Forward and Backward Analyses of Quadratic Equations (UMAP)

ABSTRACT

This Module demonstrates the methods of forward and backward analysis to prove the degree of accuracy of the computed real solutions of real quadratic equations. This Module focuses on the methods of proof but also features specific results. Applications include the computation of acidity in chemistry, yield-to-maturity (also called the internal rate of return) of one-year U.S. Treasury bills, and the Excess Related Person Indebtedness according to the U.S. corporate tax code.

Table of Contents

1. INTRODUCTION

2. ROUNDING ERRORS
2.1 The Standard Model of Floating-Point Arithmetic
2.2 Simplifications of Compounding Factors
2.3 Exercises on Rounding Errors

3. AN ACCURATE ALGORITHM

4. FORWARD ANALYSES, FOR a ∑ c < 0
4.1 The Computed Discriminant
4.2 The Computed Zeros
4.3 Computation of Acidity (pH)
4.4 Yield of Discounted Treasury Bills
4.5 Exercises on Forward Analysis

5. BACKWARD ANALYSES, FOR a ∑ c > 0
5.1 The Computed Discriminant
5.2 The Computed Zeros
5.3 The Relative Rate of Change
5.4 Excess Related Person Indebtedness
5.5 Exercises on Backward Analysis

6. ACKNOWLEDGMENTS

7. SOLUTIONS TO ODD-NUMBERED EXERCISES

REFERENCES

ABOUT THE AUTHOR