function laguerre_exactness ( quad_filename, degree_max, option )

%*****************************************************************************80
%
%% laguerre_exactness() checks a Laguerre quadrature rule for exactness.
%
%  Discussion:
%
%    This program investigates a standard Gauss-Laguerre quadrature rule
%    by using it to integrate monomials over [0,+oo), and comparing the
%    approximate result to the known exact value.
%
%    The user specifies:
%    * the "root" name of the R, W and X files that specify the rule;
%    * DEGREE_MAX, the maximum monomial degree to be checked.
%    * OPTION, whether the rule is for exp(-x)*f(x) or f(x).
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    14 May 2014
%
%  Author:
%
%    John Burkardt
%
%  Input:
%
%    character quad_filename,  the "root" name of the R, W and X files 
%    that specify the rule;
%
%    integer DEGREE_MAX, the maximum monomial degree to be checked.
%
%    integer OPTION, whether the rule is for exp(-x)*f(x) or f(x).
%
  timestamp ( );
  fprintf ( 1, '\n' );
  fprintf ( 1, 'LAGUERRE_EXACTNESS\n' );
  fprintf ( 1, '  MATLAB version\n' );
  fprintf ( 1, '\n' );
  fprintf ( 1, '  Investigate the exactness of a Gauss-Laguerre\n' );
  fprintf ( 1, '  quadrature rule for integrating monomials\n' );
  fprintf ( 1, '  with density exp(-x) or density 1\n' );
  fprintf ( 1, '  over the [0,+oo) interval.\n' );
%
%  Get the quadrature file root name:
%
  if ( 1 <= nargin )

  else

    quad_filename = input ( '  Enter the "root" name of the files, in quotes: ' );

  end
%
%  Create the names of:
%    the quadrature X file;
%    the quadrature W file;
%    the quadrature R file;
%
  quad_x_filename = strcat ( quad_filename, '_x.txt' );
  quad_w_filename = strcat ( quad_filename, '_w.txt' );
  quad_r_filename = strcat ( quad_filename, '_r.txt' );
%
%  The second command line argument is the maximum degree.
%
  if ( 2 <= nargin )

  else

    degree_max = input ( '  Enter the maximum degree: ' );

  end
%
%  The third command line argument is OPTION.
%  0 for the standard rule for integrating exp(-x)*f(x),
%  1 for a rule for integrating f(x).
%
  if ( 3 <= nargin )

  else

    fprintf ( 1, '\n' );
    fprintf ( 1, '  OPTION chooses the standard or modified rule:\n' );
    fprintf ( 1, '  0: standard rule for integrating exp(-x)*f(x);\n' );
    fprintf ( 1, '  1: modified rule for integrating         f(x).\n' );
    fprintf ( 1, '\n' );
    option = input ( '  Enter OPTION:  ' );

  end
%
%  Summarize the input.
%
  fprintf ( 1, '\n' );
  fprintf ( 1, 'LAGUERRE_EXACTNESS: User input:\n' );
  fprintf ( 1, '  Quadrature rule X file = "%s".\n', quad_x_filename );
  fprintf ( 1, '  Quadrature rule W file = "%s".\n', quad_w_filename );
  fprintf ( 1, '  Quadrature rule R file = "%s".\n', quad_r_filename );
  fprintf ( 1, '  Maximum degree to check = %d\n', degree_max );
  if ( option == 0 )
    fprintf ( 1, '  OPTION = 0, integrate exp(-x)*f(x).\n' );
  else
    fprintf ( 1, '  OPTION = 1, integrate         f(x).\n' );
  end
%
%  Read the X file.
%
  [ dim_num, order ] = r8mat_header_read ( quad_x_filename );

  if ( dim_num ~= 1 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'LAGUERRE_EXACTNESS - Fatal error!\n' );
    fprintf ( 1, '  The spatial dimension should be 1.\n');
    fprintf ( 1, '  The spatial dimension in the X file is %d\n', dim_num );
    error ( 'LAGUERRE_EXACTNESS - Fatal error!' );
  end

  fprintf ( 1, '\n' );
  fprintf ( 1, '  Spatial dimension = %d\n', dim_num );
  fprintf ( 1, '  Number of points  = %d\n', order );

  x = r8mat_data_read ( quad_x_filename, dim_num, order );
%
%  Read the W file.
%
  [ dim_num2, point_num ] = r8mat_header_read ( quad_w_filename );

  if ( dim_num2 ~= 1 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'LAGUERRE_EXACTNESS - Fatal error!\n' );
    fprintf ( 1, '  The quadrature weight file should have exactly\n');
    fprintf ( 1, '  one value on each line.\n' );
    error ( 'LAGUERRE_EXACTNESS - Fatal error!' );
  end

