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**Efficient scaling for complex division**

Douglas M. Priest

Pages: 389-401

DOI: 10.1145/1039813.1039814

We develop a simple method for scaling to avoid overflow and harmful underflow in complex division. The method guarantees that no overflow will occur unless at least one component of the quotient must overflow, otherwise the normwise error in the...

**Analysis and applications of Priest's distillation**

Yves Nievergelt

Pages: 402-433

DOI: 10.1145/1039813.1039815

Correcting an infinite loop in Douglas M. Priest's renormalization algorithm, the theory proved here supports streamlined algorithms to resolve the tablemaker's dilemma for the floating-point computation of real and complex sums and dot-products,...

**Automating the implementation of Kalman filter algorithms**

Jon Whittle, Johann Schumann

Pages: 434-453

DOI: 10.1145/1039813.1039816

autofilter is a tool that generates implementations that solve state estimation problems using Kalman filters. From a high-level, mathematics-based description of a state estimation problem, autofilter automatically generates code that computes a...

**BACOL**: B-spline adaptive collocation software for 1-D parabolic PDEs

R. Wang, P. Keast, P. Muir

Pages: 454-470

DOI: 10.1145/1039813.1039817

BACOL is a new, high quality, robust software package in Fortran 77 for solving one-dimensional parabolic PDEs, which has been shown to be significantly more efficient than any other widely available software package of the same class (to our...

**Computation of complex Airy functions and their zeros using asymptotics and the differential equation**

B. R. Fabijonas, D. W. Lozier, F. W. J. Olver

Pages: 471-490

DOI: 10.1145/1039813.1039818

We describe a method by which one can compute the solutions of Airy's differential equation, and their derivatives, both on the real line and in the complex plane. The computational methods are numerical integration of the differential equation and...

**Algorithm 838**: Airy Functions

B. R. Fabijonas

Pages: 491-501

DOI: 10.1145/1039813.1039819

We present a Fortran 90 module, which computes the solutions and their derivatives of Airy's differential equation, both on the real line and in the complex plane. The module also computes the zeros and associated values of the solutions and their...

**Algorithm 839**: FIAT, a new paradigm for computing finite element basis functions

Robert C. Kirby

Pages: 502-516

DOI: 10.1145/1039813.1039820

Much of finite element computation is constrained by the difficulty of evaluating high-order nodal basis functions. While most codes rely on explicit formulae for these basis functions, we present a new approach that allows us to construct a general...