The maximum computing time of the continued fractions method for polynomial real root isolation is at least quintic in the degree of the input polynomial. This computing time is realized for an infinite sequence of polynomials of increasing degrees, each having the same coefficients. The recursion trees for those polynomials do not depend on the use of root bounds in the continued fractions method. The trees are completely described. The height of each tree is more than half the degree. When the degree exceeds one hundred, more than one third of the nodes along the longest path are associated with primitive polynomials whose low-order and high-order coefficients are large negative integers. The length of the forty-five percent highest order coefficients and of the ten percent lowest order coefficients is at least linear in the degree of the input polynomial multiplied by the level of the node. Hence the time required to compute one node from the previous node using classical methods is at least proportional to the cube of the degree of the input polynomial multiplied by the level of the node. The intervals that the continued fractions method returns are characterized using a matrix factorization algorithm.

different weights and the sets of initial Hensel factors are the same, then the sets of resulting Hensel factors are also the same. Theorem 2: in the case that Newton polynomials are different, there is a procedure to convert one set of Hensel factors to another set, so long as there is a one-to-one correspondence among the sets of initial Hensel factors. Theorem 3: all the different sets of Hensel factors are classified by a fan in the weight space. In deriving the last theorem, we referred to the theory of polytope [Zig95]. These theorems lead us to a concept of “Hensel fan", just as the concept of Gröbner fan is introduced for classifying Gröbner bases w.r.t. different term ordering [MR88],[Stu95].

We give a new bound for the number of recursive subdivisions in the Descartes method for polynomial real root isolation. Our proof uses Ostrowski's theory of normal power series from 1950 which has so far been overlooked in the literature. We combine Ostrowski's results with a theorem of Davenport from 1985 to obtain our bound. We also characterize normality of cubic polynomials by explicit conditions on their roots and derive a generalization of one of Ostrowski's theorems. The poster is based on a paper that is to appear in the Journal of Symbolic Computation [1]. In addition to the results of the paper the poster presents facsimiles of pertinent mathematical works in French, German, and English that span a period of 400 years. We use color-coding to relate the historical results to our theory.

The Descartes method for polynomial real root isolation can be performed with respect to monomial bases and with respect to Bernstein bases. The first variant uses Taylor shift by 1 as its main subalgorithm, the second uses de Casteljau's algorithm. When applied to integer polynomials, the two variants have co-dominant, almost tight computing time bounds. Implementations of either variant can obtain speed-ups over previous state-of-the-art implementations by more than an order of magnitude if they use features of the processor architecture. We present an implementation of the Bernstein-bases variant of the Descartes method that automatically generates architecture-aware high-level code and leaves further optimizations to the compiler. We compare the performance of our implementation, algorithmically tuned implementations of the monomial and Bernstein variants, and architecture-unaware implementations of both variants on four different processor architectures and for three classes of input polynomials.

The Descartes method is an algorithm for isolating the real roots of square-free polynomials with real coefficients. We assume that coefficients are given as (potentially infinite) bit-streams. In other words, coefficients can be approximated to any desired accuracy, but are not known exactly. We show that a variant of the Descartes algorithm can cope with bit-stream coefficients. To isolate the real roots of a square-free real polynomial $q(x)=q_{n^{x^{n}}}+...+q_{0}$ with root separation ρ, coefficients |q n | ≥ 1 and $|q_{i}|\leq 2^{\tau}$ , it needs coefficient approximations to O(n(log(1/ρ) + τ)) bits after the binary point and has an expected cost of O(n 4 (log(1/ρ) + τ)2) bit operations.

We present a memory management subsystem for the computer algebra library SACLIB that removes the potential for memory leaks or double deletes in applications using the system. The system encapsulates the management of arrays that are allocated on the heap or on the system stack. The system makes arrays responsible for their own memory management, and enables the compiler to prevent other parts of SACLIB from managing array memory. To prove that our memory module and all applications using it are leak free and double delete free we introduce a new iterator concept and implement a model of that concept using generic programming techniques such as template meta-programming. Our innovations reduce the amount of code responsible for array memory management from 10,000 lines of code to 2,000 lines of code. Using hardware performance counters we show optimizing compilers are capable of avoiding any runtime overhead.

We demonstrate a software system that runs on smartphones and allows two or more geographically dispersed participants to collaborate on the solution of mathematical problems. We show how participants can create and join a collaborative session, how they can use a virtual whiteboard for the exchange of geometrical drawings and of mathematical formulas, and how they can communicate using text messages. We demonstrate the turn-management mechanism provided by our system, and we show how the system facilitates cross-referencing within the session.

The Wireless Internet Collaboration System allows wireless smartphones to be used for mathematics communication, that is, for the creation and exchange of mathematical formulas, diagrams, and text between two or more participants. The system solves the problem of providing the expressive power of mathematical notation on devices with limited input, output, and processing capabilities. The system also provides a turn-management protocol that supports mathematical dialogue between two or more participants who are geographically dispersed and thus cannot use gestures or glances for communication. The architecture of the system is extensible and can be adapted to heterogeneous and evolving hardware platforms.

We present algorithms that outperform straightforward implementations of classical Taylor shift by 1. For input poly-nomials 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 methods but it reduces the number of cycles per word addition by reducing memory traffic 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.

