function sparse_grid_hermite_dataset ( dim_num, level_max )

%*****************************************************************************80
%
%% SPARSE_GRID_HERMITE_DATASET is the main program for SPARSE_GRID_HERMITE_DATASET.
%
%  Discussion:
%
%    This program computes a sparse grid quadrature rule based on a 1D
%    Gauss-Hermite rule and writes it to a file..
%
%    The user specifies:
%    * the spatial dimension of the quadrature region,
%    * the level that defines the Smolyak grid.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    10 August 2009
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    Fabio Nobile, Raul Tempone, Clayton Webster,
%    A Sparse Grid Stochastic Collocation Method for Partial Differential
%    Equations with Random Input Data,
%    SIAM Journal on Numerical Analysis,
%    Volume 46, Number 5, 2008, pages 2309-2345.
%
  timestamp ( );
  fprintf ( 1, '\n' );
  fprintf ( 1, 'SPARSE_GRID_HERMITE_DATASET\n' );
  fprintf ( 1, '  MATLAB version\n' );
  fprintf ( 1, '\n' );
  fprintf ( 1, '  Compute the abscissas and weights of a quadrature rule\n' );
  fprintf ( 1, '  associated with a sparse grid derived from a Smolyak\n' );
  fprintf ( 1, '  construction based on 1D Gauss-Hermite rules.\n' );
  fprintf ( 1, '\n' );
  fprintf ( 1, '  Inputs to the program include:\n' );
  fprintf ( 1, '\n' );
  fprintf ( 1, '    DIM_NUM, the spatial dimension.\n' );
  fprintf ( 1, '    (typically in the range of 2 to 10)\n' );
  fprintf ( 1, '\n' );
  fprintf ( 1, '    LEVEL_MAX, the "level" of the sparse grid.\n' );
  fprintf ( 1, '    (typically in the range of 0, 1, 2, 3, ...\n' );
  fprintf ( 1, '\n' );
  fprintf ( 1, '  Output from the program includes:\n' );
  fprintf ( 1, '\n' );
  fprintf ( 1, '    A printed table of the abscissas and weights.\n' );
  fprintf ( 1, '\n' );
  fprintf ( 1, '    * A set of 3 files that define the quadrature rule.\n' );
  fprintf ( 1, '\n' );
  fprintf ( 1, '    "herm_d?_level?_r.txt", a file of the ranges.\n' );
  fprintf ( 1, '    "herm_d?_level?_w.txt", a file of the weights;\n' );
  fprintf ( 1, '    "herm_d?_level?_x.txt", a file of the abscissas;\n' );
%
%  Get the spatial dimension.
%
  if ( nargin < 1)
    fprintf ( 1, '\n' );
    dim_num = input ( '  Enter the value of DIM_NUM (1 or greater)' );
  end

  fprintf ( 1, '\n' );
  fprintf ( 1, '  Spatial dimension requested is = %d\n', dim_num );
%
%  Get the level.
%
  if ( nargin < 2 )
    fprintf ( 1, '\n' );
    level_max = input ( '  Enter the value of LEVEL_MAX (0 or greater)' );
  end

  level_min = max ( 0, level_max + 1 - dim_num );

  fprintf ( 1, '\n' );
  fprintf ( 1, '  LEVEL_MIN = %d\n', level_min );
  fprintf ( 1, '  LEVEL_MAX = %d\n', level_max );
%
%  How many distinct points will there be?
%
  point_num = sparse_grid_herm_size ( dim_num, level_max );

  fprintf ( 1, '\n' );
  fprintf ( 1, '  The number of distinct abscissas in the\n' );
  fprintf ( 1, '  quadrature rule is determined from the spatial\n' );
  fprintf ( 1, '  dimension DIM_NUM and the level LEVEL_MAX.\n' );
  fprintf ( 1, '  For the given input, this value will be = %d\n', point_num );

  r = zeros ( dim_num, 2 );
%
%  Compute the weights and points.
%
  r(1:dim_num,1) = - r8_huge ( );
  r(1:dim_num,2) = + r8_huge ( );

  [ w, x ] = sparse_grid_herm ( dim_num, level_max, point_num );

  r8mat_transpose_print_some ( dim_num, point_num, x, 1, 1, ...
    dim_num, 10, '  First 10 grid points:' );

  r8vec_print_some ( point_num, w, 1, 10, '  First 10 weights:' );

  weight_sum = sum ( w(1:point_num) );

  fprintf ( 1, '\n' );
  fprintf ( 1, '  Weights sum to   %24.16f\n', weight_sum );
  fprintf ( 1, '  Correct value is %24.16f\n', sqrt ( pi^dim_num ) );
%
%  Write the rule to files.
%
  r_filename = sprintf ( 'herm_d%d_level%d_r.txt', dim_num, level_max );
  w_filename = sprintf ( 'herm_d%d_level%d_w.txt', dim_num, level_max );
  x_filename = sprintf ( 'herm_d%d_level%d_x.txt', dim_num, level_max );

  fprintf ( 1, '\n' );
  fprintf ( 1, '  Creating R file = "%s".\n', r_filename );

  r8mat_write ( r_filename, dim_num, 2, r );

  fprintf ( 1, '  Creating W file = "%s".\n', w_filename );

  r8mat_write ( w_filename, 1, point_num, w );

  fprintf ( 1, '  Creating X file = "%s".\n', x_filename );

  r8mat_write ( x_filename, dim_num, point_num, x );

  fprintf ( 1, '\n' );
  fprintf ( 1, 'SPARSE_GRID_HERM_DATASET:\n' );
  fprintf ( 1, '  Normal end of execution.\n' );

  fprintf ( 1, '\n' );
  timestamp ( );

  return
end
function [ a, more, h, t ] = comp_next ( n, k, a, more, h, t )

