## Section: Application Domains

Keywords : arithmetic operator, hardware implementation, dedicated circuit, cypher, control, validation, proof, numerical software.

### Application Domains

Our expertise covers application domains for which the quality, such as the efficiency or safety, of the arithmetic operators is an issue. On the one hand, it can be applied to hardware oriented developments, for example to the design of arithmetic primitives which are specifically optimized for the target application and support. On the other hand, it can also be applied to software programs, when numerical reliability issues arise: these issues can consist in improving the numerical stability of an algorithm, computing guaranteed results (either exact results or certified enclosures) or certifying numerical programs.

• Developments in Coqand PVSare used to formally bound values and roundoff errorsfor safety criticalapplications such as flight control. Our automatic tool (see § 5.13 ) checks for overflows and performs forward error analysis with interval arithmetic. It generates all the necessary assessments and proofs related to each variable of a given program. Such technique has been coined as invisible formal methods . Our tool also refers to our growing library of validated properties to enhance the containment intervals.

• Developments of correctly rounded elementary functionsis critical to the reproducibilityof floating-point computations. Exponentials and logarithms, for instance, are routinely used in accounting systems for interest calculation, where roundoff errors have a financial meaning. Our current focus is on bounding the worst-case time for such computations, which is required to allow their use in safety criticalapplications.

• Arbitrary precision interval arithmetic can be used in two ways to validate a numerical result. To quickly check the accuracyof a result, one can replace the floating-point arithmetic of the numerical software that computed this result by high-precision interval arithmetic and measure the width of the interval result: a tight result corresponds to good accuracy. When getting a guaranteed enclosureof the solution is an issue, then more sophisticated procedures, such as those we develop, must be employed: this is the case of global optimization problems.

• The application domains of hardware arithmetic operators are digital signal processing, image processing, embedded applicationsand cryptography.

• The design of faster algorithms for matrix polynomials provides faster solutions to various problems in control theory, especially those involving multivariable linear systems.

Logo Inria