05-Jan-2019 19:28:05

DIFFER_TEST
  MATLAB version.
  Test DIFFERy.

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    2.82259e-16
   2    6.8125e-16
   2    5.42023e-16
   2    7.83052e-16
   2    9.87149e-16

   3    7.65747e-15
   3    1.37097e-13
   3    1.48206e-15
   3    4.49791e-14
   3    2.17359e-14

   4    2.24135e-14
   4    8.0388e-14
   4    4.50622e-14
   4    4.2476e-14
   4    2.8682e-12

   5    1.42913e-10
   5    1.17158e-12
   5    1.80607e-11
   5    6.84053e-13
   5    1.23284e-11

   6    6.21417e-10
   6    8.19882e-11
   6    6.95521e-12
   6    4.11982e-12
   6    1.62727e-11

   7    4.69827e-10
   7    7.93792e-11
   7    5.45725e-10
   7    1.9652e-11
   7    4.37147e-11

   8    7.14295e-10
   8    6.32603e-10
   8    7.14222e-09
   8    1.18461e-09
   8    1.34729e-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.

05-Jan-2019 19:28:05