%*****************************************************************************80
%
%% COMP_NEXT computes the compositions of the integer N into K parts.
%
%  Discussion:
%
%    A composition of the integer N into K parts is an ordered sequence
%    of K nonnegative integers which sum to N.  The compositions (1,2,1)
%    and (1,1,2) are considered to be distinct.
%
%    The routine computes one composition on each call until there are no more.
%    For instance, one composition of 6 into 3 parts is
%    3+2+1, another would be 6+0+0.
%
%    On the first call to this routine, set MORE = FALSE.  The routine
%    will compute the first element in the sequence of compositions, and
%    return it, as well as setting MORE = TRUE.  If more compositions
%    are desired, call again, and again.  Each time, the routine will
%    return with a new composition.
%
%    However, when the LAST composition in the sequence is computed
%    and returned, the routine will reset MORE to FALSE, signaling that
%    the end of the sequence has been reached.
%
%    This routine originally used a SAVE statement to maintain the
%    variables H and T.  I have decided that it is safer
%    to pass these variables as arguments, even though the user should
%    never alter them.  This allows this routine to safely shuffle
%    between several ongoing calculations.
%
%    There are 28 compositions of 6 into three parts.  This routine will
%    produce those compositions in the following order:
%
%     I         A
%     -     ---------
%     1     6   0   0
%     2     5   1   0
%     3     4   2   0
%     4     3   3   0
%     5     2   4   0
%     6     1   5   0
%     7     0   6   0
%     8     5   0   1
%     9     4   1   1
%    10     3   2   1
%    11     2   3   1
%    12     1   4   1
%    13     0   5   1
%    14     4   0   2
%    15     3   1   2
%    16     2   2   2
%    17     1   3   2
%    18     0   4   2
%    19     3   0   3
%    20     2   1   3
%    21     1   2   3
%    22     0   3   3
%    23     2   0   4
%    24     1   1   4
%    25     0   2   4
%    26     1   0   5
%    27     0   1   5
%    28     0   0   6
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    02 July 2008
%
%  Author:
%
%    Original FORTRAN77 version by Albert Nijenhuis, Herbert Wilf.
%    MATLAB version by John Burkardt.
%
%  Reference:
%
%    Albert Nijenhuis, Herbert Wilf,
%    Combinatorial Algorithms for Computers and Calculators,
%    Second Edition,
%    Academic Press, 1978,
%    ISBN: 0-12-519260-6,
%    LC: QA164.N54.
%
%  Parameters:
%
%    Input, integer N, the integer whose compositions are desired.
%
%    Input, integer K, the number of parts in the composition.
%
%    Input, integer A(K), the previous composition.  On the first call,
%    with MORE = FALSE, set A = [].  Thereafter, A should be the
%    value of A output from the previous call.
%
%    Input, logical MORE.  The input value of MORE on the first
%    call should be FALSE, which tells the program to initialize.
%    On subsequent calls, MORE should be TRUE, or simply the
%    output value of MORE from the previous call.
%
%    Input, integer H, T, two internal parameters needed for the
%    computation.  The user may need to initialize these before the
%    very first call, but these initial values are not important.
%    The user should not alter these parameters once the computation
%    begins.
%
%    Output, integer A(K), the next composition.
%
%    Output, logical MORE, will be TRUE unless the composition
%    that is being returned is the final one in the sequence.
%
%    Output, integer H, T, the updated values of the two internal
%    parameters.
%
  if ( ~more )

    t = n;
    h = 0;
    a(1) = n;
    a(2:k) = 0;

  else

    if ( 1 < t )
      h = 0;
    end

    h = h + 1;
    t = a(h);
    a(h) = 0;
    a(1) = t - 1;
    a(h+1) = a(h+1) + 1;

  end

  more = ( a(k) ~= n );

  return
end
function grid_point = hermite_abscissa ( dim_num, point_num, grid_index, ...
  grid_base )

