c8lib.cpp

# include <cmath>
# include <complex>
# include <cstdlib>
# include <cstring>
# include <ctime>
# include <fstream>
# include <iomanip>
# include <iostream>

using namespace std;

# include "c8lib.hpp"

//****************************************************************************80

double c8_abs ( complex <double> x )

//****************************************************************************80
//
//  Purpose:
//
//    C8_ABS returns the absolute value of a C8.
//
//  Discussion:
//
//    A C8 is a complex <double> value.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    19 June 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> X, the value whose norm is desired.
//
//    Output, double C8_ABS, the magnitude of X.
//
{
  double value;

  value = sqrt ( pow ( real ( x ), 2 ) + pow ( imag ( x ), 2 ) );

  return value;
}
//****************************************************************************80

complex <double> c8_acos ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_ACOS evaluates the inverse cosine of a C8.
//
//  Discussion:
//
//    Here we use the relationship:
//
//      C8_ACOS ( Z ) = pi/2 - C8_ASIN ( Z ).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    06 October 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_ACOS, the function value.
//
{
  complex <double> c2;
  double c2_imag;
  double c2_real;
  double r8_pi_half = 1.57079632679489661923;

  c2 = c8_asin ( c1 );

  c2_real = r8_pi_half - real ( c2 );
  c2_imag =            - imag ( c2 );

  c2 = complex <double> ( c2_real, c2_imag );

  return c2;
}
//****************************************************************************80

complex <double> c8_acosh ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_ACOSH evaluates the inverse hyperbolic cosine of a C8.
//
//  Discussion:
//
//    Here we use the relationship:
//
//      C8_ACOSH ( Z ) = i * C8_ACOS ( Z ).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    06 October 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_ACOSH, the function value.
//
{
  complex <double> c2;

  c2 = c8_i ( ) * c8_acos ( c1 );
  
  return c2;
}
//****************************************************************************80

complex <double> c8_add ( complex <double> c1, complex <double> c2 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_ADD adds two C8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    01 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, C2, the arguments.
//
//    Output, complex <double> C8_ADD, the sum of C1 and C2.
//
{
  complex <double> c3;

  c3 = c1 + c2;

  return c3;
}
//****************************************************************************80

double c8_arg ( complex <double> x )

//****************************************************************************80
//
//  Purpose:
//
//    C8_ARG returns the argument of a C8.
//
//  Discussion:
//
//    A C8 is a complex <double> value.
//
//    The value returned by this function is always between 0 and 2*PI.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    08 February 2015
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> X, the value whose argument is desired.
//
//    Output, double C8_ARG, the argument of X.
//
{
  double value;

  if ( imag ( x ) == 0.0 && real ( x ) == 0.0 )
  {
    value = 0.0;
  }
  else
  {
    value = r8_atan ( imag ( x ), real ( x ) );
  }

  return value;
}
//****************************************************************************80

complex <double> c8_asin ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_ASIN evaluates the inverse sine of a C8.
//
//  Discussion:
//
//    Here we use the relationship:
//
//      C8_ASIN ( Z ) = - i * log ( i * z + sqrt ( 1 - z * z ) )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_ASIN, the function value.
//
{
  complex <double> c2;
  complex <double> c3;
  complex <double> c4;
  complex <double> ce;

  c2 = c8_i ( );
  c3 = c8_sqrt ( 1.0 - c1 * c1 );
  c4 = c8_log ( c3 + c2 * c1 );
  ce = - c2 * c4;

  return ce;
}
//****************************************************************************80

complex <double> c8_asinh ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_ASINH evaluates the inverse hyperbolic sine of a C8.
//
//  Discussion:
//
//    Here we use the relationship:
//
//      C8_ASINH ( Z ) = - i * C8_ASIN ( i * Z ).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_ASINH, the function value.
//
{
  complex <double> c2;
  complex <double> c3;
  complex <double> c4;
  complex <double> c5;
  complex <double> c6;

  c2 = c8_i ( );
  c3 = c2 * c1;
  c4 = c8_asin ( c3 );
  c5 = c2 * c4;
  c6 = - c5;

  return c6;
}
//****************************************************************************80

complex <double> c8_atan ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_ATAN evaluates the inverse tangent of a C8.
//
//  Discussion:
//
//    Here we use the relationship:
//
//      C8_ATAN ( Z ) = ( i / 2 ) * log ( ( 1 - i * z ) / ( 1 + i * z ) )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_ATAN, the function value.
//
{
  complex <double> c2;
  complex <double> c3;
  complex <double> c4;
  complex <double> c5;
  complex <double> c6;
  complex <double> c7;
  complex <double> c8;
  complex <double> c9;
  complex <double> cx;

  c2 = c8_i ( );
  c3 = c8_one ( );
  c4 = c8_mul ( c2, c1 );
  c5 = c8_sub ( c3, c4 );
  c6 = c8_add ( c3, c4 );
  c7 = c8_div ( c5, c6 );

  c8 = c8_log ( c7 );
  c9 = c8_mul ( c2, c8 );
  cx = c9 / 2.0;

  return cx;
}
//****************************************************************************80

complex <double> c8_atanh ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_ATANH evaluates the inverse hyperbolic tangent of a C8.
//
//  Discussion:
//
//    Here we use the relationship:
//
//      C8_ATANH ( Z ) = - i * C8_ATAN ( i * Z ).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_ATANH, the function value.
//
{
  complex <double> c2;
  complex <double> c3;
  complex <double> c4;
  complex <double> c5;
  complex <double> c6;

  c2 = c8_i ( );

  c3 = c8_mul ( c2, c1 );
  c4 = c8_atan ( c3 );
  c5 = c8_mul ( c2, c4 );
  c6 = c8_neg ( c5 );

  return c6;
}
//****************************************************************************80

complex <double> c8_conj ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_CONJ conjugates a C8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_CONJ, the function value.
//
{
  complex <double> c2;

  c2 = conj ( c1 );

  return c2;
}
//****************************************************************************80

void c8_copy ( complex <double> c1, complex <double> c2 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_COPY copies a C8.
//
//  Discussion:
//
//    The order of the arguments may seem unnatural, but it is arranged so
//    that the call
//
//      c8_copy ( c1, c2 )
//
//    mimics the assignment
//
//      c1 = c2.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, complex <double> C1, the copy of C2.
//
//    Input, complex <double> C2, the value to be copied.
//
{
  c1 = c2;

  return;
}
//****************************************************************************80

complex <double> c8_cos ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_COS evaluates the cosine of a C8.
//
//  Discussion:
//
//    We use the relationship:
//
//      C8_COS ( C ) = ( C8_EXP ( i * C ) + C8_EXP ( - i * C ) ) / 2
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_COS, the function value.
//
{
  complex <double> c2;

  c2 = ( exp ( c1 * c8_i ( ) ) + exp ( - c1 * c8_i ( ) ) ) / 2.0;

  return c2;
}
//****************************************************************************80

complex <double> c8_cosh ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_COSH evaluates the hyperbolic cosine of a C8.
//
//  Discussion:
//
//    A C8 is a complex <double> value.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_COSH, the function value.
//
{
  complex <double> c2;
  complex <double> c3;
  complex <double> c4;
  complex <double> c5;
  complex <double> c6;

  c2 = c8_exp ( c1 );

  c3 = c8_neg ( c1 );
  c4 = c8_exp ( c3 );

  c5 = c8_add ( c2, c4 );
  c6 = c8_div_r8 ( c5, 2.0 );

  return c6;
}
//****************************************************************************80

complex <double> c8_cube_root ( complex <double> x )

//****************************************************************************80
//
//  Purpose:
//
//    C8_CUBE_ROOT returns the principal cube root of a C8.
//
//  Discussion:
//
//    A C8 is a complex <double> value.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    06 November 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> X, the number whose cube root is desired.
//
//    Output, complex <double> C8_CUBE_ROOT, the cube root of X.
//
{
  double argument;
  double magnitude;
  complex <double> value;

  argument = c8_arg ( x );
  magnitude = c8_mag ( x );

  if ( magnitude == 0.0 )
  {
    value = complex <double> ( 0.0, 0.0 );
  }
  else
  {
    value = pow ( magnitude, ( double ) ( 1.0 / 3.0 ) ) 
      * complex <double> ( cos ( argument / 3.0 ), sin ( argument / 3.0 ) );
  }

  return value;
}
//****************************************************************************80

complex <double> c8_div ( complex <double> c1, complex <double> c2 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_DIV divides two C8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, C2, the arguments.
//
//    Output, complex <double> C8_DIV, the function value.
//
{
  double c2_norm;
  complex <double> c3;
  double c3_imag;
  double c3_real;

  c2_norm = c8_abs ( c2 );

  c3_real = ( real ( c1 ) * real ( c2 ) 
            + imag ( c1 ) * imag ( c2 ) ) / c2_norm / c2_norm;

  c3_imag = ( imag ( c1 ) * real ( c2 ) 
            - real ( c1 ) * imag ( c2 ) ) / c2_norm / c2_norm;

  c3 = complex <double> ( c3_real, c3_imag );

  return c3;
}
//****************************************************************************80

complex <double> c8_div_r8 ( complex <double> c1, double r )

//****************************************************************************80
//
//  Purpose:
//
//    C8_DIV_R8 divides a C8 by an R8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the value to be divided.
//
//    Input, double R, the divisor.
//
//    Output, complex <double> C8_DIV_R8, the function value.
//
{
  complex <double> c2;

  c2 = c1 / r;

  return c2;
}
//****************************************************************************80

complex <double> c8_exp ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_EXP exponentiates a C8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_EXP, the function value.
//
{
  complex <double> c2;

  c2 = exp ( c1 );

  return c2;
}
//****************************************************************************80

complex <double> c8_i ( )

//****************************************************************************80
//
//  Purpose:
//
//    C8_I returns the value of the imaginary unit, i as a C8.
//
//  Discussion:
//
//    A C8 is a complex <double> value.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    06 November 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, complex <double> C8_I, the value of complex i.
//
{
  complex <double> value;

  value = complex <double> ( 0.0, 1.0 );

  return value;
}
//****************************************************************************80

double c8_imag ( complex <double> c )

//****************************************************************************80
//
//  Purpose:
//
//    C8_IMAG returns the imaginary part of a C8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C, the argument.
//
//    Output, double C8_IMAG, the function value.
//
{
  double value;

  value = imag ( c );

  return value;
}
//****************************************************************************80

complex <double> c8_inv ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_INV inverts a C8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_INV, the function value;
//
{
  complex <double> c2;

  c2 = 1.0 / c1;

  return c2;
}
//****************************************************************************80

bool c8_le_l1 ( complex <double> x, complex <double> y )

//****************************************************************************80
//
//  Purpose:
//
//    C8_LE_L1 := X <= Y for C8 values, and the L1 norm.
//
//  Discussion:
//
//    A C8 is a complex double precision value.
//
//    The L1 norm can be defined here as:
//
//      C8_NORM_L1(X) = abs ( real (X) ) + abs ( imag (X) )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    20 April 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> X, Y, the values to be compared.
//
//    Output, bool C8_LE_L1, is TRUE if X <= Y.
//
{
  bool value;

  if ( r8_abs ( real ( x ) ) + r8_abs ( imag ( x ) ) <= 
       r8_abs ( real ( y ) ) + r8_abs ( imag ( y ) ) )
  {
    value = true;
  }
  else
  {
    value = false;
  }
  return value;
}
//****************************************************************************80

bool c8_le_l2 ( complex <double> x, complex <double> y )