  if ( point_num ~= order )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'LAGUERRE_EXACTNESS - Fatal error!\n' );
    fprintf ( 1, '  The quadrature weight file should have exactly\n' );
    fprintf ( 1, '  the same number of lines as the abscissa file.\n' );
    error ( 'LAGUERRE_EXACTNESS - Fatal error!' );
  end

  w = r8mat_data_read ( quad_w_filename, 1, order );
%
%  Read the R file.
%
  [ dim_num2, point_num ] = r8mat_header_read ( quad_r_filename );

  if ( dim_num2 ~= dim_num )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'v - Fatal error!\n' );
    fprintf ( 1, '  The quadrature region file should have the same\n' );
    fprintf ( 1, '  number of values on each line as the abscissa file\n' );
    fprintf ( 1, '  does.\n' );
    error ( 'LAGUERRE_EXACTNESS - Fatal error!' );
  end

  if ( point_num ~= 2 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'LAGUERRE_EXACTNESS - Fatal error!\n' );
    fprintf ( 1, '  The quadrature region file should have two lines.\n' );
    error ( 'LAGUERRE_EXACTNESS - Fatal error!' );
  end

  r = r8mat_data_read ( quad_r_filename, dim_num, 2 );
%
%  Print the input quadrature rule.
%
  a = r(1);
  fprintf ( 1, '\n' );
  fprintf ( 1, '  The quadrature rule to be tested is\n' );
  fprintf ( 1, '  a Gauss-Laguerre quadrature rule of ORDER = %d\n', order );
  fprintf ( 1, '  for integrals of the type:\n' );
  if ( option == 0 )
    fprintf ( 1, '    Integral ( %g <= x < +oo ) f(x) exp(-x) dx\n', a );
  elseif ( option == 1 )
    fprintf ( 1, '    Integral ( %g <= x < +oo ) f(x) dx\n', a );
  end
  fprintf ( 1, '\n' );
  fprintf ( 1, '  Weights W:\n' );
  fprintf ( 1, '\n' );
  for i = 1 : order
    fprintf ( 1, '  w(%d) = %24.16f\n', i, w(i) );
  end
  fprintf ( 1, '\n' );
  fprintf ( 1, '  Abscissas X:\n' );
  fprintf ( 1, '\n' );
  for i = 1 : order
    fprintf ( 1, '  x(%d) = %24.16f\n', i, x(i) );
  end
  fprintf ( 1, '\n' );
  fprintf ( 1, '  Region R:\n' );
  fprintf ( 1, '\n' );
  for i = 1 : 2
    fprintf ( 1, '  r(%d) = %e\n', i, r(i) );
  end
%
%  Supposing the input rule is defined on [A,+oo),
%  rescale the weights, and translate the abscissas, 
%  so our revised rule is defined on [0,+oo).
%
  volume = exp ( - a );
  w(1:order) = w(1:order) / volume;
  x(1:order) = x(1:order) - a;
%
%  Explore the monomials.
%
  fprintf ( 1, '\n' );
  fprintf ( 1, '  A Gauss-Laguerre rule would be able to exactly\n' );
  fprintf ( 1, '  integrate monomials up to and including \n' );
  fprintf ( 1, '  degree = %d\n', 2 * order - 1 );
  fprintf ( 1, '\n' );
  fprintf ( 1, '  Degree      Error\n' );
  fprintf ( 1, '\n' );

  for degree = 0 : degree_max

    quad_error = laguerre_monomial_quadrature ( degree, order, option, w, x );

    fprintf ( 1, '  %2d  %24.16f\n', degree, quad_error );

  end
%
%  Terminate.
%
  fprintf ( 1, '\n' );
  fprintf ( 1, 'LAGUERRE_EXACTNESS:\n' );
  fprintf ( 1, '  Normal end of execution.\n' );
  fprintf ( 1, '\n' );
  timestamp ( );

  return
end
function column_num = file_column_count ( input_file_name )

%*****************************************************************************80
%
%% file_column_count() counts the columns in the first line of a file.
%
%  Discussion:
%
%    The file is assumed to be a simple text file.
%
%    Most lines of the file are presumed to consist of COLUMN_NUM words,
%    separated by spaces.  There may also be some blank lines, and some 
%    comment lines, which have a "#" in column 1.
%
%    The routine tries to find the first non-comment non-blank line and
%    counts the number of words in that line.
%
%    If all lines are blanks or comments, it goes back and tries to analyze
%    a comment line.
%
%  Licensing:
%
%    This code is distributed under the MIT license. 
%
%  Modified:
%
%    21 February 2004
%
%  Author:
%
%    John Burkardt
%
%  Input:
%
%    string INPUT_FILE_NAME, the name of the file.
%
%  Output:
%
%    integer COLUMN_NUM, the number of columns in the file.
%

