vali.dat

basic linear algebra classes and applications (SVD,interpolation, multivariate optimization)

./vali


		Verification of the Aitken-Lagrange interpolation

------------- Interpolation over the table with uniform grid
Function to interpolate: exp(-x)

Check to see that interpolation at a node gives an exact result

Uniform grid [0.1:1] with the step 0.1

Done

Check the precision of the interpolation at arbitrary points

Uniform grid [0.1:1] with the step 0.1

Point	Exact function value     Interpolated    Error
   0	           1		    0.999998   1.72853e-06
0.11	    0.895834		    0.895834   1.26336e-08
0.22	    0.802519		    0.802519   1.29626e-08
0.33	    0.718924		    0.718924   1.41076e-08
0.44	    0.644036		    0.644036   8.84407e-09
0.55	     0.57695		     0.57695   1.43652e-07
0.66	    0.516851		    0.516851   3.10746e-08
0.77	    0.463013		    0.463013   1.53709e-08
0.88	    0.414783		    0.414783   8.92645e-08
0.99	    0.371577		    0.371576   4.41423e-07
 1.1	    0.332871		    0.332871   2.12869e-07

Done

------------- Interpolation over the table with non-uniform grid
Function to interpolate: sin(x) * exp(-x/10)

Grids 0.1 .. 1 have the mesh 0.1, and 1..9 have the mesh 0.2

Check to see that interpolation at a node gives an exact result

Done

Check the precision of the interpolation at arbitrary points

Point	Exact function value     Interpolated    Error
   0	           0		 8.51708e-07   8.51708e-07
0.27	    0.259626		    0.259626   1.14471e-08
0.54	    0.487109		    0.487109   3.00486e-08
0.81	    0.667933		    0.667933   1.61806e-08
 1.1	     0.79167		     0.79167   6.00246e-08
 1.4	    0.852505		    0.852481   2.41944e-05
 1.6	    0.849412		    0.849412    1.1897e-08
 1.9	    0.785971		    0.785972   1.42046e-07
 2.2	    0.669875		    0.669875   9.72032e-08
 2.4	    0.512161		    0.512161   3.90728e-09
 2.7	    0.326253		    0.326253   1.96635e-08
   3	    0.126876		    0.126876   2.32042e-08
 3.2	  -0.0710583		  -0.0710583    2.1131e-08
 3.5	   -0.253526		   -0.253526   2.44474e-08
 3.8	   -0.408341		    -0.40834   1.73721e-07

Done

Check the precision of the interpolation at arbitrary points

Example from Fig. 4.11 of the book
	Numerical Methods and Software,
	by D.Kahaner, C.Moler, and S.Nash - Prentice Hall, 1989

		Interpolation nots
x      0      2      3      5      6      8      9     11     12     14     15 
y     10     10     10     10     10     10     10     15     50     60     85 

Point	Interpolated value
      0             10
   0.32             10
   0.64             10
   0.96             10
   1.28             10
    1.6             10
   1.92             10
   2.24             10
   2.56             10
   2.88             10
    3.2             10
   3.52             10
   3.84             10
   4.16             10
   4.48             10
    4.8             10
   5.12             10
   5.44             10
   5.76             10
   6.08             10
    6.4             10
   6.72             10
   7.04        9.94286
   7.36        9.93781
   7.68        9.95318
      8             10
   8.32        10.0911
   8.64        10.2309
   8.96        10.4641
   9.28        10.2771
    9.6          9.232
   9.92        8.71549
  10.24        5.79377
  10.56        9.07021
  10.88        12.9199
   11.2        20.6143
  11.52         29.442
  11.84        42.5424
  12.16        57.7971
  12.48        73.2555
   12.8        86.0643
  13.12        62.2166
  13.44        61.3373
  13.76        60.1782
  14.08        60.8715
   14.4             66
  14.72        74.3712
  15.04        86.8043
  15.36        104.118
  15.68        127.133

Done

Compilation finished at Fri Dec 25 23:20:31