Architecture-aware classical Taylor shift by 1

Abstract

Abstract. We present algorithms that outperform straightforward implementations of classical Taylor shift by 1. For input polynomials of low degrees a method of the SACLIB library is faster than straightforward implementations by a factor of at least 2; for higher degrees we develop a method that is faster than straightforward implementations by a factor of up to 7. Our Taylor shift algorithm requires more word additions than straightforward implementations but it reduces the number of cycles per word addition by reducing memory trac and the number of carry computations. The introduction of signed digits, suspended normalization, radix reduction, and delayed carry propagation enables our algorithm to take advantage of the technique of register tiling which is commonly used by optimizing compilers. While our algorithm is written in a high-level language, it depends on several parameters that can be tuned to the underlying architecture.

Cache complexity and multicore implementation for univariate real root isolation

“…We have tested our software on well-known benchmark examples, including the Bnd and Cnd polynomials from [11], the Chebychev and Mignotte polynomials from [5], as well as randomly generated polynomials. All the benchmark runs are carried out on an 8-core machine with 8 GB memory and 6 MB of L2 cache.…”

“…This paper is an extension of the two-page abstract for our poster [2] presented in ISSAC 2010. Before this work, several other pioneer works [3,5,10,11] have also been conducted on this topic with different techniques. The paper [3] reports on a first attempt on parallelizing real root isolation problem.…”

“…Their work shows that the amount of parallelism is limited if only the traversal of the binary tree associated with the isolation problem is parallelized. Later works [5,10,11] focus on improving Taylor shift via high performance computing techniques. The paper [5] proposes new scheduling algorithms for the pyramid DAG associated with Taylor shifts and reduces the communication complexity overhead to linear.…”

“…The paper [5] proposes new scheduling algorithms for the pyramid DAG associated with Taylor shifts and reduces the communication complexity overhead to linear. The paper [10,11] improves the efficiency of Taylor shift computation and real root isolation based on a technique of register tiling. This paper is organized as follows.…”

Implementation techniques for fast polynomial arithmetic in a high-level programming environment

“…Second, compiled and optimized high-level code may not take advantage of some hardware features. If writing architecture-aware code can be done in C [20], this remains a challenge in a non-imperative language like Lisp. Thus, in our second high-level programming environment, namely AXIOM, we take advantage of every component of the system, by mixing low-level code (C and assembly code), middle-level code (Lisp) and highlevel code in the AXIOM language.…”