much smaller than the set of all computed quantities; and ditto for the set of "live" symbols . In this situation, the matrix and vector implementations are too wasteful of time and space. In such situations, one should use a sparse implementation. Namely, each affine form is stored as a list of pairs (j,), containing only the terms with non-zero coefficient . For efficiency, the terms should be sorted in order of j. This representation makes the AA operations somewhat more complicated; however, the cost of each operation becomes proportional to the number of nonzero terms appearing in the operands, instead of the number of total symbols used so far. This is the representation used by LibAffa. References External links Stolfi's page on AA. LibAffa, an LGPL implementation of affine arithmetic. ASOL, a branch-and-prune method to find all solutions to systems of nonlinear equations using affine arithmetic YalAA, an object-oriented C++ based template library for affine arithmetic (AA). (C++ library which can use affine arithmetic) Numerical analysis Affine geometry