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Thesis (Ph. D. in Engineering)--University of Tsukuba, (A), no. 3426, 2004.3.25, Includes bibliographical references, In this thesis, we consider the problem of computing the zeros of analytic functions that lie inside a circular region of the complex plane. A validated method for bounding cluster of zeros of analytic functions and a validated method for locating the zeros in the cluster are proposed. First we consider the problem of computing the zeros of polynomials which have multiple zeros or clusters of zeros. An eigenvalue method is presented which consructs a new companion matrix whose eigenvalues are simple and the zeros of given polynomial. This method can caluculate multiple zeros and their mltiplicities accurately, and it is also efficient in calculating the center of the cluster of zeros and the number of zeros in the cluster. One problem in computing zeros of polynomials using eigenvalue method is that a small pertubation in the elements of a companion matrix may produce significant changes in the eigenvalues of the companion matrix calculated by the QR method. ･･･, Titlepage,Abstracts,Acknowledgements,Contents -- Chapter 1 Introduction -- Chapter 2 Interval Arithmetic -- Chapter 3 A Method for Finding the Zeros of Polynomials Using a Companion Matrix -- Chapter 4 A High Precision Companion Matrix Method to Find Multiple Zeros of Polynomials -- Chapter 5 A Verified Method for Finding Polynomial Factors of Analytic Functions -- Chapter 6 Bounding Clusters of Zeros of Analytic Functions -- Chapter 7 Conclusions and Future Work -- Bibliography -- List of Publications, http://www.tulips.tsukuba.ac.jp/limedio/dlam/B23/B2365117/1.pdf, http://www.tulips.tsukuba.ac.jp/limedio/dlam/B23/B2365117/2.pdf, http://www.tulips.tsukuba.ac.jp/limedio/dlam/B23/B2365117/3.pdf, http://www.tulips.tsukuba.ac.jp/limedio/dlam/B23/B2365117/4.pdf, http://www.tulips.tsukuba.ac.jp/limedio/dlam/B23/B2365117/5.pdf, http://www.tulips.tsukuba.ac.jp/limedio/dlam/B23/B2365117/6.pdf, http://www.tulips.tsukuba.ac.jp/limedio/dlam/B23/B2365117/7.pdf, http://www.tulips.tsukuba.ac.jp/limedio/dlam/B23/B2365117/8.pdf, http://www.tulips.tsukuba.ac.jp/limedio/dlam/B23/B2365117/9.pdf, http://www.tulips.tsukuba.ac.jp/limedio/dlam/B23/B2365117/10.pdf