//****************************************************************************80
//
//  Purpose:
//
//    C8_LE_L2 := X <= Y for C8 values, and the L2 norm.
//
//  Discussion:
//
//    A C8 is a complex double precision value.
//
//    The L2 norm can be defined here as:
//
//      C8_NORM_L2(X) = sqrt ( ( real (X) )^2 + ( imag (X) )^2 )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    20 April 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> X, Y, the values to be compared.
//
//    Output, bool C8_LE_L2, is TRUE if X <= Y.
//
{
  bool value;

  if ( pow ( real ( x ), 2 ) + pow ( imag ( x ), 2 ) <= 
       pow ( real ( y ), 2 ) + pow ( imag ( y ), 2 ) )
  {
    value = true;
  }
  else
  {
    value = false;
  }
  return value;
}
//****************************************************************************80

bool c8_le_li ( complex <double> x, complex <double> y )

//****************************************************************************80
//
//  Purpose:
//
//    C8_LE_LI := X <= Y for C8 values, and the L-oo norm.
//
//  Discussion:
//
//    A C8 is a complex double precision value.
//
//    The L-oo norm can be defined here as:
//
//      C8_NORM_LI(X) = max ( abs ( real (X) ), abs ( imag (X) ) )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    20 April 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> X, Y, the values to be compared.
//
//    Output, bool C8_LE_LI, is TRUE if X <= Y.
//
{
  bool value;

  if ( r8_max ( r8_abs ( real ( x ) ), r8_abs ( imag ( x ) ) ) <= 
       r8_max ( r8_abs ( real ( y ) ), r8_abs ( imag ( y ) ) ) )
  {
    value = true;
  }
  else
  {
    value = false;
  }
  return value;
}
//****************************************************************************80

complex <double> c8_log ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_LOG evaluates the logarithm of a C8.
//
//  Discussion:
//
//    Here we use the relationship:
//
//      C8_LOG ( Z ) = LOG ( MAG ( Z ) ) + i * ARG ( Z )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_LOG, the function value.
//
{
  double arg;
  complex <double> c2;
  double mag;

  arg = c8_arg ( c1 );
  mag = c8_mag ( c1 );

  c2 = complex <double> ( log ( mag ), arg );

  return c2;
}
//****************************************************************************80

double c8_mag ( complex <double> x )

//****************************************************************************80
//
//  Purpose:
//
//    C8_MAG returns the magnitude of a C8.
//
//  Discussion:
//
//    A C8 is a complex <double> value.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    22 October 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> X, the value whose norm is desired.
//
//    Output, double C8_MAG, the magnitude of X.
//
{
  double magnitude;

  magnitude = sqrt ( pow ( real ( x ), 2 ) + pow ( imag ( x ), 2 ) );

  return magnitude;
}
//****************************************************************************80

complex <double> c8_mul ( complex <double> c1, complex <double> c2 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_MUL multiplies two C8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, C2, the arguments.
//
//    Output, complex <double> C8_MUL, the function value.
//
{
  complex <double> c3;

  c3 = c1 * c2;

  return c3;
}
//****************************************************************************80

complex <double> c8_neg ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_NEG negates a C8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_NEG, the function value.
//
{
  complex <double> c2;

  c2 = - c1;

  return c2;
}
//****************************************************************************80

complex <double> c8_nint ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_NINT returns the nearest complex integer of a C8.
//
//  Discussion:
//
//    A C8 is a complex <double> value.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    10 May 2014
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the value to be NINT'ed.
//
//    Output, complex <double> C8_NINT, the NINT'ed value.
//
{
  double r;
  double r_min;
  double x;
  double x_min;
  double xc;
  double y;
  double y_min;
  double yc;
  complex <double> value;

  xc = real ( c1 );
  yc = imag ( c1 );
//
//  Lower left.
//
  x = r8_floor ( real ( c1 ) );
  y = r8_floor ( imag ( c1 ) );
  r = pow ( x - xc, 2 ) + pow ( y - yc, 2 );
  r_min = r;
  x_min = x;
  y_min = y;
//
//  Lower right.
//
  x = r8_floor ( real ( c1 ) ) + 1.0;
  y = r8_floor ( imag ( c1 ) );
  r = pow ( x - xc, 2 ) + pow ( y - yc, 2 );
  if ( r < r_min )
  {
    r_min = r;
    x_min = x;
    y_min = y;
  }
//
//  Upper right.
//
  x = r8_floor ( real ( c1 ) ) + 1.0;
  y = r8_floor ( imag ( c1 ) ) + 1.0;
  r = pow ( x - xc, 2 ) + pow ( y - yc, 2 );
  if ( r < r_min )
  {
    r_min = r;
    x_min = x;
    y_min = y;
  }
//
//  Upper left.
//
  x = r8_floor ( real ( c1 ) );
  y = r8_floor ( imag ( c1 ) ) + 1.0;
  r = pow ( x - xc, 2 ) + pow ( y - yc, 2 );
  if ( r < r_min )
  {
    r_min = r;
    x_min = x;
    y_min = y;
  }

  value = complex <double> ( x_min, y_min );

  return value;
}
//****************************************************************************80

double c8_norm_l1 ( complex <double> x )

//****************************************************************************80
//
//  Purpose:
//
//    C8_NORM_L1 evaluates the L1 norm of a C8.
//
//  Discussion:
//
//    A C8 is a complex <double> value.
//
//    Numbers of equal norm lie along diamonds centered at (0,0).
//
//    The L1 norm can be defined here as:
//
//      C8_NORM_L1(X) = abs ( real (X) ) + abs ( imag (X) )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    19 June 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> X, the value whose norm is desired.
//
//    Output, double C8_NORM_L1, the norm of X.
//
{
  double value;

  value = r8_abs ( real ( x ) ) + r8_abs ( imag ( x ) );

  return value;
}
//****************************************************************************80

double c8_norm_l2 ( complex <double> x )

//****************************************************************************80
//
//  Purpose:
//
//    C8_NORM_L2 evaluates the L2 norm of a C8.
//
//  Discussion:
//
//    A C8 is a complex <double> value.
//
//    Numbers of equal norm lie on circles centered at (0,0).
//
//    The L2 norm can be defined here as:
//
//      C8_NORM_L2(X) = sqrt ( ( real (X) )^2 + ( imag ( X ) )^2 )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    21 April 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> X, the value whose norm is desired.
//
//    Output, double C8_NORM_L2, the 2-norm of X.
//
{
  double value;

  value = sqrt ( pow ( real ( x ), 2 )
               + pow ( imag ( x ), 2 ) );

  return value;
}
//****************************************************************************80

double c8_norm_li ( complex <double> x )

//****************************************************************************80
//
//  Purpose:
//
//    C8_NORM_LI evaluates the L-oo norm of a C8.
//
//  Discussion:
//
//    A C8 is a complex <double> value.
//
//    Numbers of equal norm lie along squares whose centers are at (0,0).
//
//    The L-oo norm can be defined here as:
//
//      C8_NORM_LI(X) = max ( abs ( real (X) ), abs ( imag (X) ) )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    19 June 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> X, the value whose norm is desired.
//
//    Output, double C8_NORM_LI, the L-oo norm of X.
//
{
  double value;

  value = r8_max ( r8_abs ( real ( x ) ), r8_abs ( imag ( x ) ) );

  return value;
}
//****************************************************************************80

complex <double> c8_normal_01 ( int &seed )

//****************************************************************************80
//
//  Purpose:
//
//    C8_NORMAL_01 returns a unit pseudonormal C8.
//
//  Discussion:
//
//    A C8 is a complex <double> value.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    06 November 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input/output, int &SEED, a seed for the random number generator.
//
//    Output, complex <double> C8_NORMAL_01, a unit pseudornormal value.
//
{
  const double r8_pi = 3.141592653589793;
  double v1;
  double v2;
  double x_c;
  double x_r;
  complex <double> value;

  v1 = r8_uniform_01 ( seed );
  v2 = r8_uniform_01 ( seed );

  x_r = sqrt ( -2.0 * log ( v1 ) ) * cos ( 2.0 * r8_pi * v2 );
  x_c = sqrt ( -2.0 * log ( v1 ) ) * sin ( 2.0 * r8_pi * v2 );

  value = complex <double> ( x_r, x_c );

  return value;
}
//****************************************************************************80

complex <double> c8_one ( )

//****************************************************************************80
//
//  Purpose:
//
//    C8_ONE returns the value of complex 1.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    06 November 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, complex <double> C8_ONE, the value of complex 1.
//
{
  complex <double> value;

  value = complex <double> ( 1.0, 0.0);

  return value;
}
//****************************************************************************80

void c8_print ( complex <double> a, string title )

//****************************************************************************80
//
//  Purpose:
//
//    C8_PRINT prints a C8.
//
//  Discussion:
//
//    A C8 is a complex <double> value.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    10 September 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> A, the value to be printed.
//
//    Input, string TITLE, a title.
//
{
  cout << title
       << "  ( " << setw(14) << real ( a )
       << ", "   << setw(14) << imag ( a ) << " )\n";

  return;
}
//****************************************************************************80

double c8_real ( complex <double> c )

//****************************************************************************80
//
//  Purpose:
//
//    C8_REAL returns the real part of a C8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C, the complex number.
//
//    Output, double C8_REAL, the function value.
//
{
  double value;

  value = real ( c );

  return value;
}
//****************************************************************************80

complex <double> c8_sin ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_SIN evaluates the sine of a C8.
//
//  Discussion:
//
//    We use the relationship:
//
//      C8_SIN ( C ) = - i * ( C8_EXP ( i * C ) - C8_EXP ( - i * C ) ) / 2
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_SIN, the function value.
//
{
  complex <double> c2;
  complex <double> c3;
  complex <double> c4;
  complex <double> c5;
  complex <double> c6;
  complex <double> c7;
  complex <double> c8;
  complex <double> c9;
  complex <double> cx;
  double r;

  c2 = c8_i ( );

  c3 = c8_mul ( c2, c1 );
  c4 = c8_exp ( c3 );

  c5 = c8_neg ( c3 );
  c6 = c8_exp ( c5 );

  c7 = c8_sub ( c4, c6 );

  r = 2.0;
  c8 = c8_div_r8 ( c7, r );
  c9 = c8_mul ( c8, c2 );
  cx = c8_neg ( c9 );

  return cx;
}
//****************************************************************************80

complex <double> c8_sinh ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_SINH evaluates the hyperbolic sine of a C8.
//
//  Discussion:
//
//    We use the relationship:
//
//      C8_SINH ( C ) = ( C8_EXP ( C ) - C8_EXP ( - i * C ) ) / 2
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_SINH, the function value.
//
{
  complex <double> c2;
  complex <double> c3;
  complex <double> c4;
  complex <double> c5;
  complex <double> c6;
  double r;

  c2 = c8_exp ( c1 );

  c3 = c8_neg ( c1 );
  c4 = c8_exp ( c3 );

  c5 = c8_sub ( c2, c4 );

  r = 2.0;
  c6 = c8_div_r8 ( c5, r );

  return c6;
}
//****************************************************************************80

complex <double> c8_sqrt ( complex <double> x )

//****************************************************************************80
//
//  Purpose:
//
//    C8_SQRT returns the principal square root of a C8.
//
//  Discussion:
//
//    A C8 is a complex <double> value.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    06 November 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> X, the number whose square root is desired.
//
//    Output, complex <double> C8_SQRT, the square root of X.
//
{
  double argument;
  double magnitude;
  complex <double> value;

  argument = c8_arg ( x );
  magnitude = c8_mag ( x );

  if ( magnitude == 0.0 )
  {
    value = complex <double> ( 0.0, 0.0 );
  }
  else
  {
    value = sqrt ( magnitude ) 
      * complex <double> ( cos ( argument / 2.0 ), sin ( argument / 2.0 ) );
  }

  return value;
}
//****************************************************************************80

complex <double> c8_sub ( complex <double> c1, complex <double> c2 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_SUB subtracts two C8's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, C2, the arguments.
//
//    Output, complex <double> C8_SUB, the function value.
//
{
  complex <double> c3;

  c3 = c1 - c2;

  return c3;
}
//****************************************************************************80

void c8_swap ( complex <double> &x, complex <double> &y )

