
             -52.0833333333333  x^ 4  + 
             26.0416666666667  x^ 5  + 
  
  
 L_ 6 , 0 (x) =  1.00000000000000  x^0 +
             -14.7000000000000  x^ 1  + 
             81.2000000000000  x^ 2  + 
             -220.500000000000  x^ 3  + 
             315.000000000000  x^ 4  + 
             -226.800000000000  x^ 5  + 
             64.8000000000000  x^ 6  + 
  
 L_ 6 , 1 (x) =  -0.000000000000000  x^0 +
             36.0000000000000  x^ 1  + 
             -313.200000000000  x^ 2  + 
             1044.00000000000  x^ 3  + 
             -1674.00000000000  x^ 4  + 
             1296.00000000000  x^ 5  + 
             -388.800000000000  x^ 6  + 
  
 L_ 6 , 2 (x) =  0.000000000000000  x^0 +
             -45.0000000000000  x^ 1  + 
             526.500000000000  x^ 2  + 
             -2074.50000000000  x^ 3  + 
             3699.00000000000  x^ 4  + 
             -3078.00000000000  x^ 5  + 
             972.000000000000  x^ 6  + 
  
 L_ 6 , 3 (x) =  -0.000000000000000  x^0 +
             40.0000000000000  x^ 1  + 
             -508.000000000000  x^ 2  + 
             2232.00000000000  x^ 3  + 
             -4356.00000000000  x^ 4  + 
             3888.00000000000  x^ 5  + 
             -1296.00000000000  x^ 6  + 
  
 L_ 6 , 4 (x) =  0.000000000000000  x^0 +
             -22.5000000000000  x^ 1  + 
             297.000000000000  x^ 2  + 
             -1381.50000000000  x^ 3  + 
             2889.00000000000  x^ 4  + 
             -2754.00000000000  x^ 5  + 
             972.000000000000  x^ 6  + 
  
 L_ 6 , 5 (x) =  -0.000000000000000  x^0 +
             7.20000000000000  x^ 1  + 
             -97.2000000000000  x^ 2  + 
             468.000000000000  x^ 3  + 
             -1026.00000000000  x^ 4  + 
             1036.80000000000  x^ 5  + 
             -388.800000000000  x^ 6  + 
  
 L_ 6 , 6 (x) =  0.000000000000000  x^0 +
             -1.00000000000000  x^ 1  + 
             13.7000000000000  x^ 2  + 
             -67.5000000000000  x^ 3  + 
             153.000000000000  x^ 4  + 
             -162.000000000000  x^ 5  + 
             64.8000000000000  x^ 6  + 
  
  
  
 check Lagrange 5 point fit of exp(x) in 0 <= x <=1 
 L_ 5 (x) =  1.00000000000000  x^0 +
          1.00008254486209  x^ 1  + 
          0.499068782392050  x^ 2  + 
          0.170409776011455  x^ 3  + 
          0.0348664468096161  x^ 4  + 
          0.0138542783839073  x^ 5  + 
  
 check exp( 0.000000000000000 )= 1.00000000000000 , fit= 1.00000000000000 , err= 0.000000000000000
 check exp( 0.100000000000000 )= 1.10517091807565 , fit= 1.10517297727361 , err= 0.00000205919795859266
 check exp( 0.200000000000000 )= 1.22140275816017 , fit= 1.22140275816017 , err= 8.88178419700125E-16
 check exp( 0.300000000000000 )= 1.34985880757600 , fit= 1.34985810194185 , err= -7.05634150222068E-7
 check exp( 0.400000000000000 )= 1.49182469764127 , fit= 1.49182469764128 , err= 5.99520433297585E-15
 check exp( 0.500000000000000 )= 1.64872127070013 , fit= 1.64872178915559 , err= 5.18455461451239E-7
 check exp( 0.600000000000000 )= 1.82211880039051 , fit= 1.82211880039053 , err= 1.88737914186277E-14
 check exp( 0.700000000000000 )= 2.01375270747048 , fit= 2.01375196039447 , err= -7.47076005058744E-7
 check exp( 0.800000000000000 )= 2.22554092849247 , fit= 2.22554092849251 , err= 4.21884749357559E-14
 check exp( 0.900000000000000 )= 2.45960311115695 , fit= 2.45960541942050 , err= 0.00000230826354874480
 check exp( 1.00000000000000 )= 2.71828182845905 , fit= 2.71828182845912 , err= 7.77156117237610E-14
 maxerr= 0.00000230826354874480 , rmserr 9.95134299750492E-7
  
 check Lagrange 6 point fit of exp(x) in 0 <= x <=1 
 L_ 6 (x) =  1.00000000000000  x^0 +
          0.999995224533463  x^ 1  + 
          0.500069630985665  x^ 2  + 
          0.166287193149693  x^ 3  + 
          0.0426743917494075  x^ 4  + 
          0.00694954373078360  x^ 5  + 
          0.00230584431028547  x^ 6  + 
  
 check exp(%g)=%g, fit=%g, err=%g  0.000000000000000 1.00000000000000 1.00000000000000 0.000000000000000
 check exp(%g)=%g, fit=%g, err=%g  0.0833333333333333 1.08690404952123 1.08690395884967 -9.06715589277240E-8
 check exp(%g)=%g, fit=%g, err=%g  0.166666666666667 1.18136041286565 1.18136041286565 1.33226762955019E-15
 check exp(%g)=%g, fit=%g, err=%g  0.250000000000000 1.28402541668774 1.28402544191978 2.52320340177192E-8
 check exp(%g)=%g, fit=%g, err=%g  0.333333333333333 1.39561242508609 1.39561242508609 0.000000000000000
 check exp(%g)=%g, fit=%g, err=%g  0.416666666666667 1.51689679638821 1.51689678208040 -1.43078155989684E-8
 check exp(%g)=%g, fit=%g, err=%g  0.500000000000000 1.64872127070013 1.64872127070013 4.21884749357559E-15
 check exp(%g)=%g, fit=%g, err=%g  0.583333333333333 1.79200182565576 1.79200184026447 1.46087120178606E-8
 check exp(%g)=%g, fit=%g, err=%g  0.666666666666667 1.94773404105468 1.94773404105471 2.95319324550292E-14
 check exp(%g)=%g, fit=%g, err=%g  0.750000000000000 2.11700001661267 2.11699998975493 -2.68577453610419E-8
 check exp(%g)=%g, fit=%g, err=%g  0.833333333333333 2.30097589089282 2.30097589089293 1.01252339845814E-13
 check exp(%g)=%g, fit=%g, err=%g  0.916666666666667 2.50094001366213 2.50094011428137 1.00619244047806E-7
 check exp(%g)=%g, fit=%g, err=%g  1.00000000000000 2.71828182845905 2.71828182845930 2.51798581984986E-13
 maxerr= 1.00619244047806E-7  rmserr= 3.93423689715289E-8
  end lagrange.f90