Computing the real zeros of cylinder functions and the roots of the equation xC'v(x) + yCv(x)

Fast methods to compute the zeros of general cylinder functions C"@n(x)=cos@aJ"@n(x)-sin@aY"@n(x) in real intervals can be obtained from an approximate integration of the second order ODE satisfied by these functions, leading to fourth order methods with global convergence. By considering the second order ODE satisfied by the function w(x)=v^'(x), v(x)=x^@cC"@n(x), we also construct a globally convergent fourth order method for the evaluation of the roots of xC"@n^'(x)+@cC"@n(x)=0, and in particular for the first derivative of cylinder functions. The method holds for any real values of @n, @a and @c and it does not require a priori estimations for initiating the iterations. Fifteen digit accuracy is generally reached with only 1-3 iterations per each simple root. For the derivative C"@n^'(x) and the function xC"@n^'(x)+@cC"@n(x), a double root or a nearly degenerate pair of real roots may exist for some parameter values; the method computes reliably such roots.

[1]  R. Spigler,et al.  Some Remarks on Zeros of Cylinder Functions , 1984 .

[2]  Amparo Gil,et al.  ELF and GNOME: Two tiny codes to evaluate the real zeros of the Bessel functions of the first kind for real orders , 1999 .

[3]  Michael N. Vrahatis,et al.  ZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument , 1998 .

[4]  L. J. Landau.  Ratios of Bessel Functions and Roots of αJν(x) + xJ′ν(x) = 0 , 1999 .

[5]  J. Segura.  On a conjecture regarding the extrema of Bessel functions and its generalization , 2003 .

[6]  Nico M. Temme,et al.  An algorithm with ALGOL 60 program for the computation of the zeros of ordinary bessel functions and those of their derivatives , 1979 .

[7]  A. Gray.  Bessel Functions , 1899, Nature.

[8]  Javier Segura,et al.  Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros , 2001, Math. Comput..

[9]  Javier Segura.  Reliable Computation of the Zeros of Solutions of Second Order Linear ODEs Using a Fourth Order Method , 2010, SIAM J. Numer. Anal..

[10]  Lee Lorch,et al.  Monotonic sequences related to zeros of Bessel functions , 2008, Numerical Algorithms.

[11]  The real zeros of the derivatives of cylinder functions of negative order , 1999 .

[12]  D. E. Amos.  Algorithm 644: A portable package for Bessel functions of a complex argument and nonnegative order , 1986, TOMS.

[13]  A. Laforgia,et al.  ZEROS OF BESSEL FUNCTIONS: MONOTONICITY, CONCAVITY, INEQUALITIES , 2007 .

[14]  M. E. Muldoon.  On the zeros of a function related to Bessel functions , 1982 .

[15]  Robert Piessens,et al.  A series expansion for the first positive zero of the Bessel functions , 1984 .

[16]  R. Piessens,et al.  ON THE COMPUTATION OF ZEROS AND TURNING POINTS OF BESSEL FUNCTIONS , 1993 .