Many important constants, such as e and Apéry's constant ζ(3), can be approximated by a truncated hypergeometric series. The evaluation of such series to high precision has traditionally been done by binary splitting followed by fixed-point division. However, the numerator and the denominator computed by binary splitting usually contain a very large common factor. In this paper, we apply standard computer algebra techniques including modular computation and rational reconstruction to overcome the shortcomings of the binary splitting method. The space complexity of our algorithm is the same as a bound on the size of the reduced numerator and denominator of the series approximation. Moreover, if the predicted bound is small, the time complexity is better than the standard binary splitting approach. Our approach allows a series to be evaluated to a higher precision without additional memory. We show that when our algorithm is applied to compute ζ(3), the memory requirement is significantly reduced compared to the binary splitting approach.

As the availability and use of digital images increase, the efficient storage of images becomes an important area of research. Traditionally, each image in a set is compressed individually, taking advantage of the redundancies existing within the image. In the related area of video compression, a video sequence is decomposed into individual frames. Video compression algorithms take advantage of redundancy existing among consecutive frames as well as the redundancy existing within each frame. Unlike video compression, there are applications that use large image sets whose inter-image relationships are unknown. For example, a medical database may contain a large number of X-ray images of the same body part; a database of satellite images may possess "similar" characteristics; a database of facial images contains many similar images. In some applications compressed images must be identical to the original images, therefore lossless compression must be used.Compared with traditional image compression, the lossless compression of a set of similar images has received relatively little attention from researchers. These earlier schemes have only been effective on image sets with certain properties, and it is not clear which scheme is best a priori. This poster presents a framework to effectively compress sets of images in a lossless manner. We represent an image set as a graph and compute its minimum spanning tree to decide which images and differences to encode. The Centroid scheme by Karadimitriou and the previous MST scheme by Nielsen et al. can both be represented as a spanning tree in our graph. Thus, our scheme is guaranteed to be no worse than these previous schemes. In fact, our framework povides the best lossless compression for all schemes that consider interimage redundancy between two images in a set. Our experimental results show that our new MST method always produces the best result regardless of the properties of the image sets. In some cases, the first-order entropy of the image set using our scheme results in a 29% improvement over the traditional scheme of compressing each image individually.

Main parts of the algorithm are implemented in Maple and nontrivial examples are used to show that the algorithm is effective.The algorithm mainly involves computation of resultants, determination of the topology of plane curves, computation of singularities of surfaces and curves, isolating real roots of univariate equations. AbstractWe consider the evaluation of the truncated hypergeometric series