//****************************************************************************80
//
//  Purpose:
//
//    C8_SWAP swaps two C8's.
//
//  Discussion:
//
//    A C8 is a complex <double> value.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    19 October 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input/output, complex <double> &X, &Y.  On output, the values of X and
//    Y have been interchanged.
//
{
  complex <double> z;

  z = x; 
  x = y;
  y = z;

  return;
}
//****************************************************************************80

complex <double> c8_tan ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_TAN evaluates the tangent of a C8.
//
//  Discussion:
//
//    We use the relationship:
//
//      C8_TAN ( C ) = - i * ( C8_EXP ( i * C ) - C8_EXP ( - i * C ) ) 
//                         / ( C8_EXP ( I * C ) + C8_EXP ( - i * C ) )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_TAN, the function value.
//
{
  complex <double> c2;
  complex <double> c3;
  complex <double> c4;
  complex <double> c5;
  complex <double> c6;
  complex <double> c7;
  complex <double> c8;
  complex <double> c9;
  complex <double> cx;
  complex <double> ce;

  c2 = c8_i ( );
  c3 = c8_mul ( c2, c1 );
  c4 = c8_neg ( c3 );
  
  c5 = c8_exp ( c3 );
  c6 = c8_exp ( c4 );

  c7 = c8_sub ( c5, c6 );
  c8 = c8_add ( c5, c6 );

  c9 = c8_div ( c7, c8 );
  cx = c8_mul ( c2, c9 );
  ce = c8_neg ( cx );

  return ce;
}
//****************************************************************************80

complex <double> c8_tanh ( complex <double> c1 )

//****************************************************************************80
//
//  Purpose:
//
//    C8_TANH evaluates the hyperbolic tangent of a C8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C1, the argument.
//
//    Output, complex <double> C8_TANH, the function value.
//
{
  complex <double> c2;
  complex <double> c3;
  complex <double> c4;
  complex <double> c5;
  complex <double> c6;
  complex <double> c7;

  c2 = c8_exp ( c1 );

  c3 = c8_neg ( c1 );
  c4 = c8_exp ( c3 );

  c5 = c8_sub ( c2, c4 );
  c6 = c8_add ( c2, c4 );

  c7 = c8_div ( c5, c6 );

  return c7;
}
//****************************************************************************80

void c8_to_cartesian ( complex <double> c, double &x, double &y )

//****************************************************************************80
//
//  Purpose:
//
//    C8_TO_CARTESIAN converts a C8 to Cartesian form.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C, the argument.
//
//    Output, double &X, &Y, the Cartesian form.
//
{
  x = real ( c );
  y = imag ( c );

  return;
}
//****************************************************************************80

void c8_to_polar ( complex <double> c, double &r, double &theta )

//****************************************************************************80
//
//  Purpose:
//
//    C8_TO_POLAR converts a C8 to polar form.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, complex <double> C, the argument.
//
//    Output, double &R, &THETA, the polar form.
//
{
  r = c8_abs ( c );
  theta = c8_arg ( c );

  return;
}
//****************************************************************************80

complex <double> c8_uniform_01 ( int &seed )

//****************************************************************************80
//
//  Purpose:
//
//    C8_UNIFORM_01 returns a unit pseudorandom C8.
//
//  Discussion:
//
//    A C8 is a complex <double> value.
//
//    The angle should be uniformly distributed between 0 and 2 * PI,
//    the square root of the radius uniformly distributed between 0 and 1.
//
//    This results in a uniform distribution of values in the unit circle.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    21 October 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input/output, int &SEED, the "seed" value, which should NOT be 0.
//    On output, SEED has been updated.
//
//    Output, complex <double> C8_UNIFORM_01, a pseudorandom complex value.
//
{
  double r;
  int k;
  const double r8_pi = 3.141592653589793;
  double theta;
  complex <double> value;

  k = seed / 127773;

  seed = 16807 * ( seed - k * 127773 ) - k * 2836;

  if ( seed < 0 )
  {
    seed = seed + 2147483647;
  }

  r = sqrt ( ( double ) ( seed ) * 4.656612875E-10 );

  k = seed / 127773;

  seed = 16807 * ( seed - k * 127773 ) - k * 2836;

  if ( seed < 0 )
  {
    seed = seed + 2147483647;
  }

  theta = 2.0 * r8_pi * ( ( double ) ( seed ) * 4.656612875E-10 );

  value = r * complex <double> ( cos ( theta ), sin ( theta ) );

  return value;
}
//****************************************************************************80

complex <double> c8_zero ( )

//****************************************************************************80
//
//  Purpose:
//
//    C8_ZERO returns the value of 0 as a C8.
//
//  Discussion:
//
//    A C8 is a complex <double> value.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    06 November 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, complex <double> C8_ZERO, the value of complex 0.
//
{
  complex <double> value;

  value = complex <double> ( 0.0, 0.0 );

  return value;
}
//****************************************************************************80

void c8mat_add ( int m, int n, complex <double> alpha, complex <double> a[],
  complex <double> beta, complex <double> b[], complex <double> c[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_ADD combines two C8MAT's using complex scalar factors.
//
//  Discussion:
//
//    An C8MAT is a doubly dimensioned array of complex double precision values, 
//    which may be stored as a vector in column-major order.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    03 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input, complex <double> ALPHA, the first scale factor.
//
//    Input, complex <double> A[M*N], the first matrix.
//
//    Input, complex <double> BETA, the second scale factor.
//
//    Input, complex <double> B[M*N], the second matrix.
//
//    Output, complex <double> C[M*N], the result.
//
{
  int i;
  int j;

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      c[i+j*m] = alpha * a[i+j*m] + beta * b[i+j*m];
    }
  }
  return;
}
//****************************************************************************80

void c8mat_add_r8 ( int m, int n, double alpha, complex <double> a[],
  double beta, complex <double> b[], complex <double> c[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_ADD_R8 combines two C8MAT's using real scalar factors.
//
//  Discussion:
//
//    An C8MAT is a doubly dimensioned array of complex double precision values, 
//    which may be stored as a vector in column-major order.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    03 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input, double ALPHA, the first scale factor.
//
//    Input, complex <double> A[M*N], the first matrix.
//
//    Input, double BETA, the second scale factor.
//
//    Input, complex <double> B[M*N], the second matrix.
//
//    Output, complex <double> C[M*N], the result.
//
{
  int i;
  int j;

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      c[i+j*m] = alpha * a[i+j*m] + beta * b[i+j*m];
    }
  }
  return;
}
//****************************************************************************80

void c8mat_copy ( int m, int n, complex <double> a1[], complex <double> a2[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_COPY copies one C8MAT to another.
//
//  Discussion:
//
//    An C8MAT is a doubly dimensioned array of complex double precision values, 
//    which may be stored as a vector in column-major order.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    27 August 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input, complex <double> A1[M*N], the matrix to be copied.
//
//    Output, complex <double> A2[M*N], the copy of A1.
//
{
  int i;
  int j;

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      a2[i+j*m] = a1[i+j*m];
    }
  }
  return;
}
//****************************************************************************80

complex <double> *c8mat_copy_new ( int m, int n, complex <double> a1[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_COPY_NEW copies one C8MAT to a "new" C8MAT.
//
//  Discussion:
//
//    An C8MAT is a doubly dimensioned array of complex double precision values, 
//    which may be stored as a vector in column-major order.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    27 August 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input, complex <double> A1[M*N], the matrix to be copied.
//
//    Output, complex <double> C8MAT_COPY_NEW[M*N], the copy of A1.
//
{
  complex <double> *a2;
  int i;
  int j;

  a2 = new complex <double>[m*n];

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      a2[i+j*m] = a1[i+j*m];
    }
  }
  return a2;
}
//****************************************************************************80

void c8mat_fss ( int n, complex <double> a[], int nb, complex <double> x[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_FSS factors and solves a system with multiple right hand sides.
//
//  Discussion:
//
//    This routine uses partial pivoting, but no pivot vector is required.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    05 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//    N must be positive.
//
//    Input/output, complex <double> A[N*N].
//    On input, A is the coefficient matrix of the linear system.
//    On output, A is in unit upper triangular form, and
//    represents the U factor of an LU factorization of the
//    original coefficient matrix.
//
//    Input, int NB, the number of right hand sides.
//
//    Input/output, complex <double> X[N*NB], on input, the right hand sides of the
//    linear systems.  On output, the solutions of the linear systems.
//
{
  int i;
  int ipiv;
  int j;
  int jcol;
  double piv;
  complex <double> t;

  for ( jcol = 1; jcol <= n; jcol++ )
  {
//
//  Find the maximum element in column I.
//
    piv = c8_abs ( a[jcol-1+(jcol-1)*n] );
    ipiv = jcol;
    for ( i = jcol+1; i <= n; i++ )
    {
      if ( piv < c8_abs ( a[i-1+(jcol-1)*n] ) )
      {
        piv = c8_abs ( a[i-1+(jcol-1)*n] );
        ipiv = i;
      }
    }

    if ( piv == 0.0 )
    {
      cout << "\n";
      cout << "C8MAT_FSS - Fatal error!\n";
      cout << "  Zero pivot on step " << jcol << "\n";
      exit ( 1 );
    }
//
//  Switch rows JCOL and IPIV, and X.
//
    if ( jcol != ipiv )
    {
      for ( j = 1; j <= n; j++ )
      {
        t                 = a[jcol-1+(j-1)*n];
        a[jcol-1+(j-1)*n] = a[ipiv-1+(j-1)*n];
        a[ipiv-1+(j-1)*n] = t;
      }
      for ( j = 0; j < nb; j++ )
      {
        t            = x[jcol-1+j*n];
        x[jcol-1+j*n] = x[ipiv-1+j*n];
        x[ipiv-1+j*n] = t;
      }
    }
//
//  Scale the pivot row.
//
    t = a[jcol-1+(jcol-1)*n];
    a[jcol-1+(jcol-1)*n] = 1.0;
    for ( j = jcol+1; j <= n; j++ )
    {
      a[jcol-1+(j-1)*n] = a[jcol-1+(j-1)*n] / t;
    }
    for ( j = 0; j < nb; j++ )
    {
      x[jcol-1+j*n] = x[jcol-1+j*n] / t;
    }
//
//  Use the pivot row to eliminate lower entries in that column.
//
    for ( i = jcol+1; i <= n; i++ )
    {
      if ( a[i-1+(jcol-1)*n] != 0.0 )
      {
        t = - a[i-1+(jcol-1)*n];
        a[i-1+(jcol-1)*n] = 0.0;
        for ( j = jcol+1; j <= n; j++ )
        {
          a[i-1+(j-1)*n] = a[i-1+(j-1)*n] + t * a[jcol-1+(j-1)*n];
        }
        for ( j = 0; j < nb; j++ )
        {
          x[i-1+j*n] = x[i-1+j*n] + t * x[jcol-1+j*n];
        }
      }
    }
  }
//
//  Back solve.
//
  for ( jcol = n; 2 <= jcol; jcol-- )
  {
    for ( i = 1; i < jcol; i++ )
    {
      for ( j = 0; j < nb; j++ )
      {
        x[i-1+j*n] = x[i-1+j*n] - a[i-1+(jcol-1)*n] * x[jcol-1+j*n];
      }
    }
  }

  return;
}
//****************************************************************************80

