Function maxpair_spray() finds the pairwise maximum for two
sprays. Specifically, if S3 <- maxpair_spray(S1,S2), then
S3[v] == max(S1[v],S2[v]) for every index vector v.
Function pmax.spray() is the method for the generic
pmax(), which takes any number of arguments. If S3 <-
maxpair_spray(S1,S2,...), then S3[v] == max(S1[v],S2[v],...) for
every index vector v.
Function pmax.spray() operates right-associatively:
pmax(S1,S2,S3,S4) == f(S1,f(S2,f(S3,S4))) where f() is
short for maxpair_spray(). So if performance is important, put
the smallest spray (in terms of number of nonzero entries) last.
In these functions, a scalar is interpreted as a sort of global maximum.
Thus if S3 <- pmax(S,x) we have S3[v] == max(S[v],x) for
every index v. Observe that this operation is not defined if
x>0, for then there would be an infinity of v for which
S3[v] != 0, an impossibility (or at least counter to the
principles of a sparse array). Note also that x cannot have
length >1 as the elements of a spray object are stored in an
arbitrary order.
Functions minpair_spray() and pmin.spray() are analogous.
Note that minpair_spray(S1,S2) is algebraically equivalent to
-pmax_spray(-S1,-S2); see the examples.
The value of pmax(S) is problematic. Suppose
all(value(S)<0); the current implementation returns
pmax(S)==S but there is a case for returning the null polynomial.