Efficient symbolic differentiation for graphics applications

Functions with densely interconnected expression graphs, which arise in computer graphics applications such as dynamics, space-time optimization, and PRT, can be difficult to efficiently differentiate using existing symbolic or automatic differentiation techniques. Our new algorithm, D*, computes efficient symbolic derivatives for these functions by symbolically executing the expression graph at compile time to eliminate common subexpressions and by exploiting the special nature of the graph that represents the derivative of a function. This graph has a sum of products form; the new algorithm computes a factorization of this derivative graph along with an efficient grouping of product terms into subexpressions. For the problems in our test suite D* generates symbolic derivatives which are up to 4.6 x 10(3) times faster than those com uted by the symbolic math program Mathematica and up to 2.2 x 10(5) times faster than the non-symbolic automatic differentiation program CppAD. In some cases the D* derivatives rival the best manually derived solutions.