Coefficient Storage

TaylorExpansion<T, Scheme> keeps its coefficients in a single dense std::array. The Scheme parameter encodes the monomial index set (see below); TE<N, M> is the shorthand for the isotropic case TaylorExpansion<double, IsotropicScheme<N,M>>.

template <typename T, typename Scheme>   // Scheme = IsotropicScheme<N,M> for TE<N,M>
class TaylorExpansion;

• Storage: std::array<T, C(N+M, M)> — contiguous, stack-resident, no heap allocation, constexpr on every accessor.
• Coefficient order: graded-lexicographic flat index.
• All kernels iterate flat indices directly, including the unrolled (TAX_USE_UNROLL) and precomputed stencil (TAX_USE_STENCIL) Cauchy paths.
• Comparison operators, +=/-=/*=//=, in-place updates: all defined.

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The shape

\[ S = \binom{N + M}{M} \]

is the number of coefficients each expansion carries.

Reference table:

\(N \backslash M\)	1	2	3	4	5	6
1	2	3	4	5	6	7
3	4	10	20	35	56	84
5	6	21	56	126	252	462
8	9	45	165	495	1287	3003
10	11	66	286	1001	3003	8008

\(S\) grows quickly in both \(N\) and \(M\): for \(N \ge 8\) and \(M \ge 4\) an expansion starts to push the stack frame budget, so keep the truncation order and the number of live variables as low as the problem allows. Mixed-order axes are the tool for that — give each axis only the order it needs instead of paying the isotropic worst case (see Named & Mixed-Order Expansions).

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Numerical agreement

The generic loop, unrolled and stencil Cauchy paths implement the same recurrence. Round-off differences on the order of \(10^{-12}\) (double precision) are expected from ordering effects in the Cauchy sum; the agreement is tested by the test_cauchy_unroll_diff and test_cauchy_stencil_diff suites.