complex <double> *c8mat_fss_new ( int n, complex <double> a[], int nb, 
  complex <double> b[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_FSS_NEW factors and solves a system with multiple right hand sides.
//
//  Discussion:
//
//    This routine uses partial pivoting, but no pivot vector is required.
//
//    A C8MAT is a doubly dimensioned array of C8 values, stored as a vector
//    in column-major order.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//    N must be positive.
//
//    Input/output, complex <double> A[N*N].
//    On input, A is the coefficient matrix of the linear system.
//    On output, A is in unit upper triangular form, and
//    represents the U factor of an LU factorization of the
//    original coefficient matrix.
//
//    Input, int NB, the number of right hand sides.
//
//    Input, complex <double> B[N*NB], the right hand sides of the linear systems.
//
//    Output, complex <double> C8MAT_FSS_NEW[N*NB], the solutions of the 
//    linear systems.
//
{
  int i;
  int ipiv;
  int j;
  int jcol;
  double piv;
  complex <double> t;
  complex <double> *x;

  x = new complex <double>[n*nb];

  for ( j = 0; j < nb; j++ )
  {
    for ( i = 0; i < n; i++ )
    {
      x[i+j*n] = b[i+j*n];
    }
  }
  for ( jcol = 1; jcol <= n; jcol++ )
  {
//
//  Find the maximum element in column I.
//
    piv = c8_abs ( a[jcol-1+(jcol-1)*n] );
    ipiv = jcol;
    for ( i = jcol + 1; i <= n; i++ )
    {
      if ( piv < c8_abs ( a[i-1+(jcol-1)*n] ) )
      {
        piv = c8_abs ( a[i-1+(jcol-1)*n] );
        ipiv = i;
      }
    }

    if ( piv == 0.0 )
    {
      cerr << "\n";
      cerr << "C8MAT_FSS_NEW - Fatal error!\n";
      cerr << "  Zero pivot on step " << jcol << "\n";
      exit ( 1 );
    }
//
//  Switch rows JCOL and IPIV, and X.
//
    if ( jcol != ipiv )
    {
      for ( j = 1; j <= n; j++ )
      {
        t                 = a[jcol-1+(j-1)*n];
        a[jcol-1+(j-1)*n] = a[ipiv-1+(j-1)*n];
        a[ipiv-1+(j-1)*n] = t;
      }
      for ( j = 0; j < nb; j++ )
      {
        t            = x[jcol-1+j*n];
        x[jcol-1+j*n] = x[ipiv-1+j*n];
        x[ipiv-1+j*n] = t;
      }
    }
//
//  Scale the pivot row.
//
    t = a[jcol-1+(jcol-1)*n];
    a[jcol-1+(jcol-1)*n] = 1.0;
    for ( j = jcol + 1; j <= n; j++ )
    {
      a[jcol-1+(j-1)*n] = a[jcol-1+(j-1)*n] / t;
    }
    for ( j = 0; j < nb; j++ )
    {
      x[jcol-1+j*n] = x[jcol-1+j*n] / t;
    }
//
//  Use the pivot row to eliminate lower entries in that column.
//
    for ( i = jcol+1; i <= n; i++ )
    {
      if ( a[i-1+(jcol-1)*n] != 0.0 )
      {
        t = - a[i-1+(jcol-1)*n];
        a[i-1+(jcol-1)*n] = 0.0;
        for ( j = jcol+1; j <= n; j++ )
        {
          a[i-1+(j-1)*n] = a[i-1+(j-1)*n] + t * a[jcol-1+(j-1)*n];
        }
        for ( j = 0; j < nb; j++ )
        {
          x[i-1+j*n] = x[i-1+j*n] + t * x[jcol-1+j*n];
        }
      }
    }
  }
//
//  Back solve.
//
  for ( jcol = n; 2 <= jcol; jcol-- )
  {
    for ( i = 1; i < jcol; i++ )
    {
      for ( j = 0; j < nb; j++ )
      {
        x[i-1+j*n] = x[i-1+j*n] - a[i-1+(jcol-1)*n] * x[jcol-1+j*n];
      }
    }
  }

  return x;
}
//****************************************************************************80

complex <double> *c8mat_identity_new ( int n )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_IDENTITY_NEW sets a C8MAT to the identity.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    06 November 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the order of the matrix.
//
//    Output, complex <double> C8MAT_IDENTITY_NEW[N*N], the matrix.
//
{
  complex <double> *a;
  int i;
  int j;

  a = new complex <double> [n*n];

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < n; i++ )
    {
      if ( i == j )
      {
        a[i+j*n] = complex <double> ( 1.0, 0.0 );
      }
      else
      {
        a[i+j*n] = complex <double> ( 0.0, 0.0 );
      }
    }
  }
  return a;
}
//****************************************************************************80

complex <double> *c8mat_indicator_new ( int m, int n )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_INDICATOR_NEW returns the C8MAT indicator matrix.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    06 November 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Output, complex <double> C8MAT_INDICATOR_NEW[M*N], the matrix.
//
{
  complex <double> *a;
  int i;
  int j;

  a = new complex <double> [m*n];

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      a[i+j*m] = complex <double> ( i + 1, j + 1 );
    }
  }
  return a;
}
//****************************************************************************80

void c8mat_minvm ( int n1, int n2, complex <double> a[], 
  complex <double> b[], complex <double> c[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_MINVM returns inverse(A) * B for C8MAT's.
//
//  Discussion:
//
//    A C8MAT is an array of C8 values.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    05 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N1, N2, the order of the matrices.
//
//    Input, complex <double> A[N1*N1], B[N1*N2], the matrices.
//
//    Output, complex <double> C[N1*N2], the result, 
//    C = inverse(A) * B.
//
{
  complex <double> *alu;

  alu = c8mat_copy_new ( n1, n1, a );

  c8mat_copy ( n1, n2, b, c );

  c8mat_fss ( n1, alu, n2, c );
 
  delete [] alu;

  return;
}
//****************************************************************************80

complex <double> *c8mat_minvm_new ( int n1, int n2, complex <double> a[], 
  complex <double> b[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_MINVM_NEW returns inverse(A) * B for C8MAT's.
//
//  Discussion:
//
//    A C8MAT is an array of C8 values.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N1, N2, the order of the matrices.
//
//    Input, complex <double> A[N1*N1], B[N1*N2], the matrices.
//
//    Output, complex <double> C8MAT_MINVM_NEW[N1*N2], the result, 
//    C = inverse(A) * B.
//
{
  complex <double> *alu;
  complex <double> *c;

  alu = c8mat_copy_new ( n1, n1, a );
  c = c8mat_fss_new ( n1, alu, n2, b );
 
  delete [] alu;

  return c;
}
//****************************************************************************80

void c8mat_mm ( int n1, int n2, int n3, complex <double> a[], 
  complex <double> b[], complex <double> c[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_MM multiplies two matrices.
//
//  Discussion:
//
//    A C8MAT is a doubly dimensioned array of C8 values, stored as a vector
//    in column-major order.
//
//    For this routine, the result is returned as the function value.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    03 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N1, N2, N3, the order of the matrices.
//
//    Input, complex <double> A[N1*N2], complex <double> B[N2*N3], 
//    the matrices to multiply.
//
//    Output, complex <double> C[N1*N3], the product matrix C = A * B.
//
{
  complex <double> *c1;
  int i;
  int j;
  int k;

  c1 = new complex <double> [n1*n3];

  for ( i = 0; i < n1; i ++ )
  {
    for ( j = 0; j < n3; j++ )
    {
      c1[i+j*n1] = 0.0;
      for ( k = 0; k < n2; k++ )
      {
        c1[i+j*n1] = c1[i+j*n1] + a[i+k*n1] * b[k+j*n2];
      }
    }
  }

  c8mat_copy ( n1, n3, c1, c );

  delete [] c1;

  return;
}
//****************************************************************************80

complex <double> *c8mat_mm_new ( int n1, int n2, int n3, complex <double> a[], 
  complex <double> b[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_MM_NEW multiplies two matrices.
//
//  Discussion:
//
//    A C8MAT is a doubly dimensioned array of C8 values, stored as a vector
//    in column-major order.
//
//    For this routine, the result is returned as the function value.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    25 June 2011
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N1, N2, N3, the order of the matrices.
//
//    Input, complex <double> A[N1*N2], complex <double> B[N2*N3], 
//    the matrices to multiply.
//
//    Output, complex <double> C8MAT_MM_NEW[N1*N3], the product matrix C = A * B.
//
{
  complex <double> *c;
  int i;
  int j;
  int k;

  c = new complex <double> [n1*n3];

  for ( i = 0; i < n1; i ++ )
  {
    for ( j = 0; j < n3; j++ )
    {
      c[i+j*n1] = 0.0;
      for ( k = 0; k < n2; k++ )
      {
        c[i+j*n1] = c[i+j*n1] + a[i+k*n1] * b[k+j*n2];
      }
    }
  }
  return c;
}
//****************************************************************************80

void c8mat_nint ( int m, int n, complex <double> a[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_NINT rounds the entries of a C8MAT.
//
//  Discussion:
//
//    A C8MAT is an array of complex <double> values.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    06 November 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns of A.
//
//    Input/output, complex <double> A[M*N], the matrix to be NINT'ed.
//
{
  int i;
  int j;

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      a[i+j*m] = c8_nint ( a[i+j*m] );
    }
  }
  return;
}
//****************************************************************************80

double c8mat_norm_fro ( int m, int n, complex <double> a[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_NORM_FRO returns the Frobenius norm of a C8MAT.
//
//  Discussion:
//
//    A C8MAT is an array of C8 values.
//
//    The Frobenius norm is defined as
//
//      C8MAT_NORM_FRO = sqrt (
//        sum ( 1 <= I <= M ) Sum ( 1 <= J <= N ) |A(I,J)| )
//
//    The matrix Frobenius-norm is not derived from a vector norm, but
//    is compatible with the vector L2 norm, so that:
//
//      c8vec_norm_l2 ( A*x ) <= c8mat_norm_fro ( A ) * c8vec_norm_l2 ( x ).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    01 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the order of the matrix.
//
//    Input, complex <double> A[M*N], the matrix.
//
//    Output, double C8MAT_NORM_FRO, the Frobenius norm of A.
//
{
  int i;
  int j;
  double value;

  value = 0.0;
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      value = value + pow ( real ( a[i+j*m] ), 2 )
                    + pow ( imag ( a[i+j*m] ), 2 );
    }
  }
  value = sqrt ( value );

  return value;
}
//****************************************************************************80

double c8mat_norm_l1 ( int m, int n, complex <double> a[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_NORM_L1 returns the matrix L1 norm of a C8MAT.
//
//  Discussion:
//
//    A C8MAT is an MxN array of C8's, stored by (I,J) -> [I+J*M].
//
//    The matrix L1 norm is defined as:
//
//      C8MAT_NORM_L1 = max ( 1 <= J <= N )
//        sum ( 1 <= I <= M ) abs ( A(I,J) ).
//
//    The matrix L1 norm is derived from the vector L1 norm, and
//    satisifies:
//
//      c8vec_norm_l1 ( A * x ) <= c8mat_norm_l1 ( A ) * c8vec_norm_l1 ( x ).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the number of rows in A.
//
//    Input, int N, the number of columns in A.
//
//    Input, complex <double> A[M*N], the matrix whose L1 norm is desired.
//
//    Output, double C8MAT_NORM_L1, the L1 norm of A.
//
{
  double col_sum;
  int i;
  int j;
  double value;

  value = 0.0;

  for ( j = 0; j < n; j++ )
  {
    col_sum = 0.0;
    for ( i = 0; i < m; i++ )
    {
      col_sum = col_sum + c8_abs ( a[i+j*m] );
    }
    value = r8_max ( value, col_sum );
  }

