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Computing Zeros of Nonlinear Univariate Functions

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Boundary Value Problems for Engineers

Abstract

Solving nonlinear equations is one of the most important problems that has many applications in all fields of engineering , and it is one of the oldest and most basic problems in mathematics.

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Notes

  1. 1.

    Alexander Craig Aitken (1895–1967) Scottish-New Zealand origin. He was appointed to Edinburgh University in 1925 where he spent the rest of his life. Aitken wrote several books, “The theory of canonical matrices” (1932), “Determinants and matrices (1939)”, and “Statistical Mathematics (1939)”.

  2. 2.

    Lewis Fry Richardson, (1881–1953) was an English mathematician who pioneered mathematical techniques of weather forecasting. He is also known for his work for solving a system of linear equations (Modified Richardson Iteration ).

  3. 3.

    It has been pointed out that the delta squared process was first proposed by a Japanese mathematician Takakazu Seki (1642–1708) who used the method for computing π in 1712 [3, 4].

  4. 4.

    The method was first proposed by Newton in 1669 and later by Raphson in 1690.

  5. 5.

    For more detailed study on this method, readers may refer to basic textbooks on numerical analysis, such as [15,16,17,18,19].

  6. 6.

    Algorithms involving Symbolic Math commands are usually slower than those using double precision operations.

  7. 7.

    This is a simpler variant of problem described in [23].

  8. 8.

    Brent’s method builds on an earlier algorithm by Dekker [25]. The method is also known as the Van Wijngaarden–Dekker–Brent Method [26].

  9. 9.

    The Steffenson algorithm can be used to accelerate the convergence of multiple roots .

  10. 10.

    Edmond Halley (1656–1742) is best known for predicting the orbital period of the comet that bears his name.

  11. 11.

    Alexander M. Ostrowski (1893–1986) was an Ukrainian born mathematician.

  12. 12.

    Johan Frederik Steffensen was a Danish mathematician, who worked in the fields of finite differences and interpolation . He was professor at the University of Copenhagen from 1923 to 1943, and described this method in [40].

  13. 13.

    This method can be unstable if the function is not well-behaved. It can diverge or divide by zero, depending upon f(x). It is much more demanding with respect to the starting points than Newton’s method which justifies that Steffensen’s method is less used than Newton’s method to approximate solutions of equations [41].

  14. 14.

    MATLAB’s fzero is based on the work described in [43].

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Authors and Affiliations

  1. Department of Biomedical Engineering, Yeditepe University, Istanbul, Turkey

    Ali Ümit Keskin

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Keskin, A.Ü. (2019). Computing Zeros of Nonlinear Univariate Functions. In: Boundary Value Problems for Engineers. Springer, Cham. https://doi.org/10.1007/978-3-030-21080-9_1

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  • DOI: https://doi.org/10.1007/978-3-030-21080-9_1

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