We consider computing a prescribed number of smallest positive zeros of Bessel functions and of their derivatives of a prescribed order within a prescribed relative error. We also consider an inverse problem of computing the order of the Bessel function that has a zero of a prescribed order at a prescribed positive value. The case of Bessel functions of real noninteger order less than -1 is also discussed. Our emphasis in this paper is on algorithm construction and convergence analysis that will be needed for the construction of software for solving the stated problems.
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