  return value;
}
//****************************************************************************80

double c8mat_norm_li ( int m, int n, complex <double> a[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_NORM_LI returns the matrix L-oo norm of a C8MAT.
//
//  Discussion:
//
//    A C8MAT is an array of C8 values.
//
//    The matrix L-oo norm is defined as:
//
//      C8MAT_NORM_LI =  max ( 1 <= I <= M ) sum ( 1 <= J <= N ) abs ( A(I,J) ).
//
//    The matrix L-oo norm is derived from the vector L-oo norm,
//    and satisifies:
//
//      c8vec_norm_li ( A * x ) <= c8mat_norm_li ( A ) * c8vec_norm_li ( x ).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    01 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, the number of rows in A.
//
//    Input, int N, the number of columns in A.
//
//    Input, complex <double> A[M*N], the matrix whose L-oo
//    norm is desired.
//
//    Output, double C8MAT_NORM_LI, the L-oo norm of A.
//
{
  int i;
  int j;
  double row_sum;
  double value;

  value = 0.0;

  for ( i = 0; i < m; i++ )
  {
    row_sum = 0.0;
    for ( j = 0; j < n; j++ )
    {
      row_sum = row_sum + c8_abs ( a[i+j*m] );
    }
    value = r8_max ( value, row_sum );
  }
  return value;
}
//****************************************************************************80

void c8mat_print ( int m, int n, complex <double> a[], string title )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_PRINT prints a C8MAT.
//
//  Discussion:
//
//    A C8MAT is an array of complex <double> values.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    22 October 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns in the matrix.
//
//    Input, complex <double> A[M*N], the matrix.
//
//    Input, string TITLE, a title.
//
{
  c8mat_print_some ( m, n, a, 1, 1, m, n, title );

  return;
}
//****************************************************************************80

void c8mat_print_some ( int m, int n, complex <double> a[], int ilo, int jlo, 
  int ihi, int jhi, string title )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_PRINT_SOME prints some of a C8MAT.
//
//  Discussion:
//
//    A C8MAT is an array of complex <double> values.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    23 October 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns in the matrix.
//
//    Input, complex <double> A[M*N], the matrix.
//
//    Input, int ILO, JLO, IHI, JHI, the first row and
//    column, and the last row and column to be printed.
//
//    Input, string TITLE, a title.
//
{
  complex <double> c;
  int i;
  int i2hi;
  int i2lo;
  int inc;
  int incx = 4;
  int j;
  int j2;
  int j2hi;
  int j2lo;

  cout << "\n";
  cout << title << "\n";

  if ( m <= 0 || n <= 0 )
  {
    cout << "\n";
    cout << "  (None)\n";
    return;
  }
//
//  Print the columns of the matrix, in strips of INCX.
//
  for ( j2lo = jlo; j2lo <= jhi; j2lo = j2lo + incx )
  {
    j2hi = j2lo + incx - 1;
    if ( n < j2hi )
    {
      j2hi = n;
    }
    if ( jhi < j2hi )
    {
      j2hi = jhi;
    }

    inc = j2hi + 1 - j2lo;

    cout << "\n";
    cout << "  Col: ";
    for ( j = j2lo; j <= j2hi; j++ )
    {
      j2 = j + 1 - j2lo;
      cout << "     " << setw(10) << j << "     ";
    }
    cout << "\n";
    cout << "  Row\n";
    cout << "  ---\n";
//
//  Determine the range of the rows in this strip.
//
    i2lo = ilo;
    if ( i2lo < 1 )
    {
      i2lo = 1;
    }
    i2hi = ihi;
    if ( m < i2hi )
    {
      i2hi = m;
    }

    for ( i = i2lo; i <= i2hi; i++ )
    {
      cout << setw(5) << i << ":";
//
//  Print out (up to) INCX entries in row I, that lie in the current strip.
//
      for ( j2 = 1; j2 <= inc; j2++ )
      {
        j = j2lo - 1 + j2;
        c = a[i-1+(j-1)*m];
        cout << "  " << setw(8) << real ( c )
             << "  " << setw(8) << imag ( c );
      }
      cout << "\n";
    }
  }

  return;
}
//****************************************************************************80

void c8mat_scale ( int m, int n, complex <double> alpha, complex <double> a[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_SCALE scales a C8MAT by a complex scalar factor.
//
//  Discussion:
//
//    An C8MAT is a doubly dimensioned array of complex double precision values, 
//    which may be stored as a vector in column-major order.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    03 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input, complex <double> ALPHA, the scale factor.
//
//    Input/output, complex <double> A[M*N], the matrix to be scaled.
//
{
  int i;
  int j;

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      a[i+j*m] = a[i+j*m] * alpha;
    }
  }
  return;
}
//****************************************************************************80

void c8mat_scale_r8 ( int m, int n, double alpha, complex <double> a[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_SCALE_R8 scales a C8MAT by a real scalar factor.
//
//  Discussion:
//
//    An C8MAT is a doubly dimensioned array of complex double precision values, 
//    which may be stored as a vector in column-major order.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    03 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Input, double ALPHA, the scale factor.
//
//    Input/output, complex <double> A[M*N], the matrix to be scaled.
//
{
  int i;
  int j;

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      a[i+j*m] = a[i+j*m] * alpha;
    }
  }
  return;
}
//****************************************************************************80

void c8mat_uniform_01 ( int m, int n, int &seed, complex <double> c[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_UNIFORM_01 returns a unit pseudorandom C8MAT.
//
//  Discussion:
//
//    A C8MAT is an array of complex <double> values.
//
//    The angles should be uniformly distributed between 0 and 2 * PI,
//    the square roots of the radius uniformly distributed between 0 and 1.
//
//    This results in a uniform distribution of values in the unit circle.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    03 August 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns in the matrix.
//
//    Input/output, int &SEED, the "seed" value, which should NOT be 0.
//    On output, SEED has been updated.
//
//    Output, complex <double> C[M*N], the pseudorandom complex matrix.
//
{
  int i;
  const int i4_huge = 2147483647;
  int j;
  double r;
  int k;
  const double r8_pi = 3.141592653589793;
  double theta;

  if ( seed == 0 )
  {
    cerr << "\n";
    cerr << "C8MAT_UNIFORM_01 - Fatal error!\n";
    cerr << "  Input value of SEED = 0.\n";
    exit ( 1 );
  }

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      k = seed / 127773;

      seed = 16807 * ( seed - k * 127773 ) - k * 2836;

      if ( seed < 0 )
      {
        seed = seed + i4_huge;
      }

      r = sqrt ( ( double ) ( seed ) * 4.656612875E-10 );

      k = seed / 127773;

      seed = 16807 * ( seed - k * 127773 ) - k * 2836;

      if ( seed < 0 )
      {
        seed = seed + i4_huge;
      }

      theta = 2.0 * r8_pi * ( ( double ) ( seed ) * 4.656612875E-10 );

      c[i+j*m] = r * complex <double> ( cos ( theta ), sin ( theta ) );
    }
  }

  return;
}
//****************************************************************************80

complex <double> *c8mat_uniform_01_new ( int m, int n, int &seed )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_UNIFORM_01_NEW returns a unit pseudorandom C8MAT.
//
//  Discussion:
//
//    A C8MAT is an array of complex <double> values.
//
//    The angles should be uniformly distributed between 0 and 2 * PI,
//    the square roots of the radius uniformly distributed between 0 and 1.
//
//    This results in a uniform distribution of values in the unit circle.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    15 March 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns in the matrix.
//
//    Input/output, int &SEED, the "seed" value, which should NOT be 0.
//    On output, SEED has been updated.
//
//    Output, complex <double> C8MAT_UNIFORM_01_NEW[M*N], the pseudorandom 
//    complex matrix.
//
{
  complex <double> *c;
  int i;
  int j;
  double r;
  int k;
  const double r8_pi = 3.141592653589793;
  double theta;

  c = new complex <double> [m*n];

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      k = seed / 127773;

      seed = 16807 * ( seed - k * 127773 ) - k * 2836;

      if ( seed < 0 )
      {
        seed = seed + 2147483647;
      }

      r = sqrt ( ( double ) ( seed ) * 4.656612875E-10 );

      k = seed / 127773;

      seed = 16807 * ( seed - k * 127773 ) - k * 2836;

      if ( seed < 0 )
      {
        seed = seed + 2147483647;
      }

      theta = 2.0 * r8_pi * ( ( double ) ( seed ) * 4.656612875E-10 );

      c[i+j*m] = r * complex <double> ( cos ( theta ), sin ( theta ) );
    }
  }
  return c;
}
//****************************************************************************80

complex <double> *c8mat_zero_new ( int m, int n )

//****************************************************************************80
//
//  Purpose:
//
//    C8MAT_ZERO_NEW returns a new zeroed C8MAT.
//
//  Discussion:
//
//    An C8MAT is a doubly dimensioned array of complex double precision values, 
//    which may be stored as a vector in column-major order.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int M, N, the number of rows and columns.
//
//    Output, complex <double> C8MAT_ZERO_NEW[M*N], the zeroed matrix.
//
{
  complex <double> *a;
  int i;
  int j;

  a = new complex <double>[m*n];

  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      a[i+j*m] = 0.0;
    }
  }
  return a;
}
//****************************************************************************80

void c8vec_copy ( int n, complex <double> a1[], complex <double> a2[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8VEC_COPY copies a C8VEC.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    27 August 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vectors.
//
//    Input, complex <double> A1[N], the vector to be copied.
//
//    Output, complex <double> A2[N], the copy of A1.
//
{
  int i;

  for ( i = 0; i < n; i++ )
  {
    a2[i] = a1[i];
  }
  return;
}
//****************************************************************************80

complex <double> *c8vec_copy_new ( int n, complex <double> a1[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8VEC_COPY_NEW copies a C8VEC to a "new" C8VEC.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    27 August 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vectors.
//
//    Input, complex <double> A1[N], the vector to be copied.
//
//    Output, complex <double> C8VEC_COPY_NEW[N], the copy of A1.
//
{
  complex <double> *a2;
  int i;

  a2 = new complex <double>[n];

  for ( i = 0; i < n; i++ )
  {
    a2[i] = a1[i];
  }
  return a2;
}
//****************************************************************************80

complex <double> *c8vec_indicator_new ( int n )

