this paper the concept of chaotic attractor is used only as an example of a dynamic behavior that could a#ect the search process, we summarize the main characteristics and refer to [13] for a detailed theoretical analysis. Chaotic attractors are characterized by a "contraction of the areas", so that trajectories starting with di#erent initial conditions will be compressed in a limited area of the configuration space, and by a "sensitive dependence upon the initial conditions", so that di#erent trajectories will diverge. For an analytical characterization of this sensitive dependence, it is convenient to introduce the concept of Lyapunov exponent. Let us consider the function g that maps the point at step n

Earlier work by Driscoll and Healy [16] has produced an e#cient algorithm for computing the Fourier transform of band-limited functions on the 2-sphere. In this paper we present a reformulation and variation of the original algorithm which results in a greatly improved inverse transform, and consequent improved convolution algorithm for such functions. All require at most O(N log 2 N ) operations where N is the number of sample points. We also address implementation considerations and give heuristics for allowing reliable and computationally e#cient floating point implementations of slightly modified algorithms. These claims are supported by extensive numerical experiments from our implementation in C on DEC, HP and SGI platforms. These results indicate that variations of the algorithm are both reliable and e#cient for a large range of useful problem sizes. Performance appears to be architecture-dependent. The paper concludes with a brief discussion of a few potential applications. 1...

SPIRAL is a generator for libraries of fast software implementations of linear signal processing transforms. These libraries are adapted to the computing platform and can be re-optimized as the hardware is upgraded or replaced. This paper describes the main components of SPIRAL: the mathematical framework that concisely describes signal transforms and their fast algorithms; the formula generator that captures at the algorithmic level the degrees of freedom in expressing a particular signal processing transform; the formula translator that encapsulates the compilation degrees of freedom when translating a specific algorithm into an actual code implementation; and, finally, an intelligent search engine that finds within the large space of alternative formulas and implementations

In this paper we survey some recent work directed towards generalizing the fast Fourier transform (FFT). We work primarily from the point of view of group representation theory. In this setting the classical FFT can be viewed as a family of efficient algorithms for computing the Fourier transform of either a function defined on a finite abelian group, or a bandlimited function on a compact abelian group. We discuss generalizations of the FFT to arbitrary finite groups and compact Lie groups.

. Let P = fP 0 ; : : : ; Pn\Gamma1 g denote a set of polynomials with complex coefficients. Let Z = fz 0 ; : : : ; z n\Gamma1 g ae C denote any set of sample points. For any f = (f 0 ; : : : ; fn\Gamma1 ) 2 C n the discrete polynomial transform of f (with respect to P and Z) is defined as the collection of sums, f b f(P 0 ); : : : ; b f(Pn\Gamma1 )g, where f(P j ) = hf; P j i = P n\Gamma1 i=0 f i P j (z i )w(i) for some associated weight function w. These sorts of transforms find important applications in areas such as medical imaging and signal processing. In this paper we present fast algorithms for computing discrete orthogonal polynomial transforms. For a system of N orthogonal polynomials of degree at most N \Gamma 1 we give an O(N log 2 N) algorithm for computing a discrete polynomial transform at an arbitrary set of points instead of the N 2 operations required by direct evaluation. Our algorithm depends only on the fact that orthogonal polynomial sets satisfy a thre...

In this paper the task of training sub-symbolic systems is considered as a combinatorial optimization problem and solved with the heuristic scheme of the Reactive Tabu Search (RTS) proposed by the authors and based on F. Glover's Tabu Search. An iterative optimization process based on a "modified greedy search" component is complemented with a meta-strategy to realize a discrete dynamical system that discourages limit cycles and the confinement of the search trajectory in a limited portion of the search space. The possible cycles are discouraged by prohibiting (i.e., making tabu) the execution of moves that reverse the ones applied in the most recent part of the search, for a prohibition period that is adapted in an automated way. The confinement is avoided and a proper exploration is obtained by activating a diversification strategy when too many configurations are repeated excessively often. The RTS method is applicable to non-di#erentiable functions, it is robust with respect to the...

. Generalized FFTs are efficient algorithms for computing a Fourier transform of a function defined on finite group, or a bandlimited function defined on a compact group. The development of such algorithms has been accompanied and motivated by a growing number of both potential and realized applications. This paper will attempt to survey some of these applications. Appendices include some more detailed examples. 1. A brief history The now "classical" Fast Fourier Transform (FFT) has a long and interesting history. Originally discovered by Gauss, and later made famous after being rediscovered by Cooley and Tukey [21], it may be viewed as an algorithm which efficiently computes the discrete Fourier transform or DFT. In between Gauss and Cooley-Tukey others developed special cases of the algorithm, usually motivated by the need to make efficient data analysis of one sort or another. To cite but a few examples, Gauss was interested in efficiently interpolating the orbits of asteroids [43...

Abstract. This paper introduces new techniques for the efficient computation of Fourier transforms on symmetric groups and their homogeneous spaces. We replace the matrix multiplications in Clausen’s algorithm with sums indexed by combinatorial objects that generalize Young tableaux, and write the result in a form similar to Horner’s rule. The algorithm we obtain computes the Fourier transform of a function on Sn in no more than 3 n(n − 1) |Sn | multiplications 4 and the same number of additions. Analysis of our algorithm leads to several combinatorial problems that generalize path counting. We prove corresponding results for inverse transforms and transforms on homogeneous spaces. 1.

This paper introduces new techniques for the efficient computation of a Fourier transform on a finite group. We present a divide and conquer approach to the computation. The divide aspect uses factorizations of group elements to reduce the matrix sum of products for the Fourier transform to simpler sums of products. This is the separation of variables algorithm. The conquer aspect is the final computation of matrix products which we perform efficiently using a special form of the matrices. This form arises from the use of subgroup-adapted representations and their structure when evaluated at elements which lie in the centralizers of subgroups in a subgroup chain. We present a detailed analysis of the matrix multiplications arising in the calculation and obtain easy-to-use upper bounds for the complexity of our algorithm in terms of representation theoretic data for the group of interest. Our algorithm encompasses many of the known examples of fast Fourier transforms. We recover the b...

Factored Edge-Valued Binary Decision Diagrams form an extension to Edge-Valued Binary Decision Diagrams. By associating both an additive and a multiplicative weight with the edges, FEVBDDs can be used to represent a wider range of functions concisely. As a result, the computational complexity for certain operations can be significantly reduced compared to EVBDDs. Additionally, the introduction of multiplicative edge weights allows us to directly represent the so-called complement edges which are used in OBDDs, thus providing a one to one mapping of all OBDDs to FEVBDDs. Applications such as integer linear programming and logic verification that have been proposed for EVBDDs also benefit from the extension. We present a complete matrix package based on FEVBDDs and apply the package to the problem of solving the Chapman-Kolmogorov equations.