Ten Digit Algorithms

This paper was presented as the A R Mitchell Lecture at the 2005 Dundee Biennial Conference on Numerical Analysis, 27 June 2005

SWATI: Synthesizing Wordlengths Automatically Using Testing and Induction

In this paper, we present an automated technique SWATI: Synthesizing Wordlengths Automatically Using Testing and Induction, which uses a combination of Nelder-Mead optimization based testing, and induction from examples to automatically synthesize optimal fixedpoint implementation of numerical routines. The design of numerical software is commonly done using floating-point arithmetic in design-environments such as Matlab. However, these designs are often implemented using fixed-point arithmetic for speed and efficiency reasons especially in embedded systems. The fixed-point implementation reduces implementation cost, provides better performance, and reduces power consumption. The conversion from floating-point designs to fixed-point code is subject to two opposing constraints: (i) the word-width of fixed-point types must be minimized, and (ii) the outputs of the fixed-point program must be accurate. In this paper, we propose a new solution to this problem. Our technique takes the floating-point program, specified accuracy and an implementation cost model and provides the fixed-point program with specified accuracy and optimal implementation cost. We demonstrate the effectiveness of our approach on a set of examples from the domain of automated control, robotics and digital signal processing

Trusting Computations: a Mechanized Proof from Partial Differential Equations to Actual Program

Computer programs may go wrong due to exceptional behaviors, out-of-bound array accesses, or simply coding errors. Thus, they cannot be blindly trusted. Scientific computing programs make no exception in that respect, and even bring specific accuracy issues due to their massive use of floating-point computations. Yet, it is uncommon to guarantee their correctness. Indeed, we had to extend existing methods and tools for proving the correct behavior of programs to verify an existing numerical analysis program. This C program implements the second-order centered finite difference explicit scheme for solving the 1D wave equation. In fact, we have gone much further as we have mechanically verified the convergence of the numerical scheme in order to get a complete formal proof covering all aspects from partial differential equations to actual numerical results. To the best of our knowledge, this is the first time such a comprehensive proof is achieved.Comment: N° RR-8197 (2012). arXiv admin note: text overlap with arXiv:1112.179