//****************************************************************************80
//
//  Purpose:
//
//    C8VEC_INDICATOR_NEW sets a C8VEC to the indicator vector.
//
//  Discussion:
//
//    A C8VEC is a vector of complex <double> values.
//
//    X(1:N) = ( 1-1i, 2-2i, 3-3i, 4-4i, ... )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    06 November 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of elements of A.
//
//    Output, complex <double> C8VEC_INDICATOR_NEW[N], the array.
//
{
  complex <double> *a;
  int i;

  a = new complex <double> [n];

  for ( i = 0; i < n; i++ )
  {
    a[i] = complex <double> ( i+1, -i-1 );
  }

  return a;
}
//****************************************************************************80

void c8vec_nint ( int n, complex <double> a[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8VEC_NINT rounds the entries of a C8VEC.
//
//  Discussion:
//
//    A C8VEC is a vector of complex <double> values.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of components of the vector.
//
//    Input/output, complex <double> A[N], the vector to be nint'ed.
//
{
  int i;

  for ( i = 0; i < n; i++ )
  {
    a[i] = c8_nint ( a[i] );
  }

  return;
}
//****************************************************************************80

double c8vec_norm_l1 ( int n, complex <double> a[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8VEC_NORM_L1 returns the L1 norm of a C8VEC.
//
//  Discussion:
//
//    The vector L1 norm is defined as:
//
//      value = sum ( 1 <= I <= N ) abs ( A(I) )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    09 February 2015
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries.
//
//    Input, complex <double> A[N], the vector.
//
//    Output, double C8VEC_NORM_L1, the norm.
//
{
  int i;
  double value;

  value = 0.0;
  for ( i = 0; i < n; i++ )
  {
    value = value + c8_abs ( a[i] );
  }

  return value;
}
//****************************************************************************80

double c8vec_norm_l2 ( int n, complex <double> a[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8VEC_NORM_L2 returns the L2 norm of a C8VEC.
//
//  Discussion:
//
//    The vector L2 norm is defined as:
//
//      value = sqrt ( sum ( 1 <= I <= N ) conjg ( A(I) ) * A(I) ).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    08 June 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries.
//
//    Input, complex <double> A[N], the vector.
//
//    Output, double C8VEC_NORM_L2, the norm.
//
{
  int i;
  double value;

  value = 0.0;
  for ( i = 0; i < n; i++ )
  {
    value = value 
          + real ( a[i] ) * real ( a[i] ) 
          + imag ( a[i] ) * imag ( a[i] );
  }
  value = sqrt ( value );

  return value;
}
//****************************************************************************80

double c8vec_norm_li ( int n, complex <double> a[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8VEC_NORM_LI returns the Loo norm of a C8VEC.
//
//  Discussion:
//
//    The vector Loo norm is defined as:
//
//      value = max ( 1 <= I <= N ) abs ( A(I) )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    09 February 2015
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries.
//
//    Input, complex <double> A[N], the vector.
//
//    Output, double C8VEC_NORM_L2, the norm.
//
{
  int i;
  double value;

  value = 0.0;
  for ( i = 0; i < n; i++ )
  {
    value = r8_max ( value, c8_abs ( a[i] ) );
  }

  return value;
}
//****************************************************************************80

void c8vec_print ( int n, complex <double> a[], string title )

//****************************************************************************80
//
//  Purpose:
//
//    C8VEC_PRINT prints a C8VEC.
//
//  Discussion:
//
//    A C8VEC is a vector of complex <double> values.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    10 September 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of components of the vector.
//
//    Input, complex <double> A[N], the vector to be printed.
//
//    Input, string TITLE, a title.
//
{
  int i;

  cout << "\n";
  cout << title << "\n";

  cout << "\n";
  for ( i = 0; i < n; i++ )
  {
    cout << "  " << setw(8) << i
         << ": " << real ( a[i] )
         << "  " << imag ( a[i] ) << "\n";
  }

  return;
}
//****************************************************************************80

void c8vec_print_part ( int n, complex <double> a[], int max_print, 
  string title )

//****************************************************************************80
//
//  Purpose:
//
//    C8VEC_PRINT_PART prints "part" of a C8VEC.
//
//  Discussion:
//
//    The user specifies MAX_PRINT, the maximum number of lines to print.
//
//    If N, the size of the vector, is no more than MAX_PRINT, then
//    the entire vector is printed, one entry per line.
//
//    Otherwise, if possible, the first MAX_PRINT-2 entries are printed,
//    followed by a line of periods suggesting an omission,
//    and the last entry.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    22 June 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries of the vector.
//
//    Input, complex <double> A[N], the vector to be printed.
//
//    Input, int MAX_PRINT, the maximum number of lines
//    to print.
//
//    Input, string TITLE, a title.
//
{
  int i;

  if ( max_print <= 0 )
  {
    return;
  }

  if ( n <= 0 )
  {
    return;
  }

  cout << "\n";
  cout << title << "\n";
  cout << "\n";

  if ( n <= max_print )
  {
    for ( i = 0; i < n; i++ )
    {
      cout << "  " << setw(8) << i
           << "  " << setw(14) << real ( a[i] ) 
           << "  " << setw(14) << imag ( a[i] ) << "\n";
    }
  }
  else if ( 3 <= max_print )
  {
    for ( i = 0; i < max_print - 2; i++ )
    {
      cout << "  " << setw(8) << i
           << "  " << setw(14) << real ( a[i] ) 
           << "  " << setw(14) << imag ( a[i] ) << "\n";
    }
    cout << "  ........  ..............  ..............\n";
    i = n - 1;
    cout << "  " << setw(8) << i
           << "  " << setw(14) << real ( a[i] ) 
           << "  " << setw(14) << imag ( a[i] ) << "\n";
  }
  else
  {
    for ( i= 0; i < max_print - 1; i++ )
    {
      cout << "  " << setw(8) << i
           << "  " << setw(14) << real ( a[i] ) 
           << "  " << setw(14) << imag ( a[i] ) << "\n";
    }
    i = max_print - 1;
    cout << "  " << setw(8) << i
         << "  " << setw(14) << real ( a[i] ) 
         << "  " << setw(14) << imag ( a[i] )
         << "  " << "...more entries...\n";
  }

  return;
}
//****************************************************************************80

void c8vec_print_some ( int n, complex <double> a[], int i_lo, int i_hi, 
  string title )

//****************************************************************************80
//
//  Purpose:
//
//    C8VEC_PRINT_SOME prints some of a C8VEC.
//
//  Discussion:
//
//    A C8VEC is a vector of complex <double> values.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    10 June 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of components of the vector.
//
//    Input, complex <double> A[N], the vector to be printed.
//
//    Input, int I_LO, I_HI, the first and last entries to print.
//
//    Input, string TITLE, a title.
//
{
  int i;

  cout << "\n";
  cout << title << "\n";
  cout << "\n";
  for ( i = i_lo; i <= i_hi; i++ )
  {
    cout << "  " << setw(6) << i
         << ": " << real ( a[i] )
         << "  " << imag ( a[i] ) << "\n";
  }

  return;
}
//****************************************************************************80

void c8vec_sort_a_l1 ( int n, complex <double> x[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8VEC_SORT_A_L1 ascending sorts a C8VEC by L1 norm.
//
//  Discussion:
//
//    The L1 norm of A+Bi is sqrt ( A * A + B * B ).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    09 February 2015
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, length of input array.
//
//    Input/output, complex <double> X[N].
//    On input, an unsorted array.
//    On output, X has been sorted.
//
{
  int i;
  int indx;
  int isgn;
  int j;
  double normsq_i;
  double normsq_j;
  complex <double> temp;

  i = 0;
  indx = 0;
  isgn = 0;
  j = 0;

  for ( ; ; )
  {
    sort_heap_external ( n, indx, i, j, isgn );

    if ( 0 < indx )
    {
      temp   = x[i-1];
      x[i-1] = x[j-1];
      x[j-1] = temp;
    }
    else if ( indx < 0 )
    {
      if ( c8_le_l1 ( x[i-1], x[j-1] ) )
      {
        isgn = -1;
      }
      else
      {
        isgn = +1;
      }
    }
    else if ( indx == 0 )
    {
      break;
    }
  }

  return;
}
//****************************************************************************80

void c8vec_sort_a_l2 ( int n, complex <double> x[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8VEC_SORT_A_L2 ascending sorts a C8VEC by L2 norm.
//
//  Discussion:
//
//    The L2 norm of A+Bi is sqrt ( A * A + B * B ).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    03 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, length of input array.
//
//    Input/output, complex <double> X[N].
//    On input, an unsorted array.
//    On output, X has been sorted.
//
{
  int i;
  int indx;
  int isgn;
  int j;
  double normsq_i;
  double normsq_j;
  complex <double> temp;

  i = 0;
  indx = 0;
  isgn = 0;
  j = 0;

  for ( ; ; )
  {
    sort_heap_external ( n, indx, i, j, isgn );

    if ( 0 < indx )
    {
      temp = x[i-1];
      x[i-1] = x[j-1];
      x[j-1] = temp;
    }
    else if ( indx < 0 )
    {
      normsq_i = pow ( real ( x[i-1] ), 2 )
               + pow ( imag ( x[i-1] ), 2 );

      normsq_j = pow ( real ( x[j-1] ), 2 )
               + pow ( imag ( x[j-1] ), 2 );

      if ( normsq_i < normsq_j )
      {
        isgn = -1;
      }
      else
      {
        isgn = +1;
      }
    }
    else if ( indx == 0 )
    {
      break;
    }
  }

  return;
}
//****************************************************************************80

void c8vec_sort_a_li ( int n, complex <double> x[] )

//****************************************************************************80
//
//  Purpose:
//
//    C8VEC_SORT_A_LI ascending sorts a C8VEC by Loo norm.
//
//  Discussion:
//
//    The Loo norm of A+Bi is max ( abs ( A ), abs ( B ) ).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    09 February 2015
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, length of input array.
//
//    Input/output, complex <double> X[N].
//    On input, an unsorted array.
//    On output, X has been sorted.
//
{
  int i;
  int indx;
  int isgn;
  int j;
  double normsq_i;
  double normsq_j;
  complex <double> temp;

  i = 0;
  indx = 0;
  isgn = 0;
  j = 0;

  for ( ; ; )
  {
    sort_heap_external ( n, indx, i, j, isgn );

    if ( 0 < indx )
    {
      temp   = x[i-1];
      x[i-1] = x[j-1];
      x[j-1] = temp;
    }
    else if ( indx < 0 )
    {
      if ( c8_le_li ( x[i-1], x[j-1] ) )
      {
        isgn = -1;
      }
      else
      {
        isgn = +1;
      }
    }
    else if ( indx == 0 )
    {
      break;
    }
  }

  return;
}
//****************************************************************************80

complex <double> *c8vec_spiral_new ( int n, int m, complex <double> c1, 
  complex <double> c2 )

//****************************************************************************80
//
//  Purpose:
//
//    C8VEC_SPIRAL_NEW returns N points on a spiral between C1 and C2.
//
//  Discussion:
//
//    A C8VEC is a vector of C8's.
//
//    Let the polar form of C1 be ( R1, T1 ) and the polar form of C2 
//    be ( R2, T2 ) where, if necessary, we increase T2 by 2*PI so that T1 <= T2.
//    
//    Then the polar form of the I-th point C(I) is:
//
//      R(I) = ( ( N - I     ) * R1 
//             + (     I - 1 ) * R2 ) 
//              / ( N    - 1 )
//
//      T(I) = ( ( N - I     ) * T1 
//             + (     I - 1 ) * ( T2 + M * 2 * PI ) ) 
//             / ( N     - 1 )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of points on the spiral.
//
//    Input, int M, the number of full circuits the 
//    spiral makes.
//
//    Input, complex <double> C1, C2, the first and last points 
//    on the spiral.
//
//    Output, complex <double> C8VEC_SPIRAL_NEW[N], the points.
//
{
  complex <double> *c;
  int i;
  double r1;
  double r2;
  double ri;
  const double r8_pi = 3.141592653589793;
  double t1;
  double t2;
  double ti;

  c = new complex <double>[n];

  r1 = c8_abs ( c1 );
  r2 = c8_abs ( c2 );

  t1 = c8_arg ( c1 );
  t2 = c8_arg ( c2 );

  if ( m == 0 )
  {
    if ( t2 < t1 )
    {
      t2 = t2 + 2.0 * r8_pi;
    }
  }
  else if ( 0 < m )
  {
    if ( t2 < t1 )
    {
      t2 = t2 + 2.0 * r8_pi;
    }
    t2 = t2 + ( double ) ( m ) * 2.0 * r8_pi;
  }
  else if ( m < 0 )
  {
    if ( t1 < t2 )
    {
      t2 = t2 - 2.0 * r8_pi;
    }
    t2 = t2 - ( double ) ( m ) * 2.0 * r8_pi;
  }

  for ( i = 0; i < n; i++ )
  {
    ri = ( ( double ) ( n - i - 1 ) * r1
         + ( double ) (     i     ) * r2 )
         / ( double ) ( n     - 1 );

    ti = ( ( double ) ( n - i - 1 ) * t1
         + ( double ) (     i     ) * t2 )
         / ( double ) ( n     - 1 );

    c[i] = polar_to_c8 ( ri, ti );
  }

  return c;
}
//****************************************************************************80

complex <double> *c8vec_uniform_01_new ( int n, int &seed )

