Combining Numerical Analysis and Constraint Processing by Means of Controlled Propagation and Redundant Constraints

In principle, interval constraints provide tight enclosures for the solutions of several types of numerical problem. These include constrained global optimization and the solution of nonlinear systems of equalities or inequalities. Interval constraints finds these enclosures by a combination of propagation and search. The challenge is to extend the “in principle” to problems of practical interest. In this paper we describe the concept of controlled propagation. It uses this in conjunction with redundant constraints to combine numerical analysis algorithms with constraint processing. The resulting combination retains the enclosure property of constraint processing in spite of rounding errors. We apply this technique in an algorithm for solving linear algebraic equations that initially simulates interval Gaussian elimination and then proceeds to refine the result with propagation and splitting. Application of our approach to nonlinear equations yields an algorithm with a similar relation to Newton’s method.