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14 CHAPTER 1. INTRODUCTION
Definition 1 A number n, or more generally any algebraic object, is said to be
transcendental over a ring R if there is no non-zero polynomial p with coefficients
in R such that p(n) = 0.
With this definition, we can say that e π√ 163 is transcendental over the integers.
We will see throughout this book (for an early example, see page 47) that
innocent-seeming problems can give rise to numbers far greater than one would
expect. Hence there is a requirement to deal with numbers larger, or to greater
precision, than is provided by the hardware manufacturers whose chips underlie
our computers. Practically every computer algebra system, therefore, has
a package for manipulating arbitrary-sized integers (so-called bignums) or real
numbers (bigfloats). These arbitrary-sized objects, and the fact that mathematical
objects in general are unpredictably sized, means that computer algebra
systems generally need sophistucated memory management, generally garbage
collection (see [JHM11] for a general discussion of this topic).
But the fact that the systems can deal with large numbers does not mean
that we should let numbers increase without doing anything. If we have two
numbers with n digits, adding them requires a time proportional to n, or in
more formal language (see section 1.4) a time O(n). Multiplying them 4 requires
a time O(n 2 ). Calculating a g.c.d., which is fundamental in the calculation of
rational numbers, requires O(n 3 ), or O(n 2 ) with a bit of care 5 . This implies that
if the numbers become 10 times longer, the time is multiplied by 10, or by 100,
or by 1000. So it is always worth reducing the size of these integers. We will
see later (an early example is on page 113) that much ingenuity has been wellspent
in devising algorithms to compute “obvious” quantities by “non-obvious”
ways which avoid, or reduce, the use of large numbers. The phrase intermediate
expression swell is used to denote the phenomenon where intermediate quantites
are much larger than the input to, or outputs from, a calculation.
Notation 1 We write algorithms in an Algol-like notation, with the Rutishauser
symbol := to indicate assignment, and = (as opposed to C’s ==) to indicate the
equality predicate. We use indentation to indicate grouping 6 , rather than clutter
the text with begin . . . end. Comments are introduced by the # character,
running to the end of the line.
4 In principle, O(n log n log log n) is enough [AHU74, Chapter 8], but no computer algebra
system routinely uses this, for it is more like 20n log n log log n. However, most systems will
use ‘Karatsuba arithmetic’ (see [KO63, and section B.3]), which takes O(n log 2 3 ≈ n 1.585 ),
once the numbers reach an appropriate length, often 16 words [SGV94].
5 In principle, O(n log 2 n log log n) [AHU74, Chapter 8], but again no system uses it routinely,
but this combined with Karatsuba gives O(n log 2 3 log n) (section B.3.6), and this is
commonly used.
6 An idea which was tried in Axiom [JS92], but which turns out to be better in books than
in real life.