%*****************************************************************************80
%
%% HERMITE_ABSCISSA sets abscissas for multidimensional Gauss-Hermite quadrature.
%
%  Discussion:
%
%    The "nesting" as it occurs for Gauss-Hermite sparse grids simply
%    involves the use of a specified set of permissible orders for the
%    rule.
%
%    The X array lists the (complete) Gauss-Hermite abscissas for rules
%    of order 1, 3, 7, 15, 31, 63 and 127, in order.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    08 October 2007
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer DIM_NUM, the spatial dimension.
%
%    Input, integer POINT_NUM, the number of points.
%
%    Input, integer GRID_INDEX(DIM_NUM,POINT_NUM), for each
%    point and dimension, the index of the abscissa.
%
%    Input, integer GRID_BASE(DIM_NUM), the order of the
%    rule being used in each dimension.
%
%    Output, real GRID_POINT(DIM_NUM,POINT_NUM), the grid points of abscissas.
%
  skip = [ 0, 1, 4, 11, 26, 57, 120, 247 ];
  x = [ ...
    0.0E+00, ...
   -0.122474487139158904909864203735E+01, ...
    0.0E+00, ...
    0.122474487139158904909864203735E+01, ...
   -0.265196135683523349244708200652E+01, ...
   -0.167355162876747144503180139830E+01, ...
   -0.816287882858964663038710959027E+00, ...
    0.0E+00, ...
    0.816287882858964663038710959027E+00, ...
    0.167355162876747144503180139830E+01, ...
    0.265196135683523349244708200652E+01, ...
   -0.449999070730939155366438053053E+01, ...
   -0.366995037340445253472922383312E+01, ...
   -0.296716692790560324848896036355E+01, ...
   -0.232573248617385774545404479449E+01, ...
   -0.171999257518648893241583152515E+01, ...
   -0.113611558521092066631913490556E+01, ...
   -0.565069583255575748526020337198E+00, ...
    0.0E+00, ...
    0.565069583255575748526020337198E+00, ...
    0.113611558521092066631913490556E+01, ...
    0.171999257518648893241583152515E+01, ...
    0.232573248617385774545404479449E+01, ...
    0.296716692790560324848896036355E+01, ...
    0.366995037340445253472922383312E+01, ...
    0.449999070730939155366438053053E+01, ...
   -6.9956801237185402753248521473232E+00, ...
   -6.2750787049428601427036567812530E+00, ...
   -5.6739614446185883296332558789276E+00, ...
   -5.1335955771123807045862968913996E+00, ...
   -4.6315595063128599420667997654336E+00, ...
   -4.1562717558181451724831352315314E+00, ...
   -3.7007434032314694224497164589673E+00, ...
   -3.2603207323135408104645401509648E+00, ...
   -2.8316804533902054557015640151425E+00, ...
   -2.4123177054804201051740184582119E+00, ...
   -2.0002585489356389657975562598571E+00, ...
   -1.5938858604721398261388419455550E+00, ...
   -1.1918269983500464260821358649242E+00, ...
   -0.79287697691530893968593032998830E+00, ...
   -0.39594273647142311094670041663436E+00, ...
    0.0000000000000000000000000000000E+00, ...
    0.39594273647142311094670041663436E+00, ...
    0.79287697691530893968593032998830E+00, ...
    1.1918269983500464260821358649242E+00, ...
    1.5938858604721398261388419455550E+00, ...
    2.0002585489356389657975562598571E+00, ...
    2.4123177054804201051740184582119E+00, ...
    2.8316804533902054557015640151425E+00, ...
    3.2603207323135408104645401509648E+00, ...
    3.7007434032314694224497164589673E+00, ...
    4.1562717558181451724831352315314E+00, ...
    4.6315595063128599420667997654336E+00, ...
    5.1335955771123807045862968913996E+00, ...
    5.6739614446185883296332558789276E+00, ...
    6.2750787049428601427036567812530E+00, ...
    6.9956801237185402753248521473232E+00, ...
   -10.435499877854168053468115427285E+00, ...
   -9.8028759912974963635223935286507E+00, ...
   -9.2792019543050391319404745506496E+00, ...
   -8.8118581437284546442526628275570E+00, ...
   -8.3807683451863219343010651043788E+00, ...
   -7.9755950801420373181541806298501E+00, ...
   -7.5901395198641066762479783194468E+00, ...
   -7.2203167078889678461161324222529E+00, ...
   -6.8632544331795368527353285876066E+00, ...
   -6.5168348106821160605273395854042E+00, ...
   -6.1794379922705969862418461787263E+00, ...
   -5.8497884000810673462526582961482E+00, ...
   -5.5268572526403031425047575122840E+00, ...
   -5.2097979830408354861575136416263E+00, ...
   -4.8979018644975742350745099214868E+00, ...
   -4.5905665744435190229271294569091E+00, ...
   -4.2872733352824404031727616199454E+00, ...
   -3.9875699104197157485227052068068E+00, ...
   -3.6910577000963465117322810559754E+00, ...
   -3.3973817713303911852755941806287E+00, ...
   -3.1062230279282566329138616746036E+00, ...
   -2.8172919672837977750747135657355E+00, ...
   -2.5303236304712010926855221718499E+00, ...
   -2.2450734604812066298995918179330E+00, ...
   -1.9613138583081485293922008411321E+00, ...
   -1.6788312791720137520802800622638E+00, ...
   -1.3974237486049625107570752063702E+00, ...
   -1.1168987050996462690510970277840E+00, ...
   -0.83707109558947615977737795461293E+00, ...
   -0.55776166427908221668763665253822E+00, ...
   -0.27879538567115223986687628627202E+00, ...
    0.00000000000000000000000000000000E+00, ...
    0.27879538567115223986687628627202E+00, ...
    0.55776166427908221668763665253822E+00, ...
    0.83707109558947615977737795461293E+00, ...
    1.1168987050996462690510970277840E+00, ...
    1.3974237486049625107570752063702E+00, ...
    1.6788312791720137520802800622638E+00, ...
    1.9613138583081485293922008411321E+00, ...
    2.2450734604812066298995918179330E+00, ...
    2.5303236304712010926855221718499E+00, ...
    2.8172919672837977750747135657355E+00, ...
    3.1062230279282566329138616746036E+00, ...
    3.3973817713303911852755941806287E+00, ...
    3.6910577000963465117322810559754E+00, ...
    3.9875699104197157485227052068068E+00, ...
    4.2872733352824404031727616199454E+00, ...
    4.5905665744435190229271294569091E+00, ...
    4.8979018644975742350745099214868E+00, ...
    5.2097979830408354861575136416263E+00, ...
    5.5268572526403031425047575122840E+00, ...
    5.8497884000810673462526582961482E+00, ...
    6.1794379922705969862418461787263E+00, ...
    6.5168348106821160605273395854042E+00, ...
    6.8632544331795368527353285876066E+00, ...
    7.2203167078889678461161324222529E+00, ...
    7.5901395198641066762479783194468E+00, ...
    7.9755950801420373181541806298501E+00, ...
    8.3807683451863219343010651043788E+00, ...
    8.8118581437284546442526628275570E+00, ...
    9.2792019543050391319404745506496E+00, ...
    9.8028759912974963635223935286507E+00, ...
    10.435499877854168053468115427285E+00, ...
   -15.228338148167350978246954433464E+00, ...
   -14.669595158833972632746354112896E+00, ...
   -14.209085995284870755168244250887E+00, ...
   -13.799722290211676634645246746673E+00, ...
   -13.423518590070950062438258321855E+00, ...
   -13.071208660474601901583995439649E+00, ...
   -12.737235652415686338138003924072E+00, ...
   -12.417939378869715805445879624069E+00, ...
   -12.110749020947747600132123508132E+00, ...
   -11.813772198267727195134584136191E+00, ...
   -11.525565112572696599167888588564E+00, ...
   -11.244994583785543445194384194300E+00, ...
   -10.971150569840247423423040263881E+00, ...
   -10.703288201027481347670940744690E+00, ...
   -10.440787957772772867742591798027E+00, ...
   -10.183127473450343888624126450357E+00, ...
   -9.9298610495114250736847004273684E+00, ...
   -9.6806044412474728038150712732737E+00, ...
   -9.4350233389881650135019598506287E+00, ...
   -9.1928244988460305715774195052527E+00, ...
   -8.9537488108565404323807890169970E+00, ...
   -8.7175658087076307363833999548548E+00, ...
   -8.4840692689832473326097180339984E+00, ...
   -8.2530736454457156579694124243888E+00, ...
   -8.0244111514703375578594739796798E+00, ...
   -7.7979293513870105420829120455591E+00, ...
   -7.5734891556083454022834960763301E+00, ...
   -7.3509631392269052701961258043733E+00, ...
   -7.1302341220350710668064025713431E+00, ...
   -6.9111939615465713197465633109366E+00, ...
   -6.6937425208758294190074417381666E+00, ...
   -6.4777867811645365448144903821487E+00, ...
   -6.2632400742737354345609723857092E+00, ...
   -6.0500214161419845694465474482388E+00, ...
   -5.8380549248774187386601690807757E+00, ...
   -5.6272693105464816659423455794909E+00, ...
   -5.4175974259243240722848425872924E+00, ...
   -5.2089758693153983587570258372239E+00, ...
   -5.0013446320386360038520809107373E+00, ...
   -4.7946467843764925009748509930857E+00, ...
   -4.5888281947698372951606485031212E+00, ...
   -4.3838372778464736294253744407459E+00, ...
   -4.1796247675352031349421189892408E+00, ...
   -3.9761435120673355916035814195920E+00, ...
   -3.7733482881250526721004678400057E+00, ...
   -3.5711956317782180447199756485249E+00, ...
   -3.3696436841717397896643629240035E+00, ...
   -3.1686520501953630191857798261495E+00, ...
   -2.9681816685955910267761649521505E+00, ...
   -2.7681946921824058801226545958892E+00, ...
   -2.5686543769473501723144013022363E+00, ...
   -2.3695249790490401080012474645702E+00, ...
   -2.1707716587411506879498498083695E+00, ...
   -1.9723603904195020079324743227565E+00, ...
   -1.7742578780516791584676442103681E+00, ...
   -1.5764314753267801315519597621879E+00, ...
   -1.3788491099261778091441557053728E+00, ...
   -1.1814792113700685848678583598423E+00, ...
   -0.98429064194027277726568984213773E+00, ...
   -0.78725263021825034151596831878971E+00, ...
   -0.59033470680942102142230439346102E+00, ...
   -0.39350664185130136568037826200185E+00, ...
   -0.19673838392423251964272239737078E+00, ...
    0.0000000000000000000000000000000E+00, ...
    0.19673838392423251964272239737078E+00, ...
    0.39350664185130136568037826200185E+00, ...
    0.59033470680942102142230439346102E+00, ...
    0.78725263021825034151596831878971E+00, ...
    0.98429064194027277726568984213773E+00, ...
    1.1814792113700685848678583598423E+00, ...
    1.3788491099261778091441557053728E+00, ...
    1.5764314753267801315519597621879E+00, ...
    1.7742578780516791584676442103681E+00, ...
    1.9723603904195020079324743227565E+00, ...
    2.1707716587411506879498498083695E+00, ...
    2.3695249790490401080012474645702E+00, ...
    2.5686543769473501723144013022363E+00, ...
    2.7681946921824058801226545958892E+00, ...
    2.9681816685955910267761649521505E+00, ...
    3.1686520501953630191857798261495E+00, ...
    3.3696436841717397896643629240035E+00, ...
    3.5711956317782180447199756485249E+00, ...
    3.7733482881250526721004678400057E+00, ...
    3.9761435120673355916035814195920E+00, ...
    4.1796247675352031349421189892408E+00, ...
    4.3838372778464736294253744407459E+00, ...
    4.5888281947698372951606485031212E+00, ...
    4.7946467843764925009748509930857E+00, ...
    5.0013446320386360038520809107373E+00, ...
    5.2089758693153983587570258372239E+00, ...
    5.4175974259243240722848425872924E+00, ...
    5.6272693105464816659423455794909E+00, ...
    5.8380549248774187386601690807757E+00, ...
    6.0500214161419845694465474482388E+00, ...
    6.2632400742737354345609723857092E+00, ...
    6.4777867811645365448144903821487E+00, ...
    6.6937425208758294190074417381666E+00, ...
    6.9111939615465713197465633109366E+00, ...
    7.1302341220350710668064025713431E+00, ...
    7.3509631392269052701961258043733E+00, ...
    7.5734891556083454022834960763301E+00, ...
    7.7979293513870105420829120455591E+00, ...
    8.0244111514703375578594739796798E+00, ...
    8.2530736454457156579694124243888E+00, ...
    8.4840692689832473326097180339984E+00, ...
    8.7175658087076307363833999548548E+00, ...
    8.9537488108565404323807890169970E+00, ...
    9.1928244988460305715774195052527E+00, ...
    9.4350233389881650135019598506287E+00, ...
    9.6806044412474728038150712732737E+00, ...
    9.9298610495114250736847004273684E+00, ...
    10.183127473450343888624126450357E+00, ...
    10.440787957772772867742591798027E+00, ...
    10.703288201027481347670940744690E+00, ...
    10.971150569840247423423040263881E+00, ...
    11.244994583785543445194384194300E+00, ...
    11.525565112572696599167888588564E+00, ...
    11.813772198267727195134584136191E+00, ...
    12.110749020947747600132123508132E+00, ...
    12.417939378869715805445879624069E+00, ...
    12.737235652415686338138003924072E+00, ...
    13.071208660474601901583995439649E+00, ...
    13.423518590070950062438258321855E+00, ...
    13.799722290211676634645246746673E+00, ...
    14.209085995284870755168244250887E+00, ...
    14.669595158833972632746354112896E+00, ...
    15.228338148167350978246954433464E+00  ...
  ];

