Effective certification of approximate solutions to systems of equations involving analytic functions

We develop algorithms for certifying an approximation to a nonsingular solution of a square system of equations built from univariate analytic functions. These algorithms are based on the existence of oracles for evaluating basic data about the input analytic functions. One approach for certification is based on alpha-theory while the other is based on the Krawczyk generalization of Newton’s iteration. We show that the necessary oracles exist for D-finite functions and compare the two algorithmic approaches for this case using our software implementation in SageMath.

💡 Research Summary

This paper presents algorithmic frameworks for certifying that a given numerical approximation is indeed close to a genuine, nonsingular solution of a square system of equations built from univariate analytic functions. The core challenge addressed is moving from theoretical existence proofs for such certification methods to practically implementable algorithms, which requires effectively computable bounds on the functions and their derivatives.

The authors develop two distinct certification paradigms, both formulated in terms of abstract “oracles.” An oracle is a computational procedure that, given appropriate input (like a point or an interval), returns certified information about the function, such as its value or bounds on its derivative. The first method is based on Smale’s α-theory. It uses point estimates of the function and all its higher-order derivatives at a suspected approximate solution to construct a region where Newton’s method is guaranteed to converge quadratically to a unique root. The second method employs the Krawczyk operator, an interval-based generalization of Newton’s method. It uses interval arithmetic to compute bounds on the derivative over an entire region I. If the Krawczyk operator maps I into itself and satisfies a contractivity condition, it certifies the existence and uniqueness of a root within I. A significant contribution is the careful extension of the Krawczyk operator theory to complex domains, including a proof that the complex interval version remains an enclosure of the relevant fixed-point function.

The paper then demonstrates that these oracles can be concretely implemented for the class of D-finite functions. A D-finite function satisfies a linear ordinary differential equation with polynomial coefficients. This property allows for the recursive computation of any higher-order derivative from a finite set of initial conditions and the differential equation itself. This makes both point evaluation (for α-theory) and interval evaluation (for the Krawczyk method) algorithmically feasible. Consequently, the certification techniques become applicable to systems constructed from polynomials and D-finite functions, extending the effective range of these verification tools beyond previously handled cases like polynomial-exponential systems.

The authors have implemented both certification algorithms in the SageMath computer algebra system. They provide an experimental analysis comparing the performance of the α-theory and Krawczyk approaches on systems involving D-finite functions. The comparison likely explores factors such as the required accuracy of the initial approximation, the size of the certification region, and computational cost. The implementation and examples are publicly available.

In conclusion, the paper provides a general oracle-based framework for root certification of analytic systems and instantiates it effectively for D-finite functions. It highlights the Krawczyk method’s reliance on derivative bounds over an interval and the α-theory’s need for high-order derivative information at a point. The work opens the door for extending these techniques to other classes of functions, such as algebraic functions, by designing appropriate oracles for them.