Downloads & Free Reading Options - Results

We analyse the complexity of computing class polynomials, that are an important ingredient for CM constructions of elliptic curves, via complex floating point approximations of their roots. The heart of the algorithm is the evaluation of modular functions in several arguments. The fastest one of the presented approaches uses a technique devised by Dupont to evaluate modular functions by Newton iterations on an expression involving the arithmetic-geometric mean. It runs in time $O (|D| \log^5 |D| \log \log |D|) = O (|D|^{1 + \epsilon}) = O (h^{2 + \epsilon})$ for any $\epsilon > 0$, where $D$ is the CM discriminant and $h$ is the degree of the class polynomial. Another fast algorithm uses multipoint evaluation techniques known from symbolic computation; its asymptotic complexity is worse by a factor of $\log |D|$. Up to logarithmic factors, this running time matches the size of the constructed polynomials. The estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary-quadratic order and on a rigorously proven upper bound for the height of class polynomials.

From the bitsavers.org collection, a scanned-in computer-related document. components :: weitek :: XL :: XL-3232 32-Bit Graphics Floating Point Computation Unit Aug88

This thesis presents design techniques for floating-point computation in VLSI. A basis for area-time design decisions for arithmetic and memory operations is formulated from a study of computationally intensive programs. Tradeoffs in the design and implementation of an efficient coprocessor interface are studied, together with the implications of hardware support for the IEEE Floating-Point Standard. Algorithm area-time tradeoffs for basic arithmetic functions are analyzed in light of changing technology. Details of a single-chip floating-point unit designed into two micron CMOS for SPUR are described, including special design considerations for very wide datapaths. The pervasive effects of scaling technology on different levels of design are explored, from devices and circuits, through logic and micro-architecture, to algorithms and systems.

This book presents floating-point arithmetic in a somewhat generalized form which allows for variations in the radix and the word length. However, instead of striving for extreme generality, the book discusses the arithmetic of the IBM System/360 in detail and generalizes it where it is convenient to do so. The book discusses the details of what actually happens when floating-point arithmetic is performed during the execution of the program, and the emphasis is on the quality of the answers produced. It is my hope that, by making the reader more aware of the arithmetic that will be performed as a result of the FORTRAN statement he writes, the book will contribute to the production of better programs. This book is directed toward two different types of readers. First, it is addressed to the obvious audience of those who are interested in using higher-level languages to write programs which will perform floating-point computation. Second, it is also directed toward the compiler designers and machine designers who are concerned with floating-point operations. The material presented here has been found to be of interest to this group because, by illustrating the way floating-point arithmetic is used to solve problems, it leads to an understanding of the reasons for incorporating various features in the hardware and in the languages.

Our goal is to find accurate and efficient algorithms, when they exist, for evaluating rational expressions containing floating point numbers, and for computing matrix factorizations (like LU and the SVD) of matrices with rational expressions as entries. More precisely, {\em accuracy} means the relative error in the output must be less than one (no matter how tiny the output is), and {\em efficiency} means that the algorithm runs in polynomial time. Our goal is challenging because our accuracy demand is much stricter than usual.