Extended precision software packages

A description of three extended precision packages is presented along with three small conversion subroutines which can be used in conjunction with the extended precision packages. These extended packages represent software packages written in FORTRAN 4. They contain normalized or unnormalized floating point arithmetic with symmetric rounding and arbitrary mantissa lengths, and normalized floating point interval arithmetic with appropriate rounding. The purpose of an extended precision package is to enable the user to use and manipulate numbers with large decimal places as well as those with small decimal places where precision beyond double precision is required

Invariant Generation for Multi-Path Loops with Polynomial Assignments

Program analysis requires the generation of program properties expressing conditions to hold at intermediate program locations. When it comes to programs with loops, these properties are typically expressed as loop invariants. In this paper we study a class of multi-path program loops with numeric variables, in particular nested loops with conditionals, where assignments to program variables are polynomial expressions over program variables. We call this class of loops extended P-solvable and introduce an algorithm for generating all polynomial invariants of such loops. By an iterative procedure employing Gr\"obner basis computation, our approach computes the polynomial ideal of the polynomial invariants of each program path and combines these ideals sequentially until a fixed point is reached. This fixed point represents the polynomial ideal of all polynomial invariants of the given extended P-solvable loop. We prove termination of our method and show that the maximal number of iterations for reaching the fixed point depends linearly on the number of program variables and the number of inner loops. In particular, for a loop with m program variables and r conditional branches we prove an upper bound of m*r iterations. We implemented our approach in the Aligator software package. Furthermore, we evaluated it on 18 programs with polynomial arithmetic and compared it to existing methods in invariant generation. The results show the efficiency of our approach

Basic mathematical function libraries for scientific computation

Ada packages implementing selected mathematical functions for the support of scientific and engineering applications were written. The packages provide the Ada programmer with the mathematical function support found in the languages Pascal and FORTRAN as well as an extended precision arithmetic and a complete complex arithmetic. The algorithms used are fully described and analyzed. Implementation assumes that the Ada type FLOAT objects fully conform to the IEEE 754-1985 standard for single binary floating-point arithmetic, and that INTEGER objects are 32-bit entities. Codes for the Ada packages are included as appendixes

Extended Model Formulas in R. Multiple Parts and Multiple Responses.

Model formulas are the standard approach for specifying the variables in statistical models in the S language. Although being eminently useful in an extremely wide class of applications, they have certain limitations including being confined to single responses and not providing convenient support for processing formulas with multiple parts. The latter is relevant for models with two or more sets of variable, e.g., regressors/instruments in instrumental variable regressions, two-part models such as hurdle models, or alternative-specific and individual-specific variables in choice models among many others. The R package Formula addresses these two problems by providing a new class "Formula" (inheriting from "formula") that accepts an additional formula operator | separating multiple parts and by allowing all formula operators (including the new |) on the left-hand side to support multiple responses.Series: Research Report Series / Department of Statistics and Mathematic

A Subdivision Solver for Systems of Large Dense Polynomials

We describe here the package {\tt subdivision\\_solver} for the mathematical software {\tt SageMath}. It provides a solver on real numbers for square systems of large dense polynomials. By large polynomials we mean multivariate polynomials with large degrees, which coefficients have large bit-size. While staying robust, symbolic approaches to solve systems of polynomials see their performances dramatically affected by high degree and bit-size of input polynomials.Available numeric approaches suffer from the cost of the evaluation of large polynomials and their derivatives.Our solver is based on interval analysis and bisections of an initial compact domain of $\R^n$ where solutions are sought. Evaluations on intervals with Horner scheme is performed by the package {\tt fast\\_polynomial} for {\tt SageMath}.The non-existence of a solution within a box is certified by an evaluation scheme that uses a Taylor expansion at order 2, and existence and uniqueness of a solution within a box is certified with krawczyk operator.The precision of the working arithmetic is adapted on the fly during the subdivision process and we present a new heuristic criterion to decide if the arithmetic precision has to be increased

Making big steps in trajectories

We consider the solution of initial value problems within the context of hybrid systems and emphasise the use of high precision approximations (in software for exact real arithmetic). We propose a novel algorithm for the computation of trajectories up to the area where discontinuous jumps appear, applicable for holomorphic flow functions. Examples with a prototypical implementation illustrate that the algorithm might provide results with higher precision than well-known ODE solvers at a similar computation time