  if ( any ( grid_base(1:dim_num) < 0 ) )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'HERMITE_ABSCISSA - Fatal error!\n' );
    fprintf ( 1, '  Some base values are less than 0.\n' );
    error ( 'HERMITE_ABSCISSA - Fatal error!' );
  end

  if ( any ( 63 < grid_base(1:dim_num) ) )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'HERMITE_ABSCISSA - Fatal error!\n' );
    fprintf ( 1, '  Some base values are greater than 63.\n' );
    error ( 'HERMITE_ABSCISSA - Fatal error!' );
  end

  grid_point = zeros ( dim_num, point_num );

  for point = 1 : point_num
    for dim = 1 : dim_num

      level = i4_log_2 ( grid_base(dim) + 1 );

      pointer = skip(level+1) + ( grid_index(dim,point) + grid_base(dim) + 1 );

      grid_point(dim,point) = x(pointer);

    end
  end

  return
end
function weight = hermite_weights ( order )

%*****************************************************************************80
%
%% HERMITE_WEIGHTS returns weights for certain Gauss-Hermite quadrature rules.
%
%  Discussion:
%
%    The allowed orders are 1, 3, 7, 15, 31, 63 or 127.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    08 October 2007
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    Milton Abramowitz, Irene Stegun,
%    Handbook of Mathematical Functions,
%    National Bureau of Standards, 1964,
%    ISBN: 0-486-61272-4,
%    LC: QA47.A34.
%
%    Arthur Stroud, Don Secrest,
%    Gaussian Quadrature Formulas,
%    Prentice Hall, 1966,
%    LC: QA299.4G3S7.
%
%  Parameters:
%
%    Input, integer ORDER, the order of the rule.
%    ORDER must be 1, 3, 7, 15, 31, 63, or 127.
%
%    Output, real WEIGHT(1,ORDER), the weights.
%    The weights are positive, symmetric and should sum to sqrt(PI).
%
  weight = zeros ( 1, order );

  if ( order == 1 )

    weight(1) = 1.77245385090551602729816748334E+00;

  elseif ( order == 3 )

    weight(1) = 0.295408975150919337883027913890E+00;
    weight(2) = 0.118163590060367735153211165556E+01;
    weight(3) = 0.295408975150919337883027913890E+00;

  elseif ( order == 7 )

    weight(1) = 0.971781245099519154149424255939E-03;
    weight(2) = 0.545155828191270305921785688417E-01;
    weight(3) = 0.425607252610127800520317466666E+00;
    weight(4) = 0.810264617556807326764876563813E+00;
    weight(5) = 0.425607252610127800520317466666E+00;
    weight(6) = 0.545155828191270305921785688417E-01;
    weight(7) = 0.971781245099519154149424255939E-03;

  elseif ( order == 15 )

    weight(1) =  0.152247580425351702016062666965E-08;
    weight(2) =  0.105911554771106663577520791055E-05;
    weight(3) =  0.100004441232499868127296736177E-03;
    weight(4) =  0.277806884291277589607887049229E-02;
    weight(5) =  0.307800338725460822286814158758E-01;
    weight(6) =  0.158488915795935746883839384960E+00;
    weight(7) =  0.412028687498898627025891079568E+00;
    weight(8) =  0.564100308726417532852625797340E+00;
    weight(9) =  0.412028687498898627025891079568E+00;
    weight(10) = 0.158488915795935746883839384960E+00;
    weight(11) = 0.307800338725460822286814158758E-01;
    weight(12) = 0.277806884291277589607887049229E-02;
    weight(13) = 0.100004441232499868127296736177E-03;
    weight(14) = 0.105911554771106663577520791055E-05;
    weight(15) = 0.152247580425351702016062666965E-08;

  elseif ( order == 31 )

    weight(  1) =   0.46189683944498305857470556847735E-21;
    weight(  2) =   0.51106090079112519643027197715274E-17;
    weight(  3) =   0.58995564987355133075257722133966E-14;
    weight(  4) =   0.18603735214463569590294465062239E-11;
    weight(  5) =   0.23524920032013205739850619940094E-09;
    weight(  6) =   0.14611988344865057576066495091513E-07;
    weight(  7) =   0.50437125589241034841778074689627E-06;
    weight(  8) =   0.10498602757642934202945441341697E-04;
    weight(  9) =   0.13952090395003623854995664958146E-03;
    weight( 10) =   0.12336833073030489880608311394968E-02;
    weight( 11) =   0.74827999140119116765002499116934E-02;
    weight( 12) =   0.31847230731201222775249585776902E-01;
    weight( 13) =   0.96717948160569462991143316029341E-01;
    weight( 14) =   0.21213278866810461318136114862419E+00;
    weight( 15) =   0.33877265789305344906000174083214E+00;
    weight( 16) =   0.39577855609737786462923720809676E+00;
    weight( 17) =   0.33877265789305344906000174083214E+00;
    weight( 18) =   0.21213278866810461318136114862419E+00;
    weight( 19) =   0.96717948160569462991143316029341E-01;
    weight( 20) =   0.31847230731201222775249585776902E-01;
    weight( 21) =   0.74827999140119116765002499116934E-02;
    weight( 22) =   0.12336833073030489880608311394968E-02;
    weight( 23) =   0.13952090395003623854995664958146E-03;
    weight( 24) =   0.10498602757642934202945441341697E-04;
    weight( 25) =   0.50437125589241034841778074689627E-06;
    weight( 26) =   0.14611988344865057576066495091513E-07;
    weight( 27) =   0.23524920032013205739850619940094E-09;
    weight( 28) =   0.18603735214463569590294465062239E-11;
    weight( 29) =   0.58995564987355133075257722133966E-14;
    weight( 30) =   0.51106090079112519643027197715274E-17;
    weight( 31) =   0.46189683944498305857470556847735E-21;

