# include <math.h>
# include <stdbool.h>
# include <stdio.h>
# include <stdlib.h>

# include "helmholtz_exact.h"

/******************************************************************************/

double *helmholtz_exact ( double a, int m, int n, double alpha, double beta, 
  double gamma, int nxy, double x[], double y[] )

/******************************************************************************/
/*
  Purpose:

    helmholtz_exact() evaluates an exact solution of the Helmholtz equation.

  Discussion:

    We consider the Helmholtz equation for Z(x,y):
      Del^2 Z = - k^2 Z
    where k is a parameter.  We suppose we are interested in the
    problem when describing the vertical deflection of a vibrating circular
    membrane, centered at the origin, of radius a.

    In radial coordinates, the equation in Z(r,theta) becomes
      Zrr + 1/r Zr + 1/r^2 Ztt + k^2 Z = 0
    We impose the condition that Z vanishes on the circumference:
      Z(a,theta) = 0 for all theta.
    We try separation of variables:
      Z(r,theta) = R(r) T(theta)
    for which we can suppose that T is periodic
      T'' + n^2 T = 0 for some n
    whence
      T(theta) = alpha * cos ( n * theta ) + beta * sin ( n * theta )
    and
      r^2 R'' + r R' + r^2 k^2 R - n^2 R = 0
    for which the solution is
      R(r) = gamma * Jn(k*r), gamma arbitrary

    For each value of n, the Bessel function has infinitely many roots,
    denoted by rho(m,n). 

    To enforce the zero boundary condition at r=a, we must have that
    the parameter k is related to some m-th zero of some n-th Bessel
    function Jn() by:
      k(m,n) = rho(m,n)/a

    An exact solution of the Helmholtz equation, as a function of
    r and theta, can then be written as a sum of terms of the form
      gamma * Jn ( rho(m,n) * r / a ) 
      * alpha * cos ( n * theta ) + beta * sin ( n * theta )

  Licensing:

    This code is distributed under the MIT license.

  Modified:

    16 June 2025

  Author:

    John Burkardt

  Reference:

    This discussion relies on the Wikipedia article "Helmholtz Equation".

  Input:

    real a: the radius of the disk.

    integer m: the index of a zero of the Jn(x) Bessel function.
    Except when n=0, the index m=0 chooses the root at 0.
    When n=0, there is not a root at 0, and so m=0 is illegal.
    In any case, m=1 is the first root of Jn(x) greater than 0, 
    2 indexes the second smallest root and so on.

    integer n: the index of the J Bessel function.

    real alpha, beta: the coefficients of cosine(theta) and sine(theta)
    in the angular factor.

    real gamma: the multiplier for the radial factor.

    integer nxy: the number of Cartesian coordinates at which Z should
    be sampled.

    real x(nxy), y(nxy): the Cartesian coordinates at which Z should
    be sampled.

  Output:

    real z(nxy): the solution value at (x,y).
*/
{
  double bjn;
  double djn;
  int i;
  double *jz;
  double *R;
  double *rad;
  double *rho;
  double *T;
  double *theta;
  double *Z;
/*
  Convert Cartesian data to polar form.
*/
  theta = ( double * ) malloc ( nxy * sizeof ( double ) );
  for ( i = 0; i < nxy; i++ )
  {
    theta[i] = atan2 ( y[i], x[i] );
  }

  rad = ( double * ) malloc ( nxy * sizeof ( double ) );
  for ( i = 0; i < nxy; i++ )
  {
    rad[i] = sqrt ( x[i] * x[i] + y[i] * y[i] );
  }
/*
  Evaluate angular component T(theta).
*/
  T = ( double * ) malloc ( nxy * sizeof ( double ) );
  for ( i = 0; i < nxy; i++ )
  {
    T[i] = alpha * cos ( n * theta[i] ) + beta * sin ( n * theta[i] );
  }
/*
  Evaluate the m-th zero of the n-th J bessel function.

  To complicate things, Jn(x) has its first zero at zero,
  except for n = 0.  So...an input of m=0 requests the 
  special zero zero, which is illegal if n = 0.
*/
  rho = ( double * ) malloc ( nxy * sizeof ( double ) );

  if ( m == 0 )
  {
    if ( n == 0 )
    {
      printf ( "\n" );
      printf ( "helmholtz_exact(): Fatal error!\n" );
      printf ( "  m = 0 requests special first zero at 0.\n" );
      printf ( "  This is illegal, because n = 0 as well.\n" );
      exit ( 1 );
    }
    for ( i = 0; i < nxy; i++ )
    {
      rho[i] = 0.0;
    }
  }
  else
  {
    jz = jn_zeros ( n, m );
    for ( i = 0; i < nxy; i++ )
    {
      rho[i] = jz[m-1] * rad[i] / a;
    }
    free ( jz );
  }
/*
  Evaluate the radial component R(r), 
  the n-th Bessel J function evaluated at rho().
*/
  R = ( double * ) malloc ( nxy * sizeof ( double ) );
  for ( i = 0; i < nxy; i++ )
  {
    jyndd ( n, rho[i], &bjn, &djn );
    R[i] = gamma * bjn;
  }
/*
  Z is the elementwise product.
*/
  Z = ( double * ) malloc ( nxy * sizeof ( double ) );
  for ( i = 0; i < nxy; i++ )
  {
    Z[i] = R[i] * T[i];
  }

  free ( R );
  free ( rad );
  free ( rho );
  free ( T );
  free ( theta );

  return Z;
}
/******************************************************************************/

void jyndd ( int n, double x, double *bjn, double *djn )

