Residual Arithmetic, r=2, d=7
The following table illustrates the performance of residual arithmetic.
The dimension of the spline space on the generic double CloughTocher
split, for r=2 and d=7 is 99. It was computed
using three consecutive prime numbers. The table gives the top one of
those prime numbers, and the color indicates the result, as follows:

All
entries in the linear system have at least one of their residuals
equal to zero. This makes Gaussian Elimination impossible and the
matrix is considered to have rank 0.

The computed dimension is too high. Some nonzero numbers in the linear
system are treated as being zero.

The dimension is computed correctly. However, some entries in the
linear systems have a mixture of zero and nonzero residuals. The
program recognizes this as a potential problem. Numbers with mixed
residuals are considered nonzero, but they cannot serve as pivots.

The dimension is computed correctly and all nonzero entries in the linear system have three nonzero residuals. This is the expected and desired situation.
The smallest triple of primes having these properties are 2423, 2437, 2441.
5  7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79 
83  89  97  101  103  107  109  113  127  131  137  139  149  151  157  163  167  173  179  181 
191  193  197  199  211  223  227  229  233  239  241  251  257  263  269  271  277  281  283  293 
307  311  313  317  331  337  347  349  353  359  367  373  379  383  389  397  401  409  419  421 
431  433  439  443  449  457  461  463  467  479  487  491  499  503  509  521  523  541  547  557 
563  569  571  577  587  593  599  601  607  613  617  619  631  641  643  647  653  659  661  673 
677  683  691  701  709  719  727  733  739  743  751  757  761  769  773  787  797  809  811  821 
823  827  829  839  853  857  859  863  877  881  883  887  907  911  919  929  937  941  947  953 
967  971  977  983  991  997  1009  1013  1019  1021  1031  1033  1039  1049  1051  1061  1063  1069  1087  1091 
1093  1097  1103  1109  1117  1123  1129  1151  1153  1163  1171  1181  1187  1193  1201  1213  1217  1223  1229  1231 
1237  1249  1259  1277  1279  1283  1289  1291  1297  1301  1303  1307  1319  1321  1327  1361  1367  1373  1381  1399 
1409  1423  1427  1429  1433  1439  1447  1451  1453  1459  1471  1481  1483  1487  1489  1493  1499  1511  1523  1531 
1543  1549  1553  1559  1567  1571  1579  1583  1597  1601  1607  1609  1613  1619  1621  1627  1637  1657  1663  1667 
1669  1693  1697  1699  1709  1721  1723  1733  1741  1747  1753  1759  1777  1783  1787  1789  1801  1811  1823  1831 
1847  1861  1867  1871  1873  1877  1879  1889  1901  1907  1913  1931  1933  1949  1951  1973  1979  1987  1993  1997 
1999  2003  2011  2017  2027  2029  2039  2053  2063  2069  2081  2083  2087  2089  2099  2111  2113  2129  2131  2137 
2141  2143  2153  2161  2179  2203  2207  2213  2221  2237  2239  2243  2251  2267  2269  2273  2281  2287  2293  2297 
2309  2311  2333  2339  2341  2347  2351  2357  2371  2377  2381  2383  2389  2393  2399  2411  2417  2423  2437  2441 
2447  2459  2467  2473  2477  2503  2521  2531  2539  2543  2549  2551  2557  2579  2591  2593  2609  2617  2621  2633 
2647  2657  2659  2663  2671  2677  2683  2687  2689  2693  2699  2707  2711  2713  2719  2729  2731  2741  2749  2753 
2767  2777  2789  2791  2797  2801  2803  2819  2833  2837  2843  2851  2857  2861  2879  2887  2897  2903  2909  2917 
2927  2939  2953  2957  2963  2969  2971  2999  3001  3011  3019  3023  3037  3041  3049  3061  3067  3079  3083  3089 
3109  3119  3121  3137  3163  3167  3169  3181  3187  3191  3203  3209  3217  3221  3229  3251  3253  3257  3259  3271 
3299  3301  3307  3313  3319  3323  3329  3331  3343  3347  3359  3361  3371  3373  3389  3391  3407  3413  3433  3449 
3457  3461  3463  3467  3469  3491  3499  3511  3517  3527  3529  3533  3539  3541  3547  3557  3559  3571  3581  3583 
Notes

The linear system being analyzed comprises 156 equations in 233 variables. Its rank is 133.
 It's not surprising that for the small prime numbers illustrated in the table the computed dimension is not always correct.
 On the other hand, note that the dimension is often computed correctly even though the residuals suggest that there is a problem.

For large prime numbers the residuals are all nonzero, as one would expect.
[15Mar1999]