Toward high-performance computer algebra with Maple

One of the main successes of the computer algebra community in the last 30 years has been the discovery of algorithms, called modular methods, that allow to keep the swell of the intermediate expressions under control. Without these methods, many applications of computer algebra would not be possible and the impact of computer algebra in scientific computing would be severely limited. Amongst the computer algebra systems which have emerged in the 70’s and 80’s, Maple and its developers have played an essential role in this area. Another major advance in symbolic computation is the development of implementation techniques for asymptotically fast (FFT-based) polynomial arithmetic. Computer algebra systems and libraries initiated in the 90’s, such as Magma and NTL, have been key actors in this effort: they increased in a spectacular manner the range of problems solvable by computer algebra systems. In this report, we present modpn, a Maple library dedicated to fast arithmetic for multivariate polynomials. The main objective of modpn is to provide highly efficient routines for supporting the implementation of modular methods in Maple. We demonstrate in this work that modpn allows us to re-implement core operations in Maple bringing huge performance increases and offering to Maple the ability of solving problems which were previously out of reach. 1 The modpn library: Bringing Fast Polynomial Arithmetic into Maple With the need for high performance in practical problems, research in symbolic computation has entered a new era where implementation techniques have become as important as theoretical developments. This phenomenon is heavily accentuated by the revival and the democratization of parallel architectures. The previous works in parallel symbolic computing are obsolete due to the emerging programing paradigms and architectures; they had also a limited impact and, the attention of the symbolic computation community was withdrawn from parallelism during the last decade. One of the themes of our MITACS project Mathematics of Computer Algebra and Analysis (MOCAA) is to deliver the mathematical algorithms, implementation techniques and software for symbolic computation to exploit these new computing resources, from SMP to multi-core laptops. In previous work [2, 7, 9] we have investigated the integration of asymptotically fast arithmetic operations into the computer algebra system AXIOM. Since AXIOM is based today on GNU Common Lisp (GCL), we took the following approach. We realized highly optimized implementations of our fast routines in C and made them available to the AXIOM programming environment through the kernel of GCL. Therefore, library functions written in the AXIOM high-level language could be compiled down to binary code and then linked against our C code. To observe significant speed-up factors, it was sufficient to enhance existing AXIOM polynomial types with our fast routines (for univariate multiplication, division, GCD etc.) and call them in existing generic packages (for instance, for univariate square-free factorization). See [9] for details. In the present report, we investigate the integration of fast arithmetic operations implemented in C into Maple. Most of Maple library functions are high-level interpreted code. This is the case for those of the