/******************************************************************************/
/*
  Purpose:

    jyndd() evaluates a Bessel function Jn(x) and derivative.

  Licensing:

    This routine is copyrighted by Shanjie Zhang and Jianming Jin.  However,
    they give permission to incorporate this routine into a user program
    provided that the copyright is acknowledged.

  Modified:

    17 June 2025

  Author:

    Original FORTRAN77 version by Shanjie Zhang, Jianming Jin.
    This version by John Burkardt.

  Reference:

    Shanjie Zhang, Jianming Jin,
    Computation of Special Functions,
    Wiley, 1996,
    ISBN: 0-471-11963-6,
    LC: QA351.C45.

  Input:

    int N, the order.

    double X, the argument.

  Output:

    double *BJN, *DJN, the values of Jn(x), Jn'(x).
*/
{
  double *bj;
  double bs;
  double f;
  double f0;
  double f1;
  int k;
  int m;
  int mt;
  int nt;
  double su;

  if ( x <= 0.0 )
  {
    *bjn = 0.0;
    *djn = 0.0;
    return;
  }

  bj = ( double * ) malloc ( ( n + 2 ) * sizeof ( double ) );
 
  for ( nt = 1; nt <= 900; nt++ )
  {
    mt = ( int ) ( 0.5 * log10 ( 6.28 * nt ) 
      - nt * log10 ( 1.36 * fabs ( x ) / nt ) );

    if ( 20 < mt )
    {
      break;
    }

  }

  m = nt;
  bs = 0.0;
  f0 = 0.0;
  f1 = 1.0E-35;
  su = 0.0;

  for ( k = m; 0 <= k; k-- )
  {
    f = 2.0 * ( k + 1.0 ) * f1 / x - f0;

    if ( k <= n + 1 )
    {
      bj[k] = f;
    }

    if ( k == 2 * ( int ) ( k / 2 ) )
    {
      bs = bs + 2.0 * f;
      if ( k != 0 )
      {
        su = su + pow ( -1.0, k / 2 ) * f / k;
      }
    }
    f0 = f1;
    f1 = f;
  }

  for ( k = 0; k <= n + 1; k++ )
  {
    bj[k] = bj[k] / ( bs - f );
  }

  *bjn = bj[n];
  *djn = - bj[n+1] + n * bj[n] / x;

  free ( bj );

  return;
}
/******************************************************************************/

double *jn_zeros ( int n, int nt )

/******************************************************************************/
/*
  Purpose:

    jn_zeros() computes the zeros of a Bessel function Jn(x).

  Licensing:

    This routine is copyrighted by Shanjie Zhang and Jianming Jin.  However,
    they give permission to incorporate this routine into a user program
    provided that the copyright is acknowledged.

  Modified:

    12 June 2025

  Author:

    Original FORTRAN77 version by Shanjie Zhang, Jianming Jin.
    This version by John Burkardt.

  Reference:

    Shanjie Zhang, Jianming Jin,
    Computation of Special Functions,
    Wiley, 1996,
    ISBN: 0-471-11963-6,
    LC: QA351.C45.

  Input:

    int N: the order of the Bessel functions.

    int NT: the number of zeros.

  Output:

    double RJ0(NT): the first NT zeros of Jn(x).
*/
{
  double bjn;
  double djn;
  int l;
  double n_r8;
  double *rj0;
  double x;
  double x0;

  rj0 = ( double * ) malloc ( nt * sizeof ( double ) );

  n_r8 = ( double ) ( n );

  if ( n <= 20 )
  {
    x = 2.82141 + 1.15859 * n_r8;
  }
  else
  {
    x = n_r8 + 1.85576 * pow ( n_r8, 0.33333 ) 
      + 1.03315 / pow ( n_r8, 0.33333 );
  }

  l = 0;

  while ( true )
  {
    x0 = x;
    jyndd ( n, x, &bjn, &djn );
    x = x - bjn / djn;

    if ( 1.0E-09 < fabs ( x - x0 ) )
    {
      continue;
    }

    rj0[l] = x;
    l = l + 1;
    x = x + 3.1416 
      + ( 0.0972 + 0.0679 * n_r8 - 0.000354 * pow ( n_r8, 2 ) ) / l;

    if ( nt <= l )
    {
      break;
    }

  }

  return rj0;
}