//****************************************************************************80
//
//  Purpose:
//
//    C8VEC_UNIFORM_01_NEW returns a unit pseudorandom C8VEC.
//
//  Discussion:
//
//    A C8VEC is a vector of complex <double> values.
//
//    The angles should be uniformly distributed between 0 and 2 * PI,
//    the square roots of the radius uniformly distributed between 0 and 1.
//
//    This results in a uniform distribution of values in the unit circle.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    15 March 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of values to compute.
//
//    Input/output, int &SEED, the "seed" value, which should NOT be 0.
//    On output, SEED has been updated.
//
//    Output, complex <double> C8VEC_UNIFORM_01_NEW[N], the pseudorandom 
//    complex vector.
//
{
  complex <double> *c;
  int i;
  double r;
  int k;
  const double r8_pi = 3.141592653589793;
  double theta;

  c = new complex <double> [n];

  for ( i = 0; i < n; i++ )
  {
    k = seed / 127773;

    seed = 16807 * ( seed - k * 127773 ) - k * 2836;

    if ( seed < 0 )
    {
      seed = seed + 2147483647;
    }

    r = sqrt ( ( double ) ( seed ) * 4.656612875E-10 );

    k = seed / 127773;

    seed = 16807 * ( seed - k * 127773 ) - k * 2836;

    if ( seed < 0 )
    {
      seed = seed + 2147483647;
    }

    theta = 2.0 * r8_pi * ( ( double ) ( seed ) * 4.656612875E-10 );

    c[i] = r * complex <double> ( cos ( theta ), sin ( theta ) );
  }

  return c;
}
//****************************************************************************80

complex <double> *c8vec_unity_new ( int n )

//****************************************************************************80
//
//  Purpose:
//
//    C8VEC_UNITY_NEW returns the N roots of unity in a C8VEC.
//
//  Discussion:
//
//    A C8VEC is a vector of complex <double> values.
//
//    X(1:N) = exp ( 2 * PI * (0:N-1) / N )
//
//    X(1:N)^N = ( (1,0), (1,0), ..., (1,0) ).
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    06 November 2010
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of elements of A.
//
//    Output, complex <double> C8VEC_UNITY_NEW[N], the N roots of unity.
//
{
  complex <double> *a;
  int i;
  const double r8_pi = 3.141592653589793;
  double theta;

  a = new complex <double> [n];

  for ( i = 0; i < n; i++ )
  {
    theta = r8_pi * ( double ) ( 2 * i ) / ( double ) ( n );
    a[i] = complex <double> ( cos ( theta ), sin ( theta ) );
  }

  return a;
}
//****************************************************************************80

complex <double> cartesian_to_c8 ( double x, double y )

//****************************************************************************80
//
//  Purpose:
//
//    CARTESIAN_TO_C8 converts a Cartesian form to a C8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    01 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, Y, the Cartesian form.
//
//    Output, complex <double> CARTESIAN_TO_C8, the complex number.
//
{
  complex <double> c;

  c = complex <double> ( x, y );

  return c;
}
//****************************************************************************80

complex <double> polar_to_c8 ( double r, double theta )

//****************************************************************************80
//
//  Purpose:
//
//    POLAR_TO_C8 converts a polar form to a C8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 March 2013
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double R, THETA, the polar form.
//
//    Output, complex <double> POLAR_TO_C8, the complex number.
//
{
  complex <double> c;

  c = complex <double> ( r * cos ( theta ), r * sin ( theta ) );

  return c;
}
//****************************************************************************80

double r8_abs ( double x )

//****************************************************************************80
//
//  Purpose:
//
//    R8_ABS returns the absolute value of an R8.
//
//  Discussion:
//
//    The C++ math library provides the function fabs() which is preferred.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    14 November 2006
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, the quantity whose absolute value is desired.
//
//    Output, double R8_ABS, the absolute value of X.
//
{
  double value;

  if ( 0.0 <= x )
  {
    value = + x;
  }
  else
  {
    value = - x;
  }
  return value;
}
//****************************************************************************80

double r8_atan ( double y, double x )

//****************************************************************************80
//
//  Purpose:
//
//    R8_ATAN computes the inverse tangent of the ratio Y / X.
//
//  Discussion:
//
//    R8_ATAN returns an angle whose tangent is ( Y / X ), a job which
//    the built in functions ATAN and ATAN2 already do.
//
//    However:
//
//    * R8_ATAN always returns a positive angle, between 0 and 2 PI,
//      while ATAN and ATAN2 return angles in the interval [-PI/2,+PI/2]
//      and [-PI,+PI] respectively;
//
//    * R8_ATAN accounts for the signs of X and Y, (as does ATAN2).  The ATAN
//     function by contrast always returns an angle in the first or fourth
//     quadrants.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    15 August 2008
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double Y, X, two quantities which represent the tangent of
//    an angle.  If Y is not zero, then the tangent is (Y/X).
//
//    Output, double R8_ATAN, an angle between 0 and 2 * PI, whose tangent is
//    (Y/X), and which lies in the appropriate quadrant so that the signs
//    of its cosine and sine match those of X and Y.
//
{
  double abs_x;
  double abs_y;
  const double r8_pi = 3.141592653589793;
  double theta;
  double theta_0;
//
//  Special cases:
//
  if ( x == 0.0 )
  {
    if ( 0.0 < y )
    {
      theta = r8_pi / 2.0;
    }
    else if ( y < 0.0 )
    {
      theta = 3.0 * r8_pi / 2.0;
    }
    else if ( y == 0.0 )
    {
      theta = 0.0;
    }
  }
  else if ( y == 0.0 )
  {
    if ( 0.0 < x )
    {
      theta = 0.0;
    }
    else if ( x < 0.0 )
    {
      theta = r8_pi;
    }
  }
//
//  We assume that ATAN2 is correct when both arguments are positive.
//
  else
  {
    abs_y = fabs ( y );
    abs_x = fabs ( x );

    theta_0 = atan2 ( abs_y, abs_x );

    if ( 0.0 < x && 0.0 < y )
    {
      theta = theta_0;
    }
    else if ( x < 0.0 && 0.0 < y )
    {
      theta = r8_pi - theta_0;
    }
    else if ( x < 0.0 && y < 0.0 )
    {
      theta = r8_pi + theta_0;
    }
    else if ( 0.0 < x && y < 0.0 )
    {
      theta = 2.0 * r8_pi - theta_0;
    }
  }

  return theta;
}
//****************************************************************************80

complex <double> r8_csqrt ( double x )

//****************************************************************************80
//
//  Purpose:
//
//    R8_CSQRT returns the complex square root of an R8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    24 October 2005
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, the number whose square root is desired.
//
//    Output, complex <double> R8_CSQRT, the square root of X:
//
{
  double argument;
  double magnitude;
  const double r8_pi = 3.141592653589793;
  complex <double> value;

  if ( 0.0 < x )
  {
    magnitude = x;
    argument = 0.0;
  }
  else if ( 0.0 == x )
  {
    magnitude = 0.0;
    argument = 0.0;
  }
  else if ( x < 0.0 )
  {
    magnitude = -x;
    argument = r8_pi;
  }

  magnitude = sqrt ( magnitude );
  argument = argument / 2.0;

  value = magnitude * complex <double> ( cos ( argument ), sin ( argument ) );

  return value;
}
//****************************************************************************80

double r8_floor ( double x )

//****************************************************************************80
//
//  Purpose:
//
//    R8_FLOOR rounds an R8 down to the nearest integral R8.
//
//  Example:
//
//    X        R8_FLOOR(X)
//
//   -1.1      -2.0
//   -1.0      -1.0
//   -0.9      -1.0
//   -0.1      -1.0
//    0.0       0.0
//    0.1       0.0
//    0.9       0.0
//    1.0       1.0
//    1.1       1.0
//    2.9       2.0
//    3.0       3.0
//    3.14159   3.0
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    15 April 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, the number whose floor is desired.
//
//    Output, double R8_FLOOR, the floor of X.
//
{
  double value;

  value = ( double ) ( ( int ) x );

  if ( x < value )
  {
    value = value - 1.0;
  }

  return value;
}
//****************************************************************************80

double r8_max ( double x, double y )

//****************************************************************************80
//
//  Purpose:
//
//    R8_MAX returns the maximum of two R8's.
//
//  Discussion:
//
//    The C++ math library provides the function fmax() which is preferred.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    18 August 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, Y, the quantities to compare.
//
//    Output, double R8_MAX, the maximum of X and Y.
//
{
  double value;

  if ( y < x )
  {
    value = x;
  }
  else
  {
    value = y;
  }
  return value;
}
//****************************************************************************80

double r8_sign ( double x )

//****************************************************************************80
//
//  Purpose:
//
//    R8_SIGN returns the sign of an R8.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    18 October 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, double X, the number whose sign is desired.
//
//    Output, double R8_SIGN, the sign of X.
//
{
  double value;

  if ( x < 0.0 )
  {
    value = -1.0;
  }
  else
  {
    value = 1.0;
  }
  return value;
}
//****************************************************************************80

double r8_uniform_01 ( int &seed )

//****************************************************************************80
//
//  Purpose:
//
//    R8_UNIFORM_01 returns a unit pseudorandom R8.
//
//  Discussion:
//
//    This routine implements the recursion
//
//      seed = ( 16807 * seed ) mod ( 2^31 - 1 )
//      u = seed / ( 2^31 - 1 )
//
//    The integer arithmetic never requires more than 32 bits,
//    including a sign bit.
//
//    If the initial seed is 12345, then the first three computations are
//
//      Input     Output      R8_UNIFORM_01
//      SEED      SEED
//
//         12345   207482415  0.096616
//     207482415  1790989824  0.833995
//    1790989824  2035175616  0.947702
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    09 April 2012
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Paul Bratley, Bennett Fox, Linus Schrage,
//    A Guide to Simulation,
//    Second Edition,
//    Springer, 1987,
//    ISBN: 0387964673,
//    LC: QA76.9.C65.B73.
//
//    Bennett Fox,
//    Algorithm 647:
//    Implementation and Relative Efficiency of Quasirandom
//    Sequence Generators,
//    ACM Transactions on Mathematical Software,
//    Volume 12, Number 4, December 1986, pages 362-376.
//
//    Pierre L'Ecuyer,
//    Random Number Generation,
//    in Handbook of Simulation,
//    edited by Jerry Banks,
//    Wiley, 1998,
//    ISBN: 0471134031,
//    LC: T57.62.H37.
//
//    Peter Lewis, Allen Goodman, James Miller,
//    A Pseudo-Random Number Generator for the System/360,
//    IBM Systems Journal,
//    Volume 8, Number 2, 1969, pages 136-143.
//
//  Parameters:
//
//    Input/output, int &SEED, the "seed" value.  Normally, this
//    value should not be 0.  On output, SEED has been updated.
//
//    Output, double R8_UNIFORM_01, a new pseudorandom variate, 
//    strictly between 0 and 1.
//
{
  const int i4_huge = 2147483647;
  int k;
  double r;

  if ( seed == 0 )
  {
    cerr << "\n";
    cerr << "R8_UNIFORM_01 - Fatal error!\n";
    cerr << "  Input value of SEED = 0.\n";
    exit ( 1 );
  }

  k = seed / 127773;

  seed = 16807 * ( seed - k * 127773 ) - k * 2836;

  if ( seed < 0 )
  {
    seed = seed + i4_huge;
  }
  r = ( double ) ( seed ) * 4.656612875E-10;

  return r;
}
//****************************************************************************80

void r8poly2_root ( double a, double b, double c, complex <double> &r1,
  complex <double> &r2 )

