We derive a procedure for computing an upper bound on the number of equiangular lines in various Euclidean vector spaces by generalizing the classical pillar decomposition developed by Lemmens and Seidel. Applying our method, we prove new relative bounds for the angle arccos(1/5). Experiments show that our relative bounds for all possible angles are considerably smaller than the known semidefinite programming bounds for a range of larger dimensions. Our computational results also show an explicit bound on the size of a set of equiangular lines in \BbbR r regardless of angle, which is strictly less than the well-known Gerzon's bound if r + 2 is not a square of an odd number．