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Applications of Cubic Equations
Yves Nievergelt

Mathematics Topic: Calculus 1, Various 
Application Areas: Computer science, engineering, finance, mathematics, physics, sciences 
Prerequisites: Calculus through the MeanValue Theoremfor Derivatives
and Taylor series, direct and inverse trigonometric
functions, and complex numbers 

 ©2011 by COMAP, Inc.  The UMAP Journal 32.1  33 pages 

This module demonstrates the methods known as forward
analysis and backward analysis to prove in advance
the degree of accuracy of the real solutions of some
real cubicequationscomputedusingStumpff’s formula
and Kahan’s formula, with comparisons to an iterative
solver. Applications include the computation of
• yieldtomaturity (internal rate of return) of threeyear
bonds with annual interest,
• the position of a comet along a parabolic orbit, and
• the ratio of orbital radii for which the Hohmann
transfer is more fuel efficient than other transfers.
Table of Contents
1. INTRODUCTION
2. INEQUALITIES FOR PERTURBATION ANALYSIS
3. FORMULAE TO SOLVE CUBIC EQUATIONS
3.1 Cardano’s Formulae
3.2 Stumpff’s Formulae
3.3 Kahan’s Formulae
4. YIELD RATE OF THREEYEAR BONDS
4.1 The YieldtoMaturity of Bonds with Periodic Interest
4.2 The Yield Rate of ThreeYear Bonds with Annual Interest
4.3 Analysis of the Computed Yield Rate
5. POSITION OF A COMET
5.1 Kepler’s and Newton’s Laws
5.2 Parametrization of a Comet’s Parabolic Orbit with Time
5.3 Backward Analysis of a Comet’s Position
5.4 Forward Analysis of a Comet’s Position
5.5 Kirch’s Great Comet of 1680
6. COMPUTATION OF HOHMANN’S ORBITTRANSFER THRESHOLDS
6.1 Error Analysis of the Computed Solution
7. CONCLUSIONS
8. SOLUTIONS TO SELECTED EXERCISES
REFERENCES
ACKNOWLEDGMENTS
ABOUT THE AUTHOR