//****************************************************************************80
//
//  Purpose:
//
//    R8POLY2_ROOT returns the two roots of a quadratic polynomial.
//
//  Discussion:
//
//    The polynomial has the form:
//
//      A * X^2 + B * X + C = 0
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    23 October 2005
//
//  Parameters:
//
//    Input, double A, B, C, the coefficients of the polynomial.
//    A must not be zero.
//
//    Output, complex <double> &R1, &R2, the roots of the polynomial, which
//    might be real and distinct, real and equal, or complex conjugates.
//
{
  double disc;
  complex <double> q;

  if ( a == 0.0 )
  {
    cerr << "\n";
    cerr << "R8POLY2_ROOT - Fatal error!\n";
    cerr << "  The coefficient A is zero.\n";
    exit ( 1 );
  }

  disc = b * b - 4.0 * a * c;
  q = -0.5 * ( b + r8_sign ( b ) * r8_csqrt ( disc ) );
  r1 = q / a;
  r2 = c / q;

  return;
}
//****************************************************************************80

void r8poly3_root ( double a, double b, double c, double d,
  complex <double> &r1, complex <double> &r2, complex <double> &r3 )

//****************************************************************************80
//
//  Purpose:
//
//    R8POLY3_ROOT returns the three roots of a cubic polynomial.
//
//  Discussion:
//
//    The polynomial has the form
//
//      A * X^3 + B * X^2 + C * X + D = 0
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    25 October 2005
//
//  Parameters:
//
//    Input, double A, B, C, D, the coefficients of the polynomial.
//    A must not be zero.
//
//    Output, complex <double> &R1, &R2, &R3, the roots of the polynomial, which
//    will include at least one real root.
//
{
  complex <double> i;
  const double r8_pi = 3.141592653589793;
  double q;
  double r;
  double s1;
  double s2;
  double temp;
  double theta;

  if ( a == 0.0 )
  {
    cerr << "\n";
    cerr << "R8POLY3_ROOT - Fatal error!\n";
    cerr << "  A must not be zero.\n";
    exit ( 1 );
  }

  i = complex <double> ( 0.0, 1.0 );

  q = ( pow ( b / a, 2 ) - 3.0 * ( c / a ) ) / 9.0;

  r = ( 2.0 * pow ( b / a, 3 ) - 9.0 * ( b / a ) * ( c / a )
      + 27.0 * ( d / a ) ) / 54.0;

  if ( r * r < q * q * q )
  {
    theta = acos ( r / sqrt ( pow ( q, 3 ) ) );
    r1 = -2.0 * sqrt ( q ) * cos (   theta                 / 3.0 );
    r2 = -2.0 * sqrt ( q ) * cos ( ( theta + 2.0 * r8_pi ) / 3.0 );
    r3 = -2.0 * sqrt ( q ) * cos ( ( theta + 4.0 * r8_pi ) / 3.0 );
  }
  else if ( q * q * q <= r * r )
  {
    temp = -r + sqrt ( r * r - q * q * q );
    s1 = r8_sign ( temp ) * pow ( r8_abs ( temp ), 1.0 / 3.0 );

    temp = -r - sqrt ( r * r - q * q * q );
    s2 = r8_sign ( temp ) * pow ( r8_abs ( temp ), 1.0 / 3.0 );

    r1 = s1 + s2;
    r2 = -0.5 * ( s1 + s2 ) + i * 0.5 * sqrt ( 3.0 ) * ( s1 - s2 );
    r3 = -0.5 * ( s1 + s2 ) - i * 0.5 * sqrt ( 3.0 ) * ( s1 - s2 );
  }

  r1 = r1 - b / ( 3.0 * a );
  r2 = r2 - b / ( 3.0 * a );
  r3 = r3 - b / ( 3.0 * a );

  return;
}
//****************************************************************************80

void r8poly4_root ( double a, double b, double c, double d, double e,
  complex <double> &r1, complex <double> &r2, complex <double> &r3,
  complex <double> &r4 )

//****************************************************************************80
//
//  Purpose:
//
//    R8POLY4_ROOT returns the four roots of a quartic polynomial.
//
//  Discussion:
//
//    The polynomial has the form:
//
//      A * X^4 + B * X^3 + C * X^2 + D * X + E = 0
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    27 October 2005
//
//  Parameters:
//
//    Input, double A, B, C, D, the coefficients of the polynomial.
//    A must not be zero.
//
//    Output, complex <double> &R1, &R2, &R3, &R4, the roots of the polynomial.
//
{
  double a3;
  double a4;
  double b3;
  double b4;
  double c3;
  double c4;
  double d3;
  double d4;
  complex <double> p;
  complex <double> q;
  complex <double> r;
  complex <double> zero;

  zero = 0.0;

  if ( a == 0.0 )
  {
    cerr << "\n";
    cerr << "R8POLY4_ROOT - Fatal error!\n";
    cerr << "  A must not be zero.\n";
    exit ( 1 );
  }

  a4 = b / a;
  b4 = c / a;
  c4 = d / a;
  d4 = e / a;
//
//  Set the coefficients of the resolvent cubic equation.
//
  a3 = 1.0;
  b3 = -b4;
  c3 = a4 * c4 - 4.0 * d4;
  d3 = -a4 * a4 * d4 + 4.0 * b4 * d4 - c4 * c4;
//
//  Find the roots of the resolvent cubic.
//
  r8poly3_root ( a3, b3, c3, d3, r1, r2, r3 );
//
//  Choose one root of the cubic, here R1.
//
//  Set R = sqrt ( 0.25 * A4^2 - B4 + R1 )
//
  r = c8_sqrt ( 0.25 * a4 * a4 - b4  + r1 );

  if ( real ( r ) != 0.0 || imag ( r ) != 0.0 )
  {
    p = c8_sqrt ( 0.75 * a4 * a4 - r * r - 2.0 * b4
        + 0.25 * ( 4.0 * a4 * b4 - 8.0 * c4 - a4 * a4 * a4 ) / r );

    q = c8_sqrt ( 0.75 * a4 * a4 - r * r - 2.0 * b4
        - 0.25 * ( 4.0 * a4 * b4 - 8.0 * c4 - a4 * a4 * a4 ) / r );
  }
  else
  {
    p = c8_sqrt ( 0.75 * a4 * a4 - 2.0 * b4
      + 2.0 * c8_sqrt ( r1 * r1 - 4.0 * d4 ) );

    q = c8_sqrt ( 0.75 * a4 * a4 - 2.0 * b4
      - 2.0 * c8_sqrt ( r1 * r1 - 4.0 * d4 ) );
  }
//
//  Set the roots.
//
  r1 = -0.25 * a4 + 0.5 * r + 0.5 * p;
  r2 = -0.25 * a4 + 0.5 * r - 0.5 * p;
  r3 = -0.25 * a4 - 0.5 * r + 0.5 * q;
  r4 = -0.25 * a4 - 0.5 * r - 0.5 * q;

  return;
}
//****************************************************************************80

void sort_heap_external ( int n, int &indx, int &i, int &j, int isgn )

//****************************************************************************80
//
//  Purpose:
//
//    SORT_HEAP_EXTERNAL externally sorts a list of items into ascending order.
//
//  Discussion:
//
//    The actual list is not passed to the routine.  Hence it may
//    consist of integers, reals, numbers, names, etc.  The user,
//    after each return from the routine, will be asked to compare or
//    interchange two items.
//
//    The current version of this code mimics the FORTRAN version,
//    so the values of I and J, in particular, are FORTRAN indices.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    06 January 2013
//
//  Author:
//
//    Original FORTRAN77 version by Albert Nijenhuis, Herbert Wilf.
//    C++ version by John Burkardt
//
//  Reference:
//
//    Albert Nijenhuis, Herbert Wilf,
//    Combinatorial Algorithms,
//    Academic Press, 1978, second edition,
//    ISBN 0-12-519260-6.
//
//  Parameters:
//
//    Input, int N, the length of the input list.
//
//    Input/output, int &INDX.
//    The user must set INDX to 0 before the first call.
//    On return,
//      if INDX is greater than 0, the user must interchange
//      items I and J and recall the routine.
//      If INDX is less than 0, the user is to compare items I
//      and J and return in ISGN a negative value if I is to
//      precede J, and a positive value otherwise.
//      If INDX is 0, the sorting is done.
//
//    Output, int &I, &J.  On return with INDX positive,
//    elements I and J of the user's list should be
//    interchanged.  On return with INDX negative, elements I
//    and J are to be compared by the user.
//
//    Input, int ISGN. On return with INDX negative, the
//    user should compare elements I and J of the list.  If
//    item I is to precede item J, set ISGN negative,
//    otherwise set ISGN positive.
//
{
  static int i_save = 0;
  static int j_save = 0;
  static int k = 0;
  static int k1 = 0;
  static int n1 = 0;
//
//  INDX = 0: This is the first call.
//
  if ( indx == 0 )
  {

    i_save = 0;
    j_save = 0;
    k = n / 2;
    k1 = k;
    n1 = n;
  }
//
//  INDX < 0: The user is returning the results of a comparison.
//
  else if ( indx < 0 )
  {
    if ( indx == -2 )
    {
      if ( isgn < 0 )
      {
        i_save = i_save + 1;
      }
      j_save = k1;
      k1 = i_save;
      indx = -1;
      i = i_save;
      j = j_save;
      return;
    }

    if ( 0 < isgn )
    {
      indx = 2;
      i = i_save;
      j = j_save;
      return;
    }

    if ( k <= 1 )
    {
      if ( n1 == 1 )
      {
        i_save = 0;
        j_save = 0;
        indx = 0;
      }
      else
      {
        i_save = n1;
        j_save = 1;
        n1 = n1 - 1;
        indx = 1;
      }
      i = i_save;
      j = j_save;
      return;
    }
    k = k - 1;
    k1 = k;
  }
//
//  0 < INDX: the user was asked to make an interchange.
//
  else if ( indx == 1 )
  {
    k1 = k;
  }

  for ( ; ; )
  {

    i_save = 2 * k1;

    if ( i_save == n1 )
    {
      j_save = k1;
      k1 = i_save;
      indx = -1;
      i = i_save;
      j = j_save;
      return;
    }
    else if ( i_save <= n1 )
    {
      j_save = i_save + 1;
      indx = -2;
      i = i_save;
      j = j_save;
      return;
    }

    if ( k <= 1 )
    {
      break;
    }

    k = k - 1;
    k1 = k;
  }

  if ( n1 == 1 )
  {
    i_save = 0;
    j_save = 0;
    indx = 0;
    i = i_save;
    j = j_save;
  }
  else
  {
    i_save = n1;
    j_save = 1;
    n1 = n1 - 1;
    indx = 1;
    i = i_save;
    j = j_save;
  }

  return;
}
//****************************************************************************80

void timestamp ( )

//****************************************************************************80
//
//  Purpose:
//
//    TIMESTAMP prints the current YMDHMS date as a time stamp.
//
//  Example:
//
//    31 May 2001 09:45:54 AM
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    08 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    None
//
{
# define TIME_SIZE 40

  static char time_buffer[TIME_SIZE];
  const struct std::tm *tm_ptr;
  size_t len;
  std::time_t now;

  now = std::time ( NULL );
  tm_ptr = std::localtime ( &now );

  len = std::strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm_ptr );

  std::cout << time_buffer << "\n";

  return;
# undef TIME_SIZE
}