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Forward and Backward Analyses of Quadratic Equations
Yves Nievergelt

Mathematics Topic: Computer Science, Number Theory 
Application Areas: Computer Science, Engineering, Mathematics, Philosophy, Sciences 
Prerequisites: Calculus through the mean value theorem for derivatives,
Taylor series with direct and inverse trigonometric
functions, and complex variables. 

 ©2012 by COMAP, Inc.  The UMAP Journal 33.1  40 pages 

This Module demonstrates the forward and backward
analysis to prove in advance the degree of accuracy of
the computed values, from Karl Stumpff’s and W. Kahan’s
formulae, of the real solutions of some real cubic
equations. Applications include the computation
of acidity in chemistry.
Table of Contents
1. Introduction
2. Inequalities for Perturbation Analysis
3. Classical Formulae for the Solutions
3.1 Cardano’s Formulae for Solutions of Cubic Equations
3.2 Exercises on the Discriminant of Cubic Equations
3.3 Kahan’s Improvement of Accuracy
4. Stumpff’s Formulae
4.1 Stumpff’s Formulae for a Reduced Cubic Equation
4.2 Generalization of Stumpff’s Formulae
4.3 A Perturbation Analysis with Stumpff’s Formulae
4.4 Rounding Analysis with Stumpff’s Formulae
4.5 Exercises on Stumpff’s Formulae
5. Kahan’s Formulae
5.1 Kahan’s Formulae for Real Cubic Equations
5.2 A Perturbation Analysis with Kahan’s Formulae
5.3 Rounding Analysis with Kahan’s Formulae
5.4 Exercises on Kahan’s Formulae
6. Acidity in Chemical Kinetics 6.1 Stoichiometric and Kinetic Equations in Chemistry
6.2 Polynomial Equations in Chemical Kinetics
6.3 Exercises on Chemical Kinetics
7. Conclusions
8. Acknowledgments
9. Solutions to OddNumbered Exercises
About the Author



