Abstract. The Basic Polynomial Algebra Subprograms (BPAS) provides arithmetic operations (multiplication, division, root isolation, etc.) for univariate and multivariate polynomials over prime fields or with integer coefficients. The code is mainly written in CilkPlus [10] targeting multicore processors. The current distribution focuses on dense polynomials and the sparse case is work in progress. A strong emphasis is put on adaptive algorithms as the library aims at supporting a wide variety of situations in terms of problem sizes and available computing resources. One of the purposes of the BPAS project is to take advantage of hardware accelerators in the development of polynomial systems solvers. The BPAS library is publicly available in source at www.bpaslib.org.

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We show how to compute the optimal relative backward error for the numerical solution of the Dahlquist test problem by one-step methods. This is an example of a general approach that uses results from optimal control theory to compute optimal residuals, but elementary methods can also be used here because the problem is so simple. This analysis produces new insight into the numerical solution of stiff problems.Keywords stiff IVP · backward error · residual · optimal control Mathematics Subject Classification (2010) 65L04 · 65L05 · 65L20 · 49M05 IntroductionThe study of stiff differential equations and their efficient numerical solution is by now a mature field. There are several, perhaps many, efficient practical methods with freely available high quality implementations. The literature on the theory of such methods is extensive. Surprisingly, it is not yet complete: for instance, see the survey [24], which has the intriguing title Stiffness 1952-2012: Sixty years in search of a definition. That paper re-examines the fundamentals and thoroughly surveys the literature, and proposes a new stiffness indicator that they claim is useful both a priori for indicating stiffness and a posteriori for indicating varying regions of the solution that stiffness is important. This paper takes a different approach, that of optimal residuals, i.e., backward error, and uses it on the Dahlquist test problem to generate some new observations about this, the simplest of all stiff problems. Indeed, [24] calls this problem "simplistic" and with good reason, but surprisingly it still has things to teach us.Trying to study stiff problems from the point of view of backward error analysis is itself not new. For instance, there is the PhD thesis of W.L. Seward and the paper [8]. But there is an intrinsic dissonance: a stiff problem has the feature that errors are (often sharply) damped as the integration proceeds forward in time, and thus it is not obvious why one might prefer

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We provide a comprehensive presentation of algorithms, data structures, and implementation techniques for high-performance sparse multivariate polynomial arithmetic over the integers and rational numbers as implemented in the freely available Basic Polynomial Algebra Subprograms (BPAS) library. We report on an algorithm for sparse pseudo-division, based on the algorithms for division with remainder, multiplication, and addition, which are also examined herein. The pseudo-division and division with remainder operations are extended to multi-divisor pseudo-division and normal form algorithms, respectively, where the divisor set is assumed to form a triangular set. Our operations make use of two data structures for sparse distributed polynomials and sparse recursively viewed polynomials, with a keen focus on locality and memory usage for optimized performance on modern memory hierarchies. Experimentation shows that these new implementations compare favorably against competing implementations, performing between a factor of 3 better (for multiplication over the integers) to more than 4 orders of magnitude better (for pseudo-division with respect to a triangular set).