The MPFI Library: Towards IEEE 1788-2015 Compliance

International audienceThe IEEE 1788-2015 has standardized interval arithmetic. However, few libraries for interval arithmetic are compliant with this standard. The main features of the IEEE 1788-2015 standard are detailed, namely the structure into 4 levels, the possibility to accomodate a new mathematical theory of interval arithmetic through the notion of flavor, and the mechanism of decoration for handling exceptions. These features were not present in the libraries developed prior to the elaboration of the standard. MPFI is such a library: it is a C library, based on MPFR, for arbitrary precision interval arithmetic. MPFI is not (yet) compliant with the IEEE 1788-2015 standard for interval arithmetic: the planned modifications are presented. Some considerations about performance and HPC on interval computations based on this standard, or on MPFI, conclude the paper

Modal Intervals Revisited Part 2: A Generalized Interval Mean-Value Extension

In Modal Intervals Revisited Part 1, new extensions to generalized intervals (intervals whose bounds are not constrained to be ordered), called AE-extensions, have been defined. They provide the same interpretations as the extensions to modal intervals and therefore enhance the interpretations of the classical interval extensions (for example, both inner and outer approximations of function ranges are in the scope of the AE-extensions). The construction of AE-extensions is similar to the one of classical interval extensions. In particular, a natural AE-extension has been defined from the Kaucher arithmetic which simplified some central results of the modal intervals theory. Starting from this framework, the mean-value AE-extension is now defined. It represents a new way to linearize a real function, which is compatible with both inner and outer approximations of its range. With a quadratic order of convergence for real-valued functions, it allows to overcome some difficulties which were encountered using a preconditioning process together with the natural AE-extensions. Some application examples are finally presented, displaying the application potential of the mean-value AE-extension.

Modal Intervals Revisited Part 2: A Generalized Interval Mean-Value Extension

International audienceIn Modal Intervals Revisited Part 1, new extensions to generalized intervals (intervals whose bounds are not constrained to be ordered), called AE-extensions, have been defined. They provide the same interpretations as the extensions to modal intervals and therefore enhance the interpretations of the classical interval extensions (for example, both inner and outer approximations of function ranges are in the scope of the AE-extensions). The construction of AE-extensions is similar to the one of classical interval extensions. In particular, a natural AE-extension has been defined from the Kaucher arithmetic which simplified some central results of the modal intervals theory. Starting from this framework, the mean-value AE-extension is now defined. It represents a new way to linearize a real function, which is compatible with both inner and outer approximations of its range. With a quadratic order of convergence for real-valued functions, it allows to overcome some difficulties which were encountered using a preconditioning process together with the natural AE-extensions. Some application examples are finally presented, displaying the application potential of the mean-value AE-extension

Modal Intervals Revisited Part 2: A Generalized Interval Mean-Value Extension ∗

In Modal Intervals Revisited Part 1, new extensions to generalized intervals (intervals whose bounds are not constrained to be ordered), called AE-extensions, have been defined. They provide the same interpretations as modal intervals and therefore enhance the interpretations of classical interval extensions (for example, both inner and outer approximations of function ranges are in the scope of AE-extensions). The construction of AE-extensions is similar to the cnstruction of classical interval extensions. In particular, a natural AE-extension has been defined from Kaucher arithmetic which simplified some central results of modal interval theory. Starting from this framework, the mean-value AE-extension is now defined. It represents a new way to linearize a real function, which is compatible with both inner and outer approximations of its range. With a quadratic order of convergence for real-valued functions, it allows one to overcome some difficulties which were encountered using a preconditioning process together with the natural AE-extensions. Some application examples are finally presented, displaying the application potential of the mean-value AE-extension