  elseif ( order == 63 )

    weight(  1) =   0.37099206434787551197827130470031E-47;
    weight(  2) =   0.10400778615192299534481914814892E-41;
    weight(  3) =   0.19796804708258311251124226474396E-37;
    weight(  4) =   0.84687478191640015120141181138947E-34;
    weight(  5) =   0.13071305930779945903630127634063E-30;
    weight(  6) =   0.93437837175367456929765381518998E-28;
    weight(  7) =   0.36027426635173044862245783257252E-25;
    weight(  8) =   0.82963863115951789374753323156164E-23;
    weight(  9) =   0.12266629909105281472971700203949E-20;
    weight( 10) =   0.12288435628797061539461585325494E-18;
    weight( 11) =   0.86925536958188009075932426691516E-17;
    weight( 12) =   0.44857058689176221240330804981619E-15;
    weight( 13) =   0.17335817955735154599902643794700E-13;
    weight( 14) =   0.51265062385038307838565047455223E-12;
    weight( 15) =   0.11808921844532942490513037158404E-10;
    weight( 16) =   0.21508698297808025739828859845140E-09;
    weight( 17) =   0.31371929535285447801497640621672E-08;
    weight( 18) =   0.37041625984781705796752840204084E-07;
    weight( 19) =   0.35734732949879669663960738150956E-06;
    weight( 20) =   0.28393114498380927832990899215541E-05;
    weight( 21) =   0.18709113003730498008961134765721E-04;
    weight( 22) =   0.10284880800653635546698378640623E-03;
    weight( 23) =   0.47411702610173128107201781718693E-03;
    weight( 24) =   0.18409222622384813438539657470055E-02;
    weight( 25) =   0.60436044551187631655712178246467E-02;
    weight( 26) =   0.16829299199599730926458559757600E-01;
    weight( 27) =   0.39858264027692992170237391875317E-01;
    weight( 28) =   0.80467087993950415219587554532823E-01;
    weight( 29) =   0.13871950817615293377792092082674E+00;
    weight( 30) =   0.20448695346833761570957197160475E+00;
    weight( 31) =   0.25799889943058042204920467417642E+00;
    weight( 32) =   0.27876694884838411919175686949858E+00;
    weight( 33) =   0.25799889943058042204920467417642E+00;
    weight( 34) =   0.20448695346833761570957197160475E+00;
    weight( 35) =   0.13871950817615293377792092082674E+00   ;
    weight( 36) =   0.80467087993950415219587554532823E-01;
    weight( 37) =   0.39858264027692992170237391875317E-01;
    weight( 38) =   0.16829299199599730926458559757600E-01;
    weight( 39) =   0.60436044551187631655712178246467E-02;
    weight( 40) =   0.18409222622384813438539657470055E-02;
    weight( 41) =   0.47411702610173128107201781718693E-03;
    weight( 42) =   0.10284880800653635546698378640623E-03;
    weight( 43) =   0.18709113003730498008961134765721E-04;
    weight( 44) =   0.28393114498380927832990899215541E-05;
    weight( 45) =   0.35734732949879669663960738150956E-06;
    weight( 46) =   0.37041625984781705796752840204084E-07;
    weight( 47) =   0.31371929535285447801497640621672E-08;
    weight( 48) =   0.21508698297808025739828859845140E-09;
    weight( 49) =   0.11808921844532942490513037158404E-10;
    weight( 50) =   0.51265062385038307838565047455223E-12;
    weight( 51) =   0.17335817955735154599902643794700E-13;
    weight( 52) =   0.44857058689176221240330804981619E-15;
    weight( 53) =   0.86925536958188009075932426691516E-17;
    weight( 54) =   0.12288435628797061539461585325494E-18;
    weight( 55) =   0.12266629909105281472971700203949E-20;
    weight( 56) =   0.82963863115951789374753323156164E-23;
    weight( 57) =   0.36027426635173044862245783257252E-25;
    weight( 58) =   0.93437837175367456929765381518998E-28;
    weight( 59) =   0.13071305930779945903630127634063E-30;
    weight( 60) =   0.84687478191640015120141181138947E-34;
    weight( 61) =   0.19796804708258311251124226474396E-37;
    weight( 62) =   0.10400778615192299534481914814892E-41;
    weight( 63) =   0.37099206434787551197827130470031E-47;

  elseif ( order == 127 )

