11 June 2024 10:16:33 AM

companion_matrix_test():
  C++ version 
  companion_matrix() computes the companion matrix
  of a polynomial, in various bases.

companion_chebyshev_test():
  companion_chebyshev() computes the companion matrix
  of a polynomial p(x) in the Chebyshev basis.

    p(x)
         +1 * T5(x)
         +2 * T4(x)
         +3 * T3(x)
         +4 * T2(x)
         +5 * T1(x)
         +6 * T0(x)

    Monomial q(x) =
         +16 * x^5
         +16 * x^4
         -8 * x^3
         -8 * x^2
         +1 * x
         +4

  Roots of q(x):

         0: -0.520594  0.469916
         1: -0.520594  -0.469916
         2: 0.599455  0.282318
         3: 0.599455  -0.282318
         4: -1.15772  0

  Chebyshev companion matrix C(p):

  Col:          0             1             2             3             4       
  Row

    0:            0             1             0             0             0  
    1:          0.5             0           0.5             0             0  
    2:            0           0.5             0           0.5             0  
    3:            0             0           0.5             0           0.5  
    4:           -3          -2.5            -2            -1            -1  

  Eigenvalues of C(p):

         0: 0.599455  0.282318
         1: 0.599455  -0.282318
         2: -1.15772  0
         3: -0.520594  0.469916
         4: -0.520594  -0.469916

companion_gegenbauer_test():
  companion_gegenbauer() computes the companion matrix
  of a polynomial p(x) in the Gegenbauer basis.

    p(x)
         +1 * T5(x)
         +2 * T4(x)
         +3 * T3(x)
         +4 * T2(x)
         +5 * T1(x)
         +6 * T0(x)

  Gegenbauer parameter alpha will be 0.5

    Monomial q(x) =
         +7.875 * x^5
         +8.75 * x^4
         -1.25 * x^3
         -1.5 * x^2
         +2.375 * x
         +4.75

  Roots of q(x):

         0: 0.626608  0.526603
         1: 0.626608  -0.526603
         2: -0.57237  0.640794
         3: -0.57237  -0.640794
         4: -1.21959  0

  Gegenbauer companion matrix C(p):

  Col:          0             1             2             3             4       
  Row

    0:            0             1             0             0             0  
    1:     0.333333             0      0.666667             0             0  
    2:            0           0.4             0           0.6             0  
    3:            0             0      0.428571             0      0.571429  
    4:     -3.33333      -2.77778      -2.22222      -1.22222      -1.11111  

  Eigenvalues of C(p):

         0: -1.21959  0
         1: -0.57237  0.640794
         2: -0.57237  -0.640794
         3: 0.626608  0.526603
         4: 0.626608  -0.526603

  Gegenbauer parameter alpha will be 1

    Monomial q(x) =
         +32 * x^5
         +32 * x^4
         -8 * x^3
         -8 * x^2
         +4 * x
         +4

  Roots of q(x):

         0: 0.489159  0.338048
         1: 0.489159  -0.338048
         2: -0.489159  0.338048
         3: -0.489159  -0.338048
         4: -1  0

  Gegenbauer companion matrix C(p):

  Col:          0             1             2             3             4       
  Row

    0:            0           0.5             0             0             0  
    1:          0.5             0           0.5             0             0  
    2:            0           0.5             0           0.5             0  
    3:            0             0           0.5             0           0.5  
    4:           -3          -2.5            -2            -1            -1  

  Eigenvalues of C(p):

         0: 0.489159  0.338048
         1: 0.489159  -0.338048
         2: -1  0
         3: -0.489159  0.338048
         4: -0.489159  -0.338048

companion_hermite_test():
  companion_hermite() computes the companion matrix
  of a polynomial p(x) in the Hermite basis.

    p(x)
         +1 * H5(x)
         +2 * H4(x)
         +3 * H3(x)
         +4 * H2(x)
         +5 * H1(x)
         +6 * H0(x)

