Bounding roundoff errors and ranges

Many error analysis techniques are well known for obtaining a priori bounds on the global roundoff error generated by a floating-point program. Such error bounds can already be computed automatically with Gappa in the case of straight-line programs, assuming the precision of every arithmetic operation is fixed and known in advance. One of our next challenges will be to handle a priori error bounds in algorithms where the computing precision varies with each operation (this is the case in operators designed for FPGAs, or for some MPFR code for instance), and for programs involving loops of variable length.

On the other hand, techniques are also available to compute rigorous a posteriori error bounds. Interval arithmetic is probably the best established technique for this, but certified a posteriori error bounds can also be computed, at run-time, using error-free transformations and, more generally, the specifications of the IEEE-754 floating-point arithmetic. We plan to investigate and compare these two approaches.

Interval methods can also be used more generally for computing a rigorous enclosure for the range of a function on a given domain. This corresponds to the case where input data vary in a set. One can then deduce properties such as the occurrence of overflows, or the sign of the result. We will work on the automatic detection of such properties.

Higher order techniques: Taylor models, Chebyshev models

The team started developing Chebyshev models, an improvement of Taylor models which replaces Taylor approximation with Chebyshev interpolant approximation or Chebyshev truncated series approximation. The main advantage of these models is that they offer better convergence properties, leading to smaller remainders and converging on more flexible domains. We will investigate applications of Taylor and Chebyshev models to classical issues in rigorous computing, such as global optimization, certified quadrature or rigorous solving of ordinary differential equations. We also need to address the challenges of Taylor and Chebyshev models when implemented using floating-point arithmetic: combining high accuracy with the performance of hardware-supported arithmetic, possibly using error-free transformations.

Formal proof

The methods mentioned so far certify the quality of the result ...  up to a bug in their implementation. To get a higher degree of confidence, the certification should be checked by a theorem prover such as Coq. It requires that the chosen arithmetic or method is implemented and proven within the theorem prover. This is not yet the case of symbolic-numeric computation of error bounds in the case or variable precision, nor of higher order variants of interval arithmetic. We are working on these formalizations with partners from the formal proof community.

Standardization

An ongoing work is the standardization of interval arithmetic, by the IEEE 1788 working group. We will also continue to participate to the C$++$ standardization, regarding the inclusion of interval arithmetic in the STL.