scholarly journals Semi-Automatic Floating-Point Implementation of Special Functions

2015 ◽  
Author(s):  
Christoph Lauter ◽  
Marc Mezzarobba
Keyword(s):  
Special Functions ◽  
Floating Point ◽  
Floating Point Implementation

NUMERICAL STABILITY OF ALGORITHMS FOR 2D DELAUNAY TRIANGULATIONS

1995 ◽  
Vol 05 (01n02) ◽  
pp. 193-213 ◽  
Author(s):  
STEVEN FORTUNE
Keyword(s):  
Delaunay Triangulation ◽  
Floating Point ◽  
Worst Case ◽  
Delaunay Triangulations ◽  
Floating Point Arithmetic ◽  
Circular Arcs ◽  
Stability Of Algorithms ◽  
Forming Angle ◽  
Point Arithmetic ◽  
Floating Point Implementation

We consider the correctness of 2-d Delaunay triangulation algorithms implemented using floating-point arithmetic. The α-pseudocircle through points a, b, c consists of three circular arcs connecting ab, bc, and ac, each arc inside the circumcircle of a, b, c and forming angle α with the circumcircle; a triangulation is α-empty if the α-pseudocircle through the vertices of each triangle is empty. We show that a simple Delaunay triangulation algorithm—the flipping algorithm—can be implemented to produce O(n∈)-empty triangulations, where n is the number of point sites and ∈ is the relative error of floating-point arithmetic; its worst-case running time is O(n2). We also discuss floating-point implementation of other 2-d Delaunay triangulation algorithms.

scholarly journals What Can Students Learn While Solving Colebrook’s Flow Friction Equation?

Fluids ◽  
2019 ◽  
Vol 4 (3) ◽  
pp. 114 ◽  
Author(s):  
Dejan Brkić ◽  
Pavel Praks
Keyword(s):  
Artificial Intelligence ◽  
Neural Networks ◽  
Iterative Methods ◽  
Special Functions ◽  
Floating Point ◽  
Numerical Examples ◽  
Flow Friction ◽  
Newton Raphson ◽  
The Difference ◽  
Fixed Point Iterative Method

Even a relatively simple equation such as Colebrook’s offers a lot of possibilities to students to increase their computational skills. The Colebrook’s equation is implicit in the flow friction factor and, therefore, it needs to be solved iteratively or using explicit approximations, which need to be developed using different approaches. Various procedures can be used for iterative methods, such as single the fixed-point iterative method, Newton–Raphson, and other types of multi-point iterative methods, iterative methods in a combination with Padé polynomials, special functions such as Lambert W, artificial intelligence such as neural networks, etc. In addition, to develop explicit approximations or to improve their accuracy, regression analysis, genetic algorithms, and curve fitting techniques can be used too. In this learning numerical exercise, a few numerical examples will be shown along with the explanation of the estimated pedagogical impact for university students. Students can see what the difference is between the classical vs. floating-point algebra used in computers.

scholarly journals Certifying the Floating-Point Implementation of an Elementary Function Using Gappa

2011 ◽  
Vol 60 (2) ◽  
pp. 242-253 ◽  
Author(s):  
Florent de Dinechin ◽  
Christoph Lauter ◽  
Guillaume Melquiond
Keyword(s):  
Elementary Function ◽  
Floating Point ◽  
Floating Point Implementation

scholarly journals Efficient Calculation of the Genomic Relationship Matrix

2020 ◽  
Author(s):  
Martin Schlather
Keyword(s):  
Floating Point ◽  
Genomic Relationship Matrix ◽  
Relationship Matrix ◽  
Double Precision ◽  
Genomic Relationship ◽  
Arithmetic Operations ◽  
Efficient Calculation ◽  
Floating Point Implementation

Since the calculation of a genomic relationship matrix needs a large number of arithmetic operations, fast implementations are of interest. Our fastest algorithm is more accurate and 25× faster than a AVX double precision floating-point implementation.

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