    Monomial q(x) =
         +32 * x^5
         +32 * x^4
         -136 * x^3
         -80 * x^2
         +94 * x
         +22

  Roots of q(x):

         0: -0.210374  0
         1: -1.11155  0
         2: 0.799762  0
         3: -2.17107  0
         4: 1.69323  0

  Hermite companion matrix C(p):

  Col:          0             1             2             3             4       
  Row

    0:            0           0.5             0             0             0  
    1:            1             0           0.5             0             0  
    2:            0             2             0           0.5             0  
    3:            0             0             3             0           0.5  
    4:           -3          -2.5            -2           2.5            -1  

  Eigenvalues of C(p):

         0: -2.17107  0
         1: -1.11155  0
         2: -0.210374  0
         3: 0.799762  0
         4: 1.69323  0

companion_laguerre_test():
  companion_laguerre() computes the companion matrix
  of a polynomial p(x) in the Laguerre basis.

    p(x)
         +1 * L5(x)
         +2 * L4(x)
         +3 * L3(x)
         +4 * L2(x)
         +5 * L1(x)
         +6 * L0(x)

    Monomial q(x) =
         -0.00833333 * x^5
         +0.291667 * x^4
         -3.5 * x^3
         +17.5 * x^2
         -35 * x
         +21

  Roots of q(x):

         0: 1.03111  0
         1: 2.83721  0
         2: 5.62029  0
         3: 9.68291  0
         4: 15.8285  0

  Laguerre companion matrix C(p):

  Col:          0             1             2             3             4       
  Row

    0:            1            -1             0             0             0  
    1:           -1             3            -2             0             0  
    2:            0            -2             5            -3             0  
    3:            0             0            -3             7            -4  
    4:           30            25            20            11            19  

  Eigenvalues of C(p):

         0: 15.8285  0
         1: 9.68291  0
         2: 5.62029  0
         3: 1.03111  0
         4: 2.83721  0

companion_legendre_test():
  companion_legendre() computes the companion matrix
  of a polynomial p(x) in the Legendre basis.

    p(x)
         +1 * P5(x)
         +2 * P4(x)
         +3 * P3(x)
         +4 * P2(x)
         +5 * P1(x)
         +6 * P0(x)

    Monomial q(x) =
         +7.875 * x^5
         +8.75 * x^4
         -1.25 * x^3
         -1.5 * x^2
         +2.375 * x
         +4.75

  Roots of q(x):

         0: 0.626608  0.526603
         1: 0.626608  -0.526603
         2: -0.57237  0.640794
         3: -0.57237  -0.640794
         4: -1.21959  0

  Legendre companion matrix C(p):

  Col:          0             1             2             3             4       
  Row

    0:            0             1             0             0             0  
    1:     0.333333             0      0.666667             0             0  
    2:            0           0.4             0           0.6             0  
    3:            0             0      0.428571             0      0.571429  
    4:     -3.33333      -2.77778      -2.22222      -1.22222      -1.11111  

  Eigenvalues of C(p):

         0: -1.21959  0
         1: -0.57237  0.640794
         2: -0.57237  -0.640794
         3: 0.626608  0.526603
         4: 0.626608  -0.526603

companion_monomial_test():
  companion_monomial() computes the companion matrix
  of a polynomial p(x) in the monomial basis.

    p(x) =
         +1 * x^5
         +2 * x^4
         +3 * x^3
         +4 * x^2
         +5 * x
         +6

  Roots of q(x):

         0: 0.551685  1.25335
         1: 0.551685  -1.25335
         2: -1.4918  0
         3: -0.805786  1.2229
         4: -0.805786  -1.2229

  Monomial companion matrix C(p):

  Col:          0             1             2             3             4       
  Row

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

  Eigenvalues of C(p):

         0: 0.551685  1.25335
         1: 0.551685  -1.25335
         2: -1.4918  0
         3: -0.805786  1.2229
         4: -0.805786  -1.2229

companion_matrix_test():
  Normal end of execution.

11 June 2024 10:16:33 AM