  Cylindrical algebraic decomposition (CAD) is an important tool for the
investigation of semi-algebraic sets. Originally introduced by Collins in the
1970s for use in quantifier elimination it has since found numerous
applications within algebraic geometry and beyond. Following from his original
work in 1988, McCallum presented an improved algorithm, CADW, which offered a
huge increase in the practical utility of CAD. In 2009 a team based at the
University of Western Ontario presented a new and quite separate algorithm for
CAD, which was implemented and included in the computer algebra system Maple.
As part of a wider project at Bath investigating CAD and its applications,
Collins and McCallum's CAD algorithms have been implemented in Maple. This
report details these implementations and compares them to Qepcad and the
Ontario algorithm.
  The implementations were originally undertaken to facilitate research into
the connections between the algorithms. However, the ability of the code to
guarantee order-invariant output has led to its use in new research on CADs
which are minimal for certain problems. In addition, the implementation
described here is of interest as the only full implementation of CADW, (since
Qepcad does not currently make use of McCallum's delineating polynomials), and
hence can solve problems not admissible to other CAD implementations.
