Residual Arithmetic, r=3, d=10
The following table illustrates the performance of residual arithmetic.
The dimension of the spline space on the generic double Clough-Tocher
split, for r=3 and d=10 is 184. 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 non-zero 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 non-zero residuals. The
program recognizes this as a potential problem. Numbers with mixed
residuals are considered non-zero, but they cannot serve as pivots.
-
The dimension is computed correctly and all non-zero entries in the linear system have three non-zero 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 |
5003 | 5009 | 5011 | 5021 | 5023 | 5039 | 5051 | 5059 | 5077 | 5081 | 5087 | 5099 | 5101 | 5107 | 5113 | 5119 | 5147 | 5153 | 5167 | 5171 |
10007 | 10009 | 10037 | 10039 | 10061 | 10067 | 10069 | 10079 | 10091 | 10093 | 10099 | 10103 | 10111 | 10133 | 10139 | 10141 | 10151 | 10159 | 10163 | 10169 |
20011 | 20021 | 20023 | 20029 | 20047 | 20051 | 20063 | 20071 | 20089 | 20101 | 20107 | 20113 | 20117 | 20123 | 20129 | 20143 | 20147 | 20149 | 20161 | 20173 |
40009 | 40013 | 40031 | 40037 | 40039 | 40063 | 40087 | 40093 | 40099 | 40111 | 40123 | 40127 | 40129 | 40151 | 40153 | 40163 | 40169 | 40177 | 40189 | 40193 |
80021 | 80039 | 80051 | 80071 | 80077 | 80107 | 80111 | 80141 | 80147 | 80149 | 80153 | 80167 | 80173 | 80177 | 80191 | 80207 | 80209 | 80221 | 80231 | 80233 |
160001 | 160009 | 160019 | 160031 | 160033 | 160049 | 160073 | 160079 | 160081 | 160087 | 160091 | 160093 | 160117 | 160141 | 160159 | 160163 | 160169 | 160183 | 160201 | 160207 |
320009 | 320011 | 320027 | 320039 | 320041 | 320053 | 320057 | 320063 | 320081 | 320083 | 320101 | 320107 | 320113 | 320119 | 320141 | 320143 | 320149 | 320153 | 320179 | 320209 |
640007 | 640009 | 640019 | 640027 | 640039 | 640043 | 640049 | 640061 | 640069 | 640099 | 640109 | 640121 | 640127 | 640139 | 640151 | 640153 | 640163 | 640193 | 640219 | 640223 |
1000003 | 1000033 | 1000037 | 1000039 | 1000081 | 1000099 | 1000117 | 1000121 | 1000133 | 1000151 |
1000159 | 1000171 | 1000183 | 1000187 | 1000193 | 1000199 | 1000211 | 1000213 | 1000231 | 1000249 |
Notes
- The Table contains all primes through 3449, and selected ranges of larger primes.
-
The linear system being analyzed comprises 324 equations in 448 variables. Its rank is 282.
- 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 non-zero, as one would expect.
- It is remarkable that the primes for which the dimension is overestimated occur in blocks.
[15-Mar-1999]