Recent proliferation of cell phone and mobile computing technologies has yielded an opportunity to design software systems intended to facilitate collaboration and teamwork among geographically dispersed users. On-the-go collaboration on portable computers, however, cannot be facilitated with traditional Computer Mediated Collaborative Systems (CMCS)s [1] typically deployed on desktop computers. In the last couple of years, several architectures and software systems have been designed with the mobile platform in mind. The Wireless Internet Collaborative System (WICS) is one such CMCS. The design of WICS depicts an example of an architecture tailored especially to the mobile platform [2]. The WICS is furthermore designed especially for intellectual teamwork on-the-go. The architecture of the WICS integrates a dynamic services component intended as an on-the-go, plug-and-play feature where each user will be able to choose any number of available services depending on their needs. The application of services in the mobile computing domain has been previously outlined by Dustdar and Gall [3]. Dustdar and Gall introduce a peer-to-peer architecture with similar goals to the client/server architecture of WICS, namely to allow participants to collaborate with each other on multiple platforms, including the mobile platform. This paper sets to outline the service-oriented architecture of WICS and presents its unique functionalities and benefits in the on-the-go collaboration on the mobile platform.

When the nodes of a tree are visited in depth-first order there are occasional jumps from a deeper level of the tree to a higher level. On the set of all full binary trees with a given number of nodes there is about 1 jump for every 2 internal nodes, and the average jump distance is about 2 levels. These averages are close to averages for trees that arise in polynomial real root isolation.

Cylindrical algebraic decomposition requires many very time consuming operations, including resultant computation, polynomial factorization, algebraic polynomial gcd computation and polynomial real root isolation. We show how the time for algebraic polynomial real root isolation can be greatly reduced by using interval arithmetic instead of exact computation. This substantially reduces the overall time for cylindrical algebraic decomposition.

The efficiency of a parallel algorithm with input x on P ≥ 1 processors is defined as \\(E(x,P) = \\frac{{T(x,1)}}{{PT(x,P)}}\\) where T(x, P) denotes the time it takes to perform the computation using P processors and T(x, 1) is the sequential execution time. The efficiency of many parallel algorithms decreases when the number of processors increases and the sequential execution time is fixed; likewise, the efficiency increases when the sequential computing time increases and the number of processors is fixed. The term scalability refers to this change of efficiency (Sahni & Thanvantri, 1996). Intuitively, a parallel algorithm is scalable if it stays efficient when the number of processors and the sequential execution time are both increased.

Traditional numerical methods for polynomial root computation are fast but they sometimes return wrong results. Algorithms from computer algebra, on the other hand, guarantee correctness but tend to require long computing times. The computing time can, however, be dramatically reduced if exact arithmetic is carefully replaced by floating point arithmetic and if, moreover, ways are found to employ parallel computation. We applied this dual strategy to an algorithm for real root computation that is based on Descartes’ rule of signs [5]. We obtained an efficient infallible method to compute real roots of polynomials with degrees in the thousands. Such computations are needed to solve certain problems in high-energy physics.

We use computer algebra to expand the Pekeris secular determinant for two-electron atoms symbolically, to produce an explicit polynomial in the energy parameter ε, with coefficients that are polynomials in the nuclear charge Z. Repeated differentiation of the polynomial, followed by a simple transformation, gives a series for ε in decreasing powers of Z. The leading term is linear, consistent with well-known behavior that corresponds to the approximate quadratic dependence of ionization potential on atomic number (Moseley’s law). Evaluating the 12-term series for individual Z gives the roots to a precision of 10 or more digits for Z⩾2. This suggests the use of similar tactics to construct formulas for roots vs atomic, molecular, and variational parameters in other eigenvalue problems, in accordance with the general objectives of gradient theory. Matrix elements can be represented by symbols in the secular determinants, enabling the use of analytical expressions for the molecular integrals in the differentiation of the explicit polynomials. The mathematical and computational techniques include modular arithmetic to handle matrix and polynomial operations, and unrestricted precision arithmetic to overcome severe digital erosion. These are likely to find many further applications in computational chemistry.

An efficient algorithm is presented that returns the exactly rounded sum of two multiprecision floating point numbers. Depending on the input signs and exponents the algorithm distinguishes five cases. In each case, the method minimizes the number of computer words that are subject to de-normalization, addition or subtraction, and normalization. The method achieves further efficiency by trying to combine these three steps into one single pass over the mantissas. To do this, the method guesses the shift amount of the normalizing shift before the sum is known.

Two new scheduling algorithms are presented. They are used to iso- late polynomial real roots on massively parallel systems. O ne algorithm sched- ules computations modeled by a pyramid DAG. This is a directed acyclic graph isomorphic to Pascal's triangle. Pyramid DAGs are schedule d so that the com- munication overhead is linear. The other algorithm schedules parallelizable in- dependent tasks that have identical computing time functions in the number of processors. The two algorithms are combined to schedule a tree-search for poly- nomial real roots; the first algorithm schedules the computa tions associated with each node of the tree; the second algorithm schedules the nodes on each level of the tree.

Two new scheduling algorithms are presented. They are used to isolate polynomial real roots on massively parallel systems. One algorithm schedules computations modeled by a pyramid DAG. This is a directed acyclic graph isomorphic to Pascal’s triangle. Pyramid DAGs are scheduled so that the communication overhead is linear. The other algorithm schedules parallelizable independent tasks that have identical computing time functions in the number of processors. The two algorithms are combined to schedule a tree-search for polynomial real roots; the first algorithm schedules the computations associated with each node of the tree; the second algorithm schedules the nodes on each level of the tree.