# MAPLE. May 29, 2006. # # Example 5 in HCD table of spherical designs by Munemasa. d := 5; v := [1 , 2025 , 15400 , 22275 , 7128 , 275]; verts := 47104; Q := matrix([ [1 , 23 , 275 , 2277 , 12650 , 31878 ], [1 , 161/15 ,451/9 ,1771/15 , 506/9 , -3542/15 ], [1 , 23/5 , -1 , -207/5 , -46 , 414/5 ], [1 , -23/15 , -101/9 , 299/15 , 506/9 , -322/5 ], [1 , -23/3 , 175/9 , 23/3 , -1150/9, 322/3 ], [1 , -69/5 , 91 , -1863/5 , 874 , -2898/5] ]); # This tridiagonal matrix, L_1-star, allows us to fill out the cols of Q L := matrix(d+1,d+1,[ 0, 23 , 0 , 0 , 0 , 0 , 1, 0 , 22 , 0 , 0 , 0 , 0, 46/25 , 0 ,529/25 , 0 , 0 , 0, 0 , 23/9 , 0 ,184/9 , 0 , 0, 0 , 0 , 92/25 , 0 ,483/25 , 0, 0 , 0 , 0 , 23/3 , 46/3 ]); 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 2025 0 0 0 0] [ ] [1 462 1232 330 0 0] [ ] [0 162 972 810 81 0] L_1 = [ ] [0 30 560 1092 336 7] [ ] [0 0 175 1050 750 50] [ ] [0 0 0 567 1296 162] [0 0 15400 0 0 0] [ ] [0 1232 7392 6160 616 0] [ ] [1 972 5892 6885 1620 30] L_2 = [ ] [0 560 4760 7392 2576 112] [ ] [0 175 3500 8050 3500 175] [ ] [0 0 1680 9072 4536 112] [0 0 0 22275 0 0] [ ] [0 330 6160 12012 3696 77] [ ] [0 810 6885 10692 3726 162] L_3 = [ ] [1 1092 7392 10290 3360 140] [ ] [0 1050 8050 10500 2625 50] [ ] [0 567 9072 11340 1296 0] [0 0 0 0 7128 0] [ ] [0 0 616 3696 2640 176] [ ] [0 81 1620 3726 1620 81] L_4 = [ ] [0 336 2576 3360 840 16] [ ] [1 750 3500 2625 252 0] [ ] [0 1296 4536 1296 0 0] [0 0 0 0 0 275] [ ] [0 0 0 77 176 22] [ ] [0 0 30 162 81 2] L_5 = [ ] [0 7 112 140 16 0] [ ] [0 50 175 50 0 0] [ ] [1 162 112 0 0 0] [1 2025 15400 22275 7128 275] [ ] [1 945 3080 -1485 -2376 -165] [ ] [1 369 -56 -909 504 91] P = [ ] [1 105 -280 195 24 -45] [ ] [1 9 -56 99 -72 19] [ ] [1 -15 40 -45 24 -5] [1 23 275 2277 12650 31878 ] [ ] [ 161 1771 -3542 ] [1 --- 451/9 ---- 506/9 ----- ] [ 15 15 15 ] [ ] [1 23/5 -1 -207/5 -46 414/5 ] Q = [ ] [ -23 299 ] [1 --- -101/9 --- 506/9 -322/5 ] [ 15 15 ] [ ] [1 -23/3 175/9 23/3 -1150/9 322/3 ] [ ] [1 -69/5 91 -1863/5 874 -2898/5] [0 23 0 0 0 0 ] [ ] [1 0 22 0 0 0 ] [ ] [ 46 529 ] [0 -- 0 --- 0 0 ] [ 25 25 ] Ls1 = [ ] [0 0 23/9 0 184/9 0 ] [ ] [ 92 483 ] [0 0 0 -- 0 --- ] [ 25 25 ] [ ] [0 0 0 0 23/3 46/3] [0 0 275 0 0 0 ] [ ] [0 22 0 253 0 0 ] [ ] [1 0 350/9 0 2116/9 0 ] Ls2 = [ ] [0 23/9 0 520/9 0 644/3 ] [ ] [0 0 46/9 0 980/9 161 ] [ ] [0 0 0 46/3 575/9 1762/9] [0 0 0 2277 0 0 ] [ ] [0 0 253 0 2024 0 ] [ ] [ 529 44436 ] [0 --- 0 2392/5 0 ----- ] [ 25 25 ] Ls3 = [ ] [1 0 520/9 0 8372/9 1288 ] [ ] [ 92 4186 40572 ] [0 -- 0 ---- 483 ----- ] [ 25 25 25 ] [ ] [0 0 46/3 92 644 4577/3] [0 0 0 0 12650 0 ] [ ] [0 0 0 2024 0 10626 ] [ ] [0 0 2116/9 0 45080/9 7406 ] [ ] [0 184/9 0 8372/9 8050/3 9016 ] Ls4 = [ ] [1 0 980/9 483 33166/9 8372 ] [ ] [0 23/3 575/9 644 29900/9 77510/9] [0 0 0 0 0 31878 ] [ ] [0 0 0 0 10626 21252 ] [ ] [ 44436 567364 ] [0 0 0 ----- 7406 ------ ] [ 25 25 ] Ls5 = [ ] [0 0 644/3 1288 9016 64078/3 ] [ ] [ 483 40572 ] [0 --- 161 ----- 8372 108514/5] [ 25 25 ] [ ] [1 46/3 1762/9 4577/3 77510/9 21528 ]