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Flows across infinitesimals: the state a loop stepped by one infinitesimal leaves when its affine update holds values of others.
Rates of larger infinitesimals. N = T/s + δ steps of x ↦ A·x, A = I + Σ_{k≥1} u^k·A_k with
u = s^(1/q) and every A_k a matrix of values of the infinitesimals larger than s, leave
exp(N·log A)·x₀ — the recurrences of one infinitesimal’s flow (crate::germ) over those values:
log A = Σ u^k·L_k, the exponent C_0 + Σ_{j≥1} u^j·C_j with C_j = T·L_{q+j} + δ·L_j, C_0 = T·A_q,
and with commuting terms exp(Σ u^j·C_j) = Σ u^m·E_m, E_m = (1/m)·Σ k·C_k·E_{m−k}. The standard
part relative to the larger infinitesimals is e^{T·A_q}·x₀, the inner limit. Without commuting
terms, the orders below the first nonzero C_j are known, and the rest refused by name.
The exponential of a matrix of non-standard rates (matrix_exp) is read one infinitesimal at a
time: with A(ε) = Σ ε^j·B_j in the smallest infinitesimal its entries hold, the block upper-
triangular Toeplitz matrix of B_0, …, B_m has as exponential the Toeplitz matrix of the first
m + 1 coefficients of e^{T·A(ε)} (Van Loan), so each coefficient is the corner block of the
exponential of a matrix of values of larger infinitesimals — down to rational matrices, exactly.
Rates of smaller infinitesimals. N steps of A = I + Σ_{k≥1} w^k·A_k, w a root of an
infinitesimal smaller than s and the A_k values of s and the larger ones, leave
exp(N·log A)·x₀, the series Σ_m w^m·P_m with P_m = Σ_{j ≤ m} (N^j/j!)·[(log A)^j]_m: every
coefficient exact, N an ordinary number from the smaller infinitesimal’s standpoint. A scalar
update not the identity at order 0 is exp(N·ln A), the germs’ own exponential and logarithm.