# MAPLE.   May 29, 2006.
# 
# Example 4 in HCD table of spherical designs by Munemasa.
d := 4;

v := [1,  567 , 1680 , 567 , 1 ];
verts := 2816;


Q := matrix(d+1,d+1,[
1,   22, 252, 1386, 1155, 
1, 22/3,52/3,-22/3,-55/3,
1,    0, -12,    0,   11,
1,-22/3 52/3, 22/3,-55/3,
1,  -22, 252,-1386, 1155
 ]);

#  This tridiagonal  matrix, L_1-star,  allows us to fill out the cols of Q
L := matrix([
[0 ,   22  ,   0   ,    0    ,  0  ],
[1 ,   0   ,   21  ,    0    ,  0  ],
[0 ,  11/6 ,   0   ,  121/6  ,  0  ],
[0 ,   0   ,  11/3 ,    0    , 55/3],
[0 ,   0   ,   0   ,   22    ,  0  ]]);



for h from 2 to d
 do
    for i from 1 to d+1 do Q[i,h+1] := solve( Q[i,2]*Q[i,h] = L[h-1,h]*Q[i,h-1]+L[h+1,h]*x); od;
 od:


      [0    567      0      0    0]
      [                           ]
      [1    216    320     30    0]
      [                           ]
L_1 = [0    108    351    108    0]
      [                           ]
      [0     30    320    216    1]
      [                           ]
      [0      0      0    567    0]


      [0      0    1680      0    0]
      [                            ]
      [0    320    1040    320    0]
      [                            ]
L_2 = [1    351     976    351    1]
      [                            ]
      [0    320    1040    320    0]
      [                            ]
      [0      0    1680      0    0]


      [0      0      0    567    0]
      [                           ]
      [0     30    320    216    1]
      [                           ]
L_3 = [0    108    351    108    0]
      [                           ]
      [1    216    320     30    0]
      [                           ]
      [0    567      0      0    0]


      [0    0    0    0    1]
      [                     ]
      [0    0    0    1    0]
      [                     ]
L_4 = [0    0    1    0    0]
      [                     ]
      [0    1    0    0    0]
      [                     ]
      [1    0    0    0    0]



     [  1   567  1680   567    1 ]
     [                           ]
     [  1   189     0  -189   -1 ]
     [                           ]
 P = [  1    39   -80    39    1 ]
     [                           ]
     [  1    -3     0     3   -1 ]
     [                           ]
     [  1    -9    16    -9    1 ]




     [1     22      252     1386     1155 ]
     [                                    ]
     [1    22/3     52/3    -22/3    -55/3]
     [                                    ]
Q  = [1      0      -12       0       11  ]
     [                                    ]
     [1    -22/3    52/3    22/3     -55/3]
     [                                    ]
     [1     -22     252     -1386    1155 ]

     [0     22      0        0       0  ]
     [                                  ]
     [1     0       21       0       0  ]
     [                                  ]
Ls1= [0    11/6     0      121/6     0  ]
     [                                  ]
     [0     0      11/3      0      55/3]
     [                                  ]
     [0     0       0       22       0  ]


     [0     0       252       0        0  ]
     [                                    ]
     [0     21       0       231       0  ]
     [                                    ]
Ls2= [1     0      148/3      0      605/3]
     [                                    ]
     [0    11/3      0      745/3      0  ]
     [                                    ]
     [0     0       44        0       208 ]
 
 
     [0      0        0       1386       0   ]
     [                                       ]
     [0      0       231       0        1155 ]
     [                                       ]
Ls3 =[0    121/6      0      8195/6      0   ]
     [                                       ]
     [1      0      745/3      0       3410/3]
     [                                       ]
     [0     22        0       1364       0   ]
  
  
     [0     0        0        0        1155 ]
     [                                      ]
     [0     0        0       1155       0   ]
     [                                      ]
Ls4 =[0     0      605/3      0       2860/3]
     [                                      ]
     [0    55/3      0      3410/3      0   ]
     [                                      ]
     [1     0       208       0        946  ]