%
%  Open the file.
%
  input_unit = fopen ( input_file_name );

  if ( input_unit < 0 ) 
    fprintf ( 1, '\n' );
    fprintf ( 1, 'FILE_COLUMN_COUNT - Error!\n' );
    fprintf ( 1, '  Could not open the file "%s".\n', input_file_name );
    error ( 'FILE_COLUMN_COUNT - Error!' );
  end
%
%  Read one line, but skip blank lines and comment lines.
%  Use FGETL so we drop the newline character!
%
  got_one = false;

  while ( true )

    line = fgetl ( input_unit );

    if ( line == -1 )
      break;
    end

    if ( s_len_trim ( line ) == 0 )

    elseif ( line(1) == '#' )

    else
      got_one = true;
      break;
    end

  end

  fclose ( input_unit );

  if ( ~ got_one ) 
    fprintf ( 1, '\n' );
    fprintf ( 1, 'FILE_COLUMN_COUNT - Warning!\n' );
    fprintf ( 1, '  The file does not seem to contain any data.\n' );
    column_num = -1;
    return;
  end

  column_num = s_word_count ( line );

  return
end
function row_num = file_row_count ( input_file_name )

%*****************************************************************************80
%
%% file_row_count() counts the number of row records in a file.
%
%  Discussion:
%
%    Each input line is a "RECORD".
%
%    The records are divided into three groups:
%    
%    * BLANK LINES (nothing but blanks)
%    * COMMENT LINES (begin with a '#')
%    * DATA RECORDS (anything else)
%
%    The value returned by the function is the number of data records.
%
%    By the way, if the MATLAB routine FGETS is used, instead of
%    FGETL, then the variable LINE will include line termination 
%    characters, which means that a blank line would not actually
%    have zero characters.
%
%  Licensing:
%
%    This code is distributed under the MIT license. 
%
%  Modified:
%
%    31 December 2006
%
%  Author:
%
%    John Burkardt
%
%  Input:
%
%    string INPUT_FILE_NAME, the name of the input file.
%
%  Output:
%
%    integer ROW_NUM, the number of rows found. 
%
  input_unit = fopen ( input_file_name );

  if ( input_unit < 0 ) 
    fprintf ( 1, '\n' );
    fprintf ( 1, 'FILE_ROW_COUNT - Error!\n' );
    fprintf ( 1, '  Could not open the file "%s".\n', input_file_name );
    error ( 'FILE_ROW_COUNT - Error!' );
  end

  blank_num = 0;
  comment_num = 0;
  row_num = 0;
  
  record_num = 0;

  while ( true )

    line = fgetl ( input_unit );

    if ( line == -1 )
      break;
    end

    record_num = record_num + 1;
    record_length = s_len_trim ( line );
    
    if ( record_length <= 0 )
      blank_num = blank_num + 1;
    elseif ( line(1) == '#' )
      comment_num = comment_num + 1;
    else
      row_num = row_num + 1;
    end

  end

  fclose ( input_unit );

  return
end
function value = laguerre_integral ( n )

%*****************************************************************************80
%
%% laguerre_integral() evaluates a monomial Laguerre integral.
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    08 June 2013
%
%  Author:
%
%    John Burkardt
%
%  Input:
%
%    integer N, the order of the integral.
%    0 <= N.
%
%  Output:
%
%    real VALUE, the value of the integral.
%
  value = r8_factorial ( n );

  return
end
function quad_error = laguerre_monomial_quadrature ( expon, order, option, ...
  w, x )

%*****************************************************************************80
%
%% laguerre_monomial_quadrature() applies a quadrature rule to a monomial.
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    08 June 2013
%
%  Author:
%
%    John Burkardt
%
%  Input:
%
%    integer EXPON, the exponent.
%
%    intege ORDER, the number of points in the rule.
%
%    integer OPTION, indicates standard or modified rule.
%    0, standard Gauss-Laguerre rule for integrand exp(-x)*f(x).
%    1, modified Gauss-Laguerre rule for integrand         f(x).
%
%    real W(ORDER), the quadrature weights.
%
%    real X(ORDER), the quadrature points.
%
%  Output:
%
%    real QUAD_ERROR, the quadrature error.
%

%
%  Get the exact value of the integral of the unscaled monomial.
%
  exact = laguerre_integral ( expon );

  if ( option == 0 )
    value(1:order) =                         x(1:order).^expon;
  else
    value(1:order) = exp ( - x(1:order) ) .* x(1:order).^expon;
  end
%
%  Compute the weighted sum.
%
  quad = w(1:order) * value(1:order)';
%
%  Absolute error for cases where exact integral is zero,
%  Relative error otherwise.
%
  if ( exact == 0.0 )
    quad_error = abs ( quad );
  else
    quad_error = abs ( quad - exact ) / abs ( exact );
  end

  return
end
function value = r8_factorial ( n )

%*****************************************************************************80
%
%% r8_factorial() returns N factorial.
%
%  Discussion:
%
%    factorial ( N ) = Product ( 1 <= I <= N ) I
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    02 February 1999
%
%  Author:
%
%    John Burkardt
%
%  Input:
%
%    integer N, the argument of the function.
%    0 <= N.
%
%  Output:
%
%    real VALUE, the factorial of N.
%
  if ( n < 0 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'R8_FACTORIAL - Fatal error!\n' );
    fprintf ( 1, '  N < 0.\n' );
    error ( 'R8_FACTORIAL - Fatal error!' );
  end

  value = 1.0;

  for i = 2 : n
    value = value * i;
  end

  return
end
function table = r8mat_data_read ( input_filename, m, n )

%*****************************************************************************80
%
%% r8mat_data_read() reads data from an R8MAT file.
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    27 January 2006
%
%  Author:
%
%    John Burkardt
%
%  Input:
%
%    string INPUT_FILENAME, the name of the input file.
%
%    integer M, N, the number of rows and columns of data.
%
%  Output:
%
%    real TABLE(M,N), the point coordinates.
%
  table = zeros ( m, n );
%
%  Build up the format string for reading M real numbers.
%
  string = ' ';

  for i = 0 : m
    string = strcat ( string, ' %f' );
  end

  input_unit = fopen ( input_filename );

  if ( input_unit < 0 ) 
    fprintf ( 1, '\n' );
    fprintf ( 1, 'R8MAT_DATA_READ - Error!\n' );
    fprintf ( 1, '  Could not open the file.\n' );
    error ( 'R8MAT_DATA_READ - Error!' );
  end

  i = 0;

  while ( i < n )

    line = fgets ( input_unit );

    if ( line == -1 )
      break;
    end

    if ( line(1) == '#' )

    elseif ( s_len_trim ( line ) == 0 )
      
    else

      [ x, count ] = sscanf ( line, string );

      if ( count == m )
        i = i + 1;
        table(1:m,i) = x(1:m);
      end

    end

  end

  fclose ( input_unit );

  return
end
function [ m, n ] = r8mat_header_read ( input_filename )

%*****************************************************************************80
%
%% r8mat_header_read() reads the header from an R8MAT file.
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    22 October 2004
%
%  Author:
%
%    John Burkardt
%
%  Input:
%
%    string INPUT_FILENAME, the name of the input file.
%
%  Output:
%
%    integer M, the spatial dimension.
%
%    integer N, the number of points.
%
  m = file_column_count ( input_filename );

  if ( m <= 0 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'R8MAT_HEADER_READ - Fatal error!\n' );
    fprintf ( 1, '  There was some kind of I/O problem while trying\n' );
    fprintf ( 1, '  to count the number of data columns in\n' );
    fprintf ( 1, '  the file %s.\n', input_filename );
  end

  n = file_row_count ( input_filename );

  if ( n <= 0 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'R8MAT_HEADER_READ - Fatal error!\n' );
    fprintf ( 1, '  There was some kind of I/O problem while trying\n' );
    fprintf ( 1, '  to count the number of data rows in\n' );
    fprintf ( 1, '  the file %s\n', input_filename );
  end

  return
end
function len = s_len_trim ( s )

%*****************************************************************************80
%
%% s_len_trim() returns the length of a character string to the last nonblank.
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    14 June 2003
%
%  Author:
%
%    John Burkardt
%
%  Input:
%
%    string S, the string to be measured.
%
%  Output:
%
%    integer LENGTH, the length of the string up to the last nonblank.
%
  len = length ( s );

  while ( 0 < len )
    if ( s(len) ~= ' ' )
      return
    end
    len = len - 1;
  end
end
function word_num = s_word_count ( s )

%*****************************************************************************80
%
%% s_word_count() counts the number of "words" in a string.
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    30 January 2006
%
%  Author:
%
%    John Burkardt
%
%  Input:
%
%    string S, the string to be examined.
%
%  Output:
%
%    integer WORD_NUM, the number of "words" in the string.
%    Words are presumed to be separated by one or more blanks.
%
  word_num = 0;
  s_length = length ( s );

  if ( s_length <= 0 )
    return;
  end

  blank = true;

  for i = 1 : s_length

    if ( s(i) == ' ' )
      blank = true;
    elseif ( blank )
      word_num = word_num + 1;
      blank = false;
    end

  end

  return
end
function timestamp ( )

%*****************************************************************************80
%
%% timestamp() prints the current YMDHMS date as a timestamp.
%
%  Licensing:
%
%    This code is distributed under the MIT license.
%
%  Modified:
%
%    14 February 2003
%
%  Author:
%
%    John Burkardt
%
  t = now;
  c = datevec ( t );
  s = datestr ( c, 0 );
  fprintf ( 1, '%s\n', s );

  return
end
 
