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1.5. SOME MAPLE 27
Definition 18 (Following [BFSS06]) We say that an algorithm producing
output of size N is optimal if its running time is O(N), and almost optimal if
its running time is Õ(N).
Addition of numbers is therefore optimal, and multiplication almost optimal.
However, the reader should be warned that Õ expressions are often far from
the reality experienced in computer algebra, where data are small enough that
the limiting processes in equations (1.8) and (1.9) have not really taken hold
(see note 4 ), or are quantised (in practice integer lengths are measured in words,
not bits, for example).
When it comes to measuring the intrinstic difficulty of a problem, rather
than the efficiency of a particular algorithm, we need lower bounds rather than
the upper bounds implied in (1.8) and (1.9).
Notation 7 (Lower bounds) Consider a problem P , and a given encoding,
e.g. “dense polynomials (Definition 22) with coefficients in binary”. Let N be
the size of a problem instance, and C a particular computing paradigm, and way
of counting operations. If we can prove that there is a c such that any algorithm
solving this problem must take at least cf(N) operations on at least one problem
instance of size N, then we say that this problem has cost at least of the order
of f(n), written
Again “=” really ought to be “∈”.
P C = Ω(f(N)) or loosely P = Ω(f(N)). (1.10)
In some instances (sorting is one of these) we can match upper and lower bounds.
Notation 8 (Θ) If P C = Ω(f(N)) and P C = O(f(N)), then we say that P C is
of order exactly f(N), and write P C = Θ(f(N)).
For example if C is the paradigm in which we only count comparison operations,
sorting N objects is Θ(N log N).
Notation 9 (Further Abuse) We will sometimes abuse these notations further,
and write, say, f(N) = 2 O(N) , which can be understood as either of the
equivalent forms log 2 f(N) = O(N) or ∃C ∈ R, M : ∀N > Mf(N) < 2 CN .
Note that 2 O(N) and O(2 N ) are very different things. 4 N = 2 2N is 2 O(N) but
not O(2 N ).
1.5 Some Maple
1.5.1 The RootOf construct
Note that Maple indexes the reult of RootOf according to the rules at http://
www.maplesoft.com/support/help/Maple/view.aspx?path=RootOf/indexed.