## Section: Application Domains

### 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.

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

• Developments of correctly rounded elementary functions is critical to the reproducibility of 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 critical applications, and in proving the correct rounding property for a complete implementation.

• Certifying a numerical application usually requires bounds on rounding errors and ranges of variables. Our automatic tool (see § 5.6 ) computes or verifies such bounds. For increased confidence in the numerical applications, it also generates formal proofs of the arithmetic properties. These proofs can then be used and machine-checked by proof assistants like Coq.

• Arbitrary precision interval arithmetic can be used in two ways to validate a numerical result. To quickly check the accuracy of 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 enclosure of 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 design of faster algorithms for matrix polynomials provides faster solutions to various problems in control theory, especially those involving multivariable linear systems.

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