The compression and decompression of continuous-tone images is important in document management and transmission systems. This paper considers an alternative image representation scheme, based on Gaussian derivatives, to the standard discrete cosine transformation (DCT), within a Joint Photographic Experts Group (JPEG) framework. Depending on the computer arithmetic hardware used, the approach developed might yield a compression/decompression technique twice as fast as the DCT and of (essentially) equal quality.

HOMPACK90 is a Fortran 90 version of the Fortran 77 package HOMPACK (Algorithm 652), a collection of codes for finding zeros or fixed points of nonlinear systems using globally convergent probability-one homotopy algorithms.Three qualitatively different algorithmsordinary differential equation based, normal flow, quasi-Newton augmented Jacobian matrix-are provided for tracking homotopy zero curves, as well as separate routines for dense and sparse Jacobian matrices.A high level driver for the special case of polynomial systems is also provided.Changes to HOMPACK include numerous minor improvements, simpler and more elegant interfaces, use of modules, new end games, support for several sparse matrix data structures, and new iterative algorithms for large sparse Jacobian matrices.

A system of nonlinear equations has been used as a test case by at least two authors. This system is purported to describe the equilibrium of the products of hydrocarbon combustion. The given system does not describe the stated physical problem, a fact which invalidates it as a test of solution methods for chemical equilibrium systems. In this note, the problem is correctly stated and then solved by the method of element variables.

Although the theory of polynomial continuation has been established for over a decade (following the work of Garcia, Zangwill, and Drexler), it is difficult to solve polynomial systems using continuation in practice. Divergent paths (solutions at infinity), singular solutions, and extreme scaling of coefficients can create catastrophic numerical problems. Further, the large number of paths that typically arise can be discouraging. In this paper we summarize polynomial-solving homotopy continuation and report on the performance of three standard path-tracking algorithms (as implemented in HOMPACK) in solving three physical problems of varying degrees of difficulty. Our purpose is to provide useful information on solving polynomial systems, including specific guidelines for homotopy construction and parameter settings. The m -homogeneous strategy for constructing polynomial homotopies is outlined, along with more traditional approaches. Computational comparisons are included to illustrate and contrast the major HOMPACK options. The conclusions summarize our numerical experience and discuss areas for future research.

There are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points, i.e., with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map. HOMPACK provides three qualitatively different algorithms for tracking the homotopy zero curve: ordinary differential equation-based, normal flow, and augmented Jacobian matrix. Separate routines are also provided for dense and sparse Jacobian matrices. A high-level driver is included for the special case of polynomial systems.

Small polynomial systems of equations arise in many apphcations areas: computer-aided design, mechanical design, chemmal kinetics modeling, and nonlinear circuit analysis.Use of local iterative methods, such as Newton's method, can be a hit-or-miss process.Purely algebraic schemes, such as the method of resultants, can lead to severe numerical difficulties.Imbedding methods provide the most reliable techniques for computing all solutmns to small polynomial systems.One such method is described and computational experience with it is reported on.