ACM Transactions on Mathematical Software (TOMS)

Flores, Juan

doi: 10.1145/317275.317277pmid: N/A

If we allow the magnitude and angle of a complex number (expressed in polar form) to range over an interval, it describes a semicircular region, similar to a fan; these regions are what we call complex fans. Complex numbers are a special case of complex fans, where the magnitude and angle are point intervals. Operations (especially addition) with complex numbers in polar form are complicated. What most applications do is to convert them to rectangular form, perform operations, and return the result to polar form. However, if the complex number is a Complex Fan, that transformation increases ambiguity in the result. That is, the resulting Fan is not the smallest Fan that contains all possible results. The need for minimal results took us to develop algorithms to perform the basic arithmetic operations with complex fans, ensuring the result will always be the smallest possible complex fan. We have developed the arithmetic operations of addition, negation, subtraction, product, and division of complex fans. The algorithms presented in this article are written in pseudocode, and the programs in Common Lisp, making use of CLOS (Common Lisp Object System). Translation to any other high-level programming language should be straightforward.

Heinkenschloss, Matthias; Vicente, Luís N.

doi: 10.1145/317275.317278pmid: N/A

An interface between the application problem and the nonlinear optimization algorithm is proposed for the numerical solution of distributed optimal control problems. By using this interface, numerical optimization algorithms can be designed to take advantage of inherent problem features like the splitting of the variables into states and controls and the scaling inherited from the functional scalar products. Further, the interface allows the optimization algorithm to make efficient use of user-provided function evaluations and derivative calculations.

Gockenbach, Mark S.; Petro, Matthew J.; Symes, William W.

doi: 10.1145/317275.317280pmid: N/A

The object-oriented programming paradigm can be used to overcome the incompatibilities between off-the-shelf optimization software and application software. The Hilbert Class Library (HCL) defines the fundamental mathematical objects arising in optimization problems, such as vectors, linear operators, and so forth, as C++ classes, making it possible to write optimization code in a natural fashion, while allowing application software such as simulators to use the most convenient data structures and programming style. In spite of the poor reputation C++ has for runtime performance, the use of mixed-language programming allows performance equal to that achieved by standard Fortran packages, as comparisons with the popular code LBFGS and ARPACK demonstrate.

Gautschi, Walter

doi: 10.1145/317275.317282pmid: N/A

The concern here is with Gauss-type quadrature rules that are exact for a mixture of polynomials and rational functions, the latter being selected so as to simulate poles that may be present in the integrand. The underlying theory is presented as well as methods for constructing such rational Gauss formulae. Relevant computer routines are provided and applied to a number examples, including Fermi-Dirac and Bose-Einstein integrals of interest in solid state physics.

Wieder, Thomas

doi: 10.1145/317275.317284pmid: N/A

The numerical evaluation of the Hankel transform poses the problems of both infinite integration and Bessel function calculation. Using the corresponding numerical program routines from the literature, a Fortran program has been written to perform the Hankel transform for real functions, given either in analytical form as subroutines or in discrete form as tabulated data.

Verschelde, Jan

doi: 10.1145/317275.317286pmid: N/A

Polynomial systems occur in a wide variety of application domains. Homotopy continuation methods are reliable and powerful methods to compute numerically approximations to all isolated complex solutions. During the last decade considerable progress has been accomplished on exploiting structure in a polynomial system, in particular its sparsity. In this article the structure and design of the software package PHC is described. The main program operates in several modes, is menu driven, and is file oriented. This package features great variety of root-counting methods among its tools. The outline of one black-box solver is sketched, and a report is given on its performance on a large database of test problems. The software has been developed on four different machine architectures. Its portability is ensured by the gnu-ada compiler.