Werner Krandick

Interval Arithmetic in Cylindrical Algebraic Decomposition

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.

Polynomial real root isolation using approximate arithmetic

A method is presented for isolating and refining the real roots of polynomials with either integer or real algebraic number coefficients. For root isolation the method uses a well-known algorithm that is baaed on Descartes’ rule of signs. However, exact arithmetic is replaced as far as possible by validated double precision floating point arithmetic. The resulting method is powerful and very fast.

An efficient algorithm for infallible polynomial complex root isolation

Applying the principle of the argument to rectangles provides an efficient algorithm for polynomial complex root isolation. For example, the roots of a real integral polynomial of degree 10 (with 10-bit coefficients) can be isolated in about 1 second. Furthermore, although the algorithm is not designed for efficient refinement of isolating rectangles, it does nevertheless refine all c]f these rectangles to width 10-50 in less than 4 minutes.

Parallel Real Root Isolation Using the Descartes Method

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.

High-performance implementations of the Descartes method

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 Casteljaus 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.

Architecture-aware classical Taylor shift by 1

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.

Power series expansion of the roots of a secular equation containing symbolic elements: Computer algebra and Moseley’s law

We use computer algebra to expand the Pekeris secular determinant for two-electron atoms symbolically, to produce an explicit polynomial in the energy parameter e, with coefficients that are polynomials in the nuclear charge Z. Repeated differentiation of the polynomial, followed by a simple transformation, gives a series for e 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 different...

On the computing time of the continued fractions method

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.

Isoefficiency and the Parallel Descartes Method

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.

New bounds for the Descartes method

We give a new bound for the number of recursive subdivisions in the Descartes method for polynomial real root isolation. Our proof uses Ostrowskis theory of normal power series from 1950 which has so far been overlooked in the literature. We combine Ostrowskis 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 Ostrowskis 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.