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Product No. 99806 Supplementary Print Price: FREE with membership

Forward and Backward Analyses of Quadratic Equations

Yves Nievergelt

Mathematics Topic:
Computer Science, Number Theory
Application Areas:
Computer Science, Engineering, Mathematics, Philosophy, Sciences
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 Odd-Numbered Exercises

About the Author