    weight(  1) =   0.12504497577050595552677230002883E-100;
    weight(  2) =   0.17272798059419131415318615789672E-93;
    weight(  3) =   0.89321681571986548608031150791499E-88;
    weight(  4) =   0.77306185240893578449625186483810E-83;
    weight(  5) =   0.20143957652648255497735460506196E-78;
    weight(  6) =   0.21503714733610239701351039429345E-74;
    weight(  7) =   0.11341924208594594813715533569504E-70;
    weight(  8) =   0.33489139011795051950683388483136E-67;
    weight(  9) =   0.60486548964016681064424451668405E-64;
    weight( 10) =   0.71375092946352177824971347343892E-61;
    weight( 11) =   0.57884563374885556636801095624030E-58;
    weight( 12) =   0.33581166223858230300409326551248E-55;
    weight( 13) =   0.14394641949253923568603163698953E-52;
    weight( 14) =   0.46821808383216117724080263903889E-50;
    weight( 15) =   0.11817054440684264071348471955361E-47;
    weight( 16) =   0.23581659156008927203181682045005E-45;
    weight( 17) =   0.37814427940797540210712758405540E-43;
    weight( 18) =   0.49411031115771638145610738414006E-41;
    weight( 19) =   0.53255303775425059266087298458297E-39;
    weight( 20) =   0.47854390680131484999315199332765E-37;
    weight( 21) =   0.36191883445952356128627543209554E-35;
    weight( 22) =   0.23232083386343554805352497446119E-33;
    weight( 23) =   0.12753331411008716683688974281454E-31;
    weight( 24) =   0.60277753850758742112436095241270E-30;
    weight( 25) =   0.24679773241777200207460855084439E-28;
    weight( 26) =   0.88019567691698482573264198727415E-27;
    weight( 27) =   0.27482489212040561315005725890593E-25;
    weight( 28) =   0.75468218903085486125222816438456E-24;
    weight( 29) =   0.18303134636280466270545996891835E-22;
    weight( 30) =   0.39355990860860813085582448449811E-21;
    weight( 31) =   0.75293161638581191068419292570042E-20;
    weight( 32) =   0.12857997786722855037584105682618E-18;
    weight( 33) =   0.19659326888445857792541925311450E-17;
    weight( 34) =   0.26986511907214101894995783364250E-16;
    weight( 35) =   0.33344414303198856330118301113874E-15;
    weight( 36) =   0.37173303125150639885726463109574E-14;
    weight( 37) =   0.37473954472839737091885387788983E-13;
    weight( 38) =   0.34230094493397259538669512076007E-12;
    weight( 39) =   0.28385303724993373166810860630552E-11;
    weight( 40) =   0.21406920290454669208938772802828E-10;
    weight( 41) =   0.14706331273431716244229273183839E-09;
    weight( 42) =   0.92173940967434659264335883218167E-09;
    weight( 43) =   0.52781663936972714041837056042506E-08;
    weight( 44) =   0.27650497044951117835905283127679E-07;
    weight( 45) =   0.13267855842539464770913063113371E-06;
    weight( 46) =   0.58380944276113062188573331195042E-06;
    weight( 47) =   0.23581561724775629112332165335800E-05;
    weight( 48) =   0.87524468034280444703919485644809E-05;
    weight( 49) =   0.29876790535909012274846532159647E-04;
    weight( 50) =   0.93874435720072545206729594267039E-04;
    weight( 51) =   0.27170762627931172053444716883938E-03;
    weight( 52) =   0.72493929742498358979684249380921E-03;
    weight( 53) =   0.17841208326763432884316727108264E-02;
    weight( 54) =   0.40524855186046131499765636276283E-02;
    weight( 55) =   0.85000263041544110385806705526917E-02;
    weight( 56) =   0.16471142241609687824005585301760E-01;
    weight( 57) =   0.29499296248213632269675010319119E-01;
    weight( 58) =   0.48847387114300011006959603975676E-01;
    weight( 59) =   0.74807989768583731416517226905270E-01;
    weight( 60) =   0.10598520508090929403834368934301E+00;
    weight( 61) =   0.13893945309051540832066283010510E+00;
    weight( 62) =   0.16856236074207929740526975049765E+00;
    weight( 63) =   0.18927849580120432177170145550076E+00;
    weight( 64) =   0.19673340688823289786163676995151E+00;
    weight( 65) =   0.18927849580120432177170145550076E+00;
    weight( 66) =   0.16856236074207929740526975049765E+00;
    weight( 67) =   0.13893945309051540832066283010510E+00;
    weight( 68) =   0.10598520508090929403834368934301E+00;
    weight( 69) =   0.74807989768583731416517226905270E-01;
    weight( 70) =   0.48847387114300011006959603975676E-01;
    weight( 71) =   0.29499296248213632269675010319119E-01;
    weight( 72) =   0.16471142241609687824005585301760E-01;
    weight( 73) =   0.85000263041544110385806705526917E-02;
    weight( 74) =   0.40524855186046131499765636276283E-02;
    weight( 75) =   0.17841208326763432884316727108264E-02;
    weight( 76) =   0.72493929742498358979684249380921E-03;
    weight( 77) =   0.27170762627931172053444716883938E-03;
    weight( 78) =   0.93874435720072545206729594267039E-04;
    weight( 79) =   0.29876790535909012274846532159647E-04;
    weight( 80) =   0.87524468034280444703919485644809E-05;
    weight( 81) =   0.23581561724775629112332165335800E-05;
    weight( 82) =   0.58380944276113062188573331195042E-06;
    weight( 83) =   0.13267855842539464770913063113371E-06;
    weight( 84) =   0.27650497044951117835905283127679E-07;
    weight( 85) =   0.52781663936972714041837056042506E-08;
    weight( 86) =   0.92173940967434659264335883218167E-09;
    weight( 87) =   0.14706331273431716244229273183839E-09;
    weight( 88) =   0.21406920290454669208938772802828E-10;
    weight( 89) =   0.28385303724993373166810860630552E-11;
    weight( 90) =   0.34230094493397259538669512076007E-12;
    weight( 91) =   0.37473954472839737091885387788983E-13;
    weight( 92) =   0.37173303125150639885726463109574E-14;
    weight( 93) =   0.33344414303198856330118301113874E-15;
    weight( 94) =   0.26986511907214101894995783364250E-16;
    weight( 95) =   0.19659326888445857792541925311450E-17;
    weight( 96) =   0.12857997786722855037584105682618E-18;
    weight( 97) =   0.75293161638581191068419292570042E-20;
    weight( 98) =   0.39355990860860813085582448449811E-21;
    weight( 99) =   0.18303134636280466270545996891835E-22;
    weight(100) =   0.75468218903085486125222816438456E-24;
    weight(101) =   0.27482489212040561315005725890593E-25;
    weight(102) =   0.88019567691698482573264198727415E-27;
    weight(103) =   0.24679773241777200207460855084439E-28;
    weight(104) =   0.60277753850758742112436095241270E-30;
    weight(105) =   0.12753331411008716683688974281454E-31;
    weight(106) =   0.23232083386343554805352497446119E-33;
    weight(107) =   0.36191883445952356128627543209554E-35;
    weight(108) =   0.47854390680131484999315199332765E-37;
    weight(109) =   0.53255303775425059266087298458297E-39;
    weight(110) =   0.49411031115771638145610738414006E-41;
    weight(111) =   0.37814427940797540210712758405540E-43;
    weight(112) =   0.23581659156008927203181682045005E-45;
    weight(113) =   0.11817054440684264071348471955361E-47;
    weight(114) =   0.46821808383216117724080263903889E-50;
    weight(115) =   0.14394641949253923568603163698953E-52;
    weight(116) =   0.33581166223858230300409326551248E-55;
    weight(117) =   0.57884563374885556636801095624030E-58;
    weight(118) =   0.71375092946352177824971347343892E-61;
    weight(119) =   0.60486548964016681064424451668405E-64;
    weight(120) =   0.33489139011795051950683388483136E-67;
    weight(121) =   0.11341924208594594813715533569504E-70;
    weight(122) =   0.21503714733610239701351039429345E-74;
    weight(123) =   0.20143957652648255497735460506196E-78;
    weight(124) =   0.77306185240893578449625186483810E-83;
    weight(125) =   0.89321681571986548608031150791499E-88;
    weight(126) =   0.17272798059419131415318615789672E-93;
    weight(127) =   0.12504497577050595552677230002883E-100;

  else

    fprintf ( 1, '\n' );
    fprintf ( 1, 'HERMITE_WEIGHTS - Fatal error!\n' );
    fprintf ( 1, '  Illegal value of ORDER = %d\n', order );
    fprintf ( 1, '  Legal values are 1, 3, 7, 15, 31, 63 or 127.\n' );
    error ( 'HERMITE_WEIGHTS - Fatal error!' );

  end

  return
end
function value = i4_choose ( n, k )

%*****************************************************************************80
%
%% I4_CHOOSE computes the binomial coefficient C(N,K).
%
%  Discussion:
%
%    The value is calculated in such a way as to avoid overflow and
%    roundoff.  The calculation is done in integer arithmetic.
%
%    The formula used is:
%
%      C(N,K) = N! / ( K! * (N-K)! )
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    02 June 2007
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    ML Wolfson, HV Wright,
%    Algorithm 160:
%    Combinatorial of M Things Taken N at a Time,
%    Communications of the ACM,
%    Volume 6, Number 4, April 1963, page 161.
%
%  Parameters:
%
%    Input, integer N, K, are the values of N and K.
%
%    Output, integer VALUE, the number of combinations of N
%    things taken K at a time.
%
  mn = min ( k, n - k );

  if ( mn < 0 )

    value = 0;

  elseif ( mn == 0 )

    value = 1;

  else

    mx = max ( k, n - k );
    value = mx + 1;

    for i = 2 : mn
      value = ( value * ( mx + i ) ) / i;
    end

  end

  return
end
function value = i4_log_2 ( i )

%*****************************************************************************80
%
%% I4_LOG_2 returns the integer part of the logarithm base 2 of |I|.
%
%  Discussion:
%
%    For positive I4_LOG_2(I), it should be true that
%      2**I4_LOG_2(X) <= |I| < 2**(I4_LOG_2(I)+1).
%    The special case of I4_LOG_2(0) returns -HUGE().
%
%  Example:
%
%     I  Value
%
%     0  -1
%     1,  0
%     2,  1
%     3,  1
%     4,  2
%     5,  2
%     6,  2
%     7,  2
%     8,  3
%     9,  3
%    10,  3
%   127,  6
%   128,  7
%   129,  7
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    09 April 2004
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer I, the number whose logarithm base 2 is desired.
%
%    Output, integer VALUE, the integer part of the logarithm base 2 of
%    the absolute value of I.
%
  i = floor ( i );

  if ( i == 0 )

    value = -inf;

  else

    value = 0;

    i = abs ( i );

    while ( 2 <= i )
      i = i / 2;
      value = value + 1;
    end

  end

  return
end
function grid_level = index_level_herm ( level, level_max, dim_num, ...
  point_num, grid_index, grid_base )

%*****************************************************************************80
%
%% INDEX_LEVEL_HERM: determine first level at which given index is generated.
%
%  Discussion:
%
%    We are constructing a sparse grid of Gauss-Hermite points.  The grid
%    is built up of product grids, with a characteristic LEVEL.
%
%    We are concerned with identifying points in this product grid which
%    have actually been generated previously, on a lower value of LEVEL.
%
%    This routine determines the lowest value of LEVEL at which each of
%    the input points would be generated.
%
%    In 1D, given LEVEL, the number of points is ORDER = 2**(LEVEL+1) + 1,
%    (except that LEVEL = 0 implies ORDER = 1), the BASE is (ORDER-1)/2,
%    and the point INDEX values range from -BASE to +BASE.
%
%    The values of INDEX and BASE allow us to determine the abstract
%    properties of the point.  In particular, if INDEX is 0, the corresponding
%    Gauss-Hermite abscissa is 0, the special "nested" value we need
%    to take care of.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    08 October 2007
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    Fabio Nobile, Raul Tempone, Clayton Webster,
%    A Sparse Grid Stochastic Collocation Method for Partial Differential
%    Equations with Random Input Data,
%    SIAM Journal on Numerical Analysis,
%    Volume 46, Number 5, 2008, pages 2309-2345.
%
%  Parameters:
%
%    Input, integer LEVEL, the level at which these points were
%    generated.  LEVEL_MIN <= LEVEL <= LEVEL_MAX.
%
%    Input, integer LEVEL_MAX, the maximum level.
%
%    Input, integer DIM_NUM, the spatial dimension.
%
%    Input, integer POINT_NUM, the number of points to be tested.
%
%    Input, integer GRID_INDEX(DIM_NUM,POINT_NUM), the indices of the
%    points to be tested.
%
%    Input, integer GRID_BASE(DIM_NUM), the "base", which is essentially
%    the denominator of the index.
%
%    Output, integer GRID_LEVEL(POINT_NUM,1), the value of LEVEL at
%    which the point would first be generated.  This will be the same as
%    the input value of LEVEL, unless the point has an INDEX of 0 and
%    a corresponding BASE that is NOT zero.
%
  level_min = max ( 0, level_max + 1 - dim_num );
%
%  If a point has a DIM-th component whose INDEX is 0, then the
%  value of LEVEL at which this point would first be generated is
%  less than LEVEL, unless the DIM-th component of GRID_BASE is 0.
%
  grid_level = zeros ( point_num, 1 );

  for point = 1 : point_num

    grid_level(point) = max ( level, level_min );

    for dim = 1 : dim_num
      if ( grid_index(dim,point) == 0 )
        grid_level(point) = max ( grid_level(point) - grid_base(dim), level_min );
      end
    end

  end

  return
end
function order = level_to_order_open ( dim_num, level )

%*****************************************************************************80
%
%% LEVEL_TO_ORDER converts a level to an order for open rules.
%
%  Discussion:
%
%    Sparse grids can naturally be nested.  A natural scheme is to use
%    a series of one-dimensional rules arranged in a series of "levels"
%    whose order roughly doubles with each step.
%
%    The arrangement described here works naturally for the Fejer Type 1,
%    Fejer Type 2, Newton Cotes Open, Newton Cotes Half Open,
%    and Gauss-Patterson rules.  It also can be used, partially, to describe
%    the growth of Gauss-Legendre rules.
%
%    The idea is that we start with LEVEL = 0, ORDER = 1 indicating the single
%    point at the center, and for all values afterwards, we use the relationship
%
%      ORDER = 2**(LEVEL+1) - 1.
%
%    The following table shows how the growth will occur:
%
%    Level    Order
%
%    0          1
%    1          3 =  4 - 1
%    2          7 =  8 - 1
%    3         15 = 16 - 1
%    4         31 = 32 - 1
%    5         63 = 64 - 1
%
%    For the Fejer Type 1, Fejer Type 2, Newton Cotes Open,
%    Newton Cotes Open Half, and Gauss-Patterson rules, the point growth is
%    nested.  If we have ORDER points on a particular LEVEL, the next level
%    includes all these old points, plus ORDER+1 new points, formed in the
%    gaps between successive pairs of old points plus an extra point at each
%    end.
%
%    Level    Order = New + Old
%
%    0          1   =  1  +  0
%    1          3   =  2  +  1
%    2          7   =  4  +  3
%    3         15   =  8  +  7
%    4         31   = 16  + 15
%    5         63   = 32  + 31
%
%    If we use a series of Gauss-Legendre rules, then there is almost no
%    nesting, except that the central point is shared.  If we insist on
%    producing a comparable series of such points, then the "nesting" behavior
%    is as follows:
%
%    Level    Order = New + Old
%
%    0          1   =  1  +  0
%    1          3   =  2  +  1
%    2          7   =  6  +  1
%    3         15   = 14  +  1
%    4         31   = 30  +  1
%    5         63   = 62  +  1
%
%    Moreover, if we consider ALL the points used in such a set of "nested"
%    Gauss-Legendre rules, then we must sum the "NEW" column, and we see that
%    we get roughly twice as many points as for the truly nested rules.
%
%    In this routine, we assume that a vector of levels is given,
%    and the corresponding orders are desired.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    18 April 2007
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    Fabio Nobile, Raul Tempone, Clayton Webster,
%    A Sparse Grid Stochastic Collocation Method for Partial Differential
%    Equations with Random Input Data,
%    SIAM Journal on Numerical Analysis,
%    Volume 46, Number 5, 2008, pages 2309-2345.
%
%  Parameters:
%
%    Input, integer DIM_NUM, the spatial dimension.
%
%    Input, integer LEVEL(DIM_NUM), the nesting level.
%
%    Output, integer ORDER(DIM_NUM,1), the order (number of points) of the rule.
%
  order = zeros ( dim_num, 1 );

  for dim = 1 : dim_num

    if ( level(dim) < 0 )
      order(dim) = -1;
    elseif ( level(dim) == 0 )
      order(dim) = 1;
    else
      order(dim) = 2^( level(dim) + 1 ) - 1;
    end

  end

  return
end
function indx = multigrid_index_z ( dim_num, order_1d, order_nd )

%*****************************************************************************80
%
%% MULTIGRID_INDEX_Z returns an indexed multidimensional grid.
%
%  Discussion:
%
%    For dimension DIM, the number of points is ORDER_1D(DIM).
%
%    We assume that ORDER_1D(DIM) is an odd number,
%      ORDER_1D(DIM) = N = 2 * M + 1
%    so that the points have coordinates
%      -M/M, -(M-1)/M, ..., -1/M, 0/M, 1/M, 2/M, 3/M, ..., (M-1)/M, M/M.
%    and we index them as
%      -M,   -(M-1),        -1,   0,   1,   2,   3,   ...,  M-1,    M.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    05 February 2011
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    Fabio Nobile, Raul Tempone, Clayton Webster,
%    A Sparse Grid Stochastic Collocation Method for Partial Differential
%    Equations with Random Input Data,
%    SIAM Journal on Numerical Analysis,
%    Volume 46, Number 5, 2008, pages 2309-2345.
%
%  Parameters:
%
%    Input, integer DIM_NUM, the spatial dimension of the points.
%
%    Input, integer ORDER_1D(DIM_NUM), the order of the
%    rule in each dimension.
%
%    Input, integer ORDER_ND, the product of the entries of ORDER_1D.
%
%    Output, integer INDX(DIM_NUM,ORDER_ND), the indices of the points in
%    the grid.  The second dimension of this array is equal to the
%    product of the entries of ORDER_1D.
%
  indx = zeros ( dim_num, order_nd );

  a = zeros ( dim_num, 1 );

  more = 0;
  p = 0;

  while ( 1 )

    [ a, more ] = vec_colex_next2 ( dim_num, order_1d, a, more );

    if ( ~more )
      break
    end

    p = p + 1;
%
%  The values of A(DIM) are between 0 and ORDER_1D(DIM)-1 = N - 1 = 2 * M.
%  Subtracting M sets the range to -M to +M, as we wish.
%
    indx(1:dim_num,p) = a(1:dim_num) - round ( ( order_1d(1:dim_num) - 1 ) / 2 );

  end

  return
end
function w_nd = product_weight_herm ( dim_num, order_1d, order_nd )

%*****************************************************************************80
%
%% PRODUCT_WEIGHT_HERM: weights for a product Gauss-Hermite rule.
%
%  Discussion:
%
%    This routine computes the weights for a quadrature rule which is
%    a product of 1D Gauss-Hermite rules of varying order.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    08 October 2007
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer DIM_NUM, the spatial dimension.
%
%    Input, integer ORDER_1D(DIM_NUM), the order of the 1D rules.
%
%    Input, integer ORDER_ND, the order of the product rule.
%
%    Output, real W_ND(1,ORDER_ND), the product rule weights.
%
  w_nd = ones ( 1, order_nd );

  for dim = 1 : dim_num

    w_1d = hermite_weights ( order_1d(dim) );

    w_nd = r8vec_direct_product2 ( dim, order_1d(dim), w_1d, dim_num, ...
      order_nd, w_nd );

  end

  return
end
function value = r8_huge ( )

%*****************************************************************************80
%
%% R8_HUGE returns a "huge" real number.
%
%  Discussion:
%
%    The value returned by this function is NOT required to be the
%    maximum representable R8.  This value varies from machine to machine,
%    from compiler to compiler, and may cause problems when being printed.
%    We simply want a "very large" but non-infinite number.
%
%    MATLAB provides a built-in symbolic constant "inf" that can be used
%    if a huge number is really what you want!
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    27 January 2008
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Output, real VALUE, a huge number.
%
  value = 1.0E+30;

  return
end
function r8mat_transpose_print_some ( m, n, a, ilo, jlo, ihi, jhi, title )

%*****************************************************************************80
%
%% R8MAT_TRANSPOSE_PRINT_SOME prints some of an R8MAT, transposed.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    23 May 2005
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer M, N, the number of rows and columns.
%
%    Input, real A(M,N), an M by N matrix to be printed.
%
%    Input, integer ILO, JLO, the first row and column to print.
%
%    Input, integer IHI, JHI, the last row and column to print.
%
%    Input, string TITLE, a title.
%
  incx = 5;

  fprintf ( 1, '\n' );
  fprintf ( 1, '%s\n', title );

  for i2lo = max ( ilo, 1 ) : incx : min ( ihi, m )

    i2hi = i2lo + incx - 1;
    i2hi = min ( i2hi, m );
    i2hi = min ( i2hi, ihi );

    inc = i2hi + 1 - i2lo;

    fprintf ( 1, '\n' );
    fprintf ( 1, '  Row: ' );
    for i = i2lo : i2hi
      fprintf ( 1, '%7d       ', i );
    end
    fprintf ( 1, '\n' );
    fprintf ( 1, '  Col\n' );

    j2lo = max ( jlo, 1 );
    j2hi = min ( jhi, n );

    for j = j2lo : j2hi

      fprintf ( 1, '%5d ', j );
      for i2 = 1 : inc
        i = i2lo - 1 + i2;
        fprintf ( 1, '%12f', a(i,j) );
      end
      fprintf ( 1, '\n' );

    end

  end

  return
end
function r8mat_write ( output_filename, m, n, table )

%*****************************************************************************80
%
%% R8MAT_WRITE writes an R8MAT file.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    09 August 2009
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, string OUTPUT_FILENAME, the output filename.
%
%    Input, integer M, the spatial dimension.
%
%    Input, integer N, the number of points.
%
%    Input, real TABLE(M,N), the points.
%

%
%  Open the file.
%
  output_unit = fopen ( output_filename, 'wt' );

  if ( output_unit < 0 )
    fprintf ( 1, '\n' );
    fprintf ( 1, 'R8MAT_WRITE - Error!\n' );
    fprintf ( 1, '  Could not open the output file.\n' );
    error ( 'R8MAT_WRITE - Error!' );
  end
%
%  Write the data.
%
  for j = 1 : n
    for i = 1 : m
      fprintf ( output_unit, '  %24.16f', table(i,j) );
    end
    fprintf ( output_unit, '\n' );
  end
%
%  Close the file.
%
  fclose ( output_unit );

  return
end
function w = r8vec_direct_product2 ( factor_index, factor_order, ...
  factