06-Oct-2022 07:41:51

differ_test():
  MATLAB/Octave version 4.2.2
  Test differ().

differ_matrix_test():
  Demonstrate that the DIFFER matrix is "really"
  a Vandermonde matrix.

  Stencil matrix:

  Col:       1             2             3             4       
  Row
      1 :         2.5           3.3          -1.3           0.5  
      2 :        6.25         10.89          1.69          0.25  
      3 :      15.625        35.937        -2.197         0.125  
      4 :     39.0625       118.592        2.8561        0.0625  

  Solution of DIFFER system:

     1:            1
     2:            2
     3:            3
     4:            4

  Solution of VANDERMONDE system:

     1:          2.5
     2:          6.6
     3:         -3.9
     4:            2

  Transformed solution of VANDERMONDE system:

     1:            1
     2:            2
     3:            3
     4:            4

differ_test02():
  differ_inverse() returns the inverse of a DIFFER matrix;

   N    Inverse error

   2    5.34053e-16
   2    4.00288e-16
   2    8.5705e-16
   2    2.03024e-15
   2    4.97112e-15

   3    8.02052e-15
   3    4.65043e-15
   3    5.9146e-14
   3    4.36116e-14
   3    9.26155e-14

   4    6.56758e-14
   4    5.63945e-11
   4    3.01372e-14
   4    4.57894e-14
   4    7.19669e-14

   5    6.65399e-13
   5    1.19083e-12
   5    3.79539e-13
   5    1.33247e-11
   5    3.87688e-12

   6    1.10113e-09
   6    9.66704e-11
   6    1.21571e-12
   6    3.87772e-11
   6    4.32587e-12

   7    4.44778e-08
   7    1.01368e-09
   7    1.66699e-11
   7    2.66519e-10
   7    1.08634e-10

   8    1.91213e-08
   8    4.05168e-08
   8    4.75355e-10
   8    1.42053e-09
   8    3.24869e-09

differ_test03():
  Reproduce a specific example.

  Solution of DIFFER system:

     1:   -0.0833333
     2:          0.5
     3:         -1.5
     4:         0.25

  DFDX =         3.66931
  d exp(x) /dx = 3.6693

differ_test04():
  differ_forward(),
  differ_backward(), and
  differ_central() produce coefficients for difference
  approximations of the O-th derivative, 
  with error of order H^P, for a uniform spacing of H.

  Use a spacing of H = 1 for all examples.

  Forward difference coefficients, O = 3, P = 1

       1:              0            -1
       2:              1             3
       3:              2            -3
       4:              3             1

  Backward difference coefficients, O = 3, P = 1

       1:             -3            -1
       2:             -2             3
       3:             -1            -3
       4:              0             1

  Central difference coefficients, O = 3, P = 2

       1:             -2          -0.5
       2:             -1             1
       3:              0             0
       4:              1            -1
       5:              2           0.5

  Central difference coefficients, O = 3, P = 4

       1:             -3         0.125
       2:             -2            -1
       3:             -1         1.625
       4:              0   2.12062e-14
       5:              1        -1.625
       6:              2             1
       7:              3        -0.125

  Forward difference coefficients, O = 4, P = 1

       1:              0             1
       2:              1            -4
       3:              2             6
       4:              3            -4
       5:              4             1

  Backward difference coefficients, O = 4, P = 1

       1:             -4             1
       2:             -3            -4
       3:             -2             6
       4:             -1            -4
       5:              0             1

  Central difference coefficients, O = 4, P = 3

       1:             -3     -0.166667
       2:             -2             2
       3:             -1          -6.5
       4:              0       9.33333
       5:              1          -6.5
       6:              2             2
       7:              3     -0.166667

differ_test05():
  differ_stencil() produces coefficients for difference
  approximations of the O-th derivative, 
  using arbitrarily spaced data, with maximum spacing H
  with error of order H^P.

  For all tests, let X0 = 0
  and use a uniformly spacing of 1, so we can compare
  with previous forward, backward and central differences.

  Finite difference coefficients, O = 3, P = 1

       1:              0            -1
       2:              1             3
       3:              2            -3
       4:              3             1

  Backward difference coefficients, O = 3, P = 1

       1:             -3            -1
       2:             -2             3
       3:             -1            -3
       4:              0             1

  Central difference coefficients, O = 3, P = 2

       1:             -2          -0.5
       2:             -1             1
       3:              0             0
       4:              1            -1
       5:              2           0.5

  Central difference coefficients, O = 3, P = 4

       1:             -3         0.125
       2:             -2            -1
       3:             -1         1.625
       4:              0   2.12062e-14
       5:              1        -1.625
       6:              2             1
       7:              3        -0.125

  Finite difference coefficients, O = 4, P = 1

       1:              0             1
       2:              1            -4
       3:              2             6
       4:              3            -4
       5:              4             1

  Backward difference coefficients, O = 4, P = 1

       1:             -4             1
       2:             -3            -4
       3:             -2             6
       4:             -1            -4
       5:              0             1

  Central difference coefficients, O = 4, P = 3

       1:             -3     -0.166667
       2:             -2             2
       3:             -1          -6.5
       4:              0       9.33333
       5:              1          -6.5
       6:              2             2
       7:              3     -0.166667

differ_test():
  Normal end of execution.

06-Oct-2022 07:41:51