Evaluates a sequence of Bessel functions of the second kind with real nonnegative order and real positive arguments.

Required Arguments

XNU — Real argument which is the lowest order desired.   (Input)
It must be at least zero and less than one.

X — Real positive argument for which the sequence of Bessel functions is to be evaluated.   (Input)

N — Number of elements in the sequence.   (Input)

BSY — Vector of length N containing the values of the function through the series.   (Output)
BSY(I) contains the value of the Bessel function of order I − 1 + XNU at x for I = 1
to N.

FORTRAN 90 Interface

Generic:                              CALL BSYS (XNU, X, N, BSY)

Specific:                             The specific interface names are S_BSYS and D_BSYS.

FORTRAN 77 Interface

Single:                                CALL BSYS (XNU, X, N, BSY)

Double:                              The double precision name is DBSYS.

Description

The Bessel function Yv(x) is defined to be

The variable ν must satisfy 0 ≤ ν < 1. If this condition is not met, then BSY is set to −b. In addition, x must be in  where  and . If , then −b (b = AMACH(2), the largest representable number) is returned; and if , then zero is returned.

The algorithm is based on work of Cody and others, (see Cody et al. 1976; Cody 1969; NATS FUNPACK 1976). It uses a special series expansion for small arguments. For moderate arguments, an analytic continuation in the argument based on Taylor series with special rational minimax approximations providing starting values is employed. An asymptotic expansion is used for large arguments.

Example

In this example, Y0.015625+v−1(0.0078125), ν = 1, 2, 3 is computed and printed.

 

      USE BSYS_INT

      USE UMACH_INT

 

      IMPLICIT   NONE

!                                 Declare variables

      INTEGER    N

      PARAMETER  (N=3)

!    

      INTEGER    K, NOUT

      REAL       BSY(N), X, XNU

!                                 Compute

      XNU = 0.015625

      X   = 0.0078125

      CALL BSYS (XNU, X, N, BSY)

!                                 Print the results

      CALL UMACH (2, NOUT)

      DO 10  K=1, N

         WRITE (NOUT,99999) XNU+K-1, X, BSY(K)

   10 CONTINUE

99999 FORMAT (' Y sub ', F6.3, ' (', F6.3, ') = ', F10.3)

      END

Output

 

 Y sub  0.016 ( 0.008) =     -3.189

 Y sub  1.016 ( 0.008) =    -88.096

 Y sub  2.016 ( 0.008) = -22901.732

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