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On the condition of algebraic equations

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Summary

Given an algebraic equation, in which the polynomial in question is expressed in terms of any set of basis polynomials, we study the sensitivity of the roots with respect to small perturbations in the coefficients of the equation. The degree of sensitivity of each root is measured by an appropriate condition number. We analyze this condition number first in the case where the basis polynomials are the powers, and then, in less detail, in the case where the basis is a set of orthogonal polynomials. Several examples are treated, allowing for a comparative study.

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References

  1. Dejon, B., Nickel, K.: A never failing, fast convergent rootfinding algorithm, Constructive Aspects of the Fundamental Theorem of Algebra (Proc. Sympos. Zürich-Rüschlikon, 1967), 1–35. New York: Wiley-Interscience 1969

    Google Scholar 

  2. Freud, G.: Orthogonal polynomials. Oxford: Pergamon Press 1971

    Google Scholar 

  3. Gautschi, W.: On inverses of Vandermonde and confluent Vandermonde matrices. Numer. Math.4, 117–123 (1962)

    Article  Google Scholar 

  4. Iverson, K. E.: The zeros of the partial sums ofe z. Math. Tables Aids Comput.7, 165–168 (1953)

    Google Scholar 

  5. Specht, W.: Die Lage der Nullstellen eines Polynoms. Math. Nachr.15, 353–374 (1956); ibid. Specht, W.: Die Lage der Nullstellen eines Polynoms. Math. Nachr.,16, 257–263 (1957); ibid. Specht, W.: Die Lage der Nullstellen eines Polynoms. Math. Nachr.,16, 369–389 (1957); ibid. Specht, W.: Die Lage der Nullstellen eines Polynoms. Math. Nachr.21, 201–222 (1960)

    Google Scholar 

  6. Wilkinson, J. H.: Rounding errors in algebraic processes Englewood Cliffs, N. J.: Prentice-Hall 1963

    Google Scholar 

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Work performed in part at the U.S.A.F. Aerospace Research Laboratories under contract F33615-71-C-1463 with Technology Incorporated.

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Gautschi, W. On the condition of algebraic equations. Numer. Math. 21, 405–424 (1973). https://doi.org/10.1007/BF01436491

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  • DOI: https://doi.org/10.1007/BF01436491

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