Our purpose is to compare how much the F5 algorithm can gain in efficiency compared to the F4 algorithm. This can be achieve as the F5 algorithm uses the concept of signatures to foresee potential useless computation which the F4 algorithm might make represented by zero rows in the reduction of a large matrix. We experimentally show that this is a modest increase in efficiency for the parameters we tested.

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Let$L$=−Δ+$V$be a Schrödinger operator on ℝ^{$d$},$d$≥3. We assume that$V$is a nonnegative, compactly supported potential that belongs to$L$^{$p$}(ℝ^{$d$}), for some$p$>$d$$/$2. Let$K$_{$t$}be the semigroup generated by −$L$. We say that an$L$^{1}(ℝ^{$d$})-function$f$belongs to the Hardy space $H^{1}_{L}$ associated with$L$if sup_{$t$>0}|$K$_{$t$}$f$| belongs to$L$^{1}(ℝ^{$d$}). We prove that $f\in H^{1}_{L}$ if and only if$R$_{$j$}$f$∈$L$^{1}(ℝ^{$d$}) for$j$=1,…,$d$, where$R$_{$j$}=($∂$/$∂$$x$_{$j$})$L$^{−1$/$2}are the Riesz transforms associated with$L$.