  In well-known work, Kazhdan and Lusztig (1979) defined a new set of Hecke
algebra basis elements (actually two such sets) associated to elements in any
Coxeter group. Often these basis elements are computed by a standard recursive
algorithm which, for Coxeter group elements of long length, generally involves
computing most basis elements corresponding to Coxeter group elements of
smaller length. Thus, many calculations simply compute all basis elements
associated to a given length or less, even if the interest is in a specific
Kazhdan-Lusztig basis element. Similar remarks apply to "parabolic" versions of
these basis elements defined later by Deodhar (1987,1990), though the lengths
involved are the (smaller) lengths of distinguished coset representatives. We
give an algorithm which targets any given Kazhdan-Lusztig basis element or
parabolic analog and does not precompute any other Kazhdan-Lusztig basis
elements. In particular it does not have to store them. This results in a
considerable saving in memory usage, enabling new calculations in an important
case (for finite and algebraic group 1-cohomology with irreducible
coefficients) analyzed by Scott-Xi (2010).
