Work and storage counts for an M-node SDC sweep

With $K=0$, the algorithm evaluates and stores the $M$ predictor derivatives, then returns iteration-budget-exhausted without an accepted endpoint. If the run stops after $s>0$ completed sweeps with no incomplete sweep, each adds exactly $M$ evaluations because its $M$ new stage states are evaluated once and the previous $F(U^j)$ values are reused. Thus the total is $M+sM=M(s+1)$, and for $s\le K$ this is at most $M(K+1)$, attained if all $K$ sweeps complete. A failed run with a partially evaluated sweep is counted by its actual prefix as in [F2], and still obeys that bound. Acceptance after $s\ge 1$ sweeps returns $U^{s-1}$ and uses its cached endpoint derivatives with no additional calls.

The stored dense matrices $Q,Q_{\mathrm{\Delta }}$ each have $M^2$ real entries, so their storage is $O(M^2)$; retaining old/current stage vectors and their derivative values takes $O(Md)$ additional scalar storage. Each forward-recursion stage has two sums with at most $2M$ vector terms in total, for at most $O(M^{2}d)$ scalar arithmetic per sweep. Residual formation adds $O(Md)$ scalar arithmetic and at most $M$ square roots; an accepted endpoint adds $O(Md)$ arithmetic and no new field evaluations. These are the proved counts in [F2]. They omit the cost of evaluating $f$, constructing the polynomial quadrature data (which depends on their representation), supplied predictor generation, and external admissibility/invariance certification and their internal storage. The model uses exact real arithmetic and gives no floating-point roundoff guarantee.