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:

• Red:

  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.

• Purple:

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

• Cyan:

  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.

• Green:

  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.

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.