Solution of Nonlinear Equations

Part of the Statistics and Computing book series (SCO)


Solving linear and nonlinear equations is a major preoccupation of applied mathematics and statistics. For nonlinear equations, closed-form solutions are the exception rather than the rule. Here we will concentrate on three simple techniques—bisection, functional iteration, and Newton’s method— for solving equations in one variable. Insight into how these methods operate can be gained by a combination of theory and examples. Since functional iteration and Newton’s method generalize to higher-dimensional problems, it is particularly important to develop intuition about their strengths and weaknesses. Equipped with this intuition, we can tackle harder problems with more confidence and understanding.


Nonlinear Equation Golden Section Extinction Probability Fractional Linear Transformation Incomplete Gamma Function 
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  1. 1.
    Box GEP, Tiao G (1973) Bayesian Inference in Statistical Analysis. Addison-Wesley, Reading, MAMATHGoogle Scholar
  2. 2.
    Dennis JE Jr, Schnabel RB (1996) Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, PhiladelphiaMATHGoogle Scholar
  3. 3.
    Feller W (1968) An Introduction to Probability Theory and Its Applications, Volume 1, 3rd ed. Wiley, New YorkGoogle Scholar
  4. 4.
    Henrici P (1982) Essentials of Numerical Analysis with Pocket Calculator Demonstrations. Wiley, New YorkMATHGoogle Scholar
  5. 5.
    Juola RC (1993) More on shortest confidence intervals. Amer Statistician 47:117–119CrossRefMathSciNetGoogle Scholar
  6. 6.
    Lotka AJ (1931) Population analysis—the extinction of families I. J Wash Acad Sci 21:377–380Google Scholar
  7. 7.
    Lotka AJ (1931) Population analysis—the extinction of families II. J Wash Acad Sci 21:453–459Google Scholar
  8. 8.
    Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. Cambridge University Press, CambridgeGoogle Scholar
  9. 9.
    Stoer J, Bulirsch R (2002) Introduction to Numerical Analysis, 3rd ed. Springer, New YorkMATHGoogle Scholar
  10. 10.
    Strang G (1986) Introduction to Applied Mathematics. Wellesley-Cambridge Press, Wellesley, MAMATHGoogle Scholar
  11. 11.
    Tate RF, Klett GW (1969) Optimal confidence intervals for the variance of a normal distribution. J Amer Stat Assoc 54:674–682CrossRefMathSciNetGoogle Scholar
  12. 12.
    Wall HS (1948) Analytic Theory of Continued Fractions. Van Nostrand, New YorkMATHGoogle Scholar

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© Springer New York 2010

Authors and Affiliations

  1. 1.Departments of Biomathematics, Human Genetics, and Statistics David Geffen School of MedicineUniversity of California, Los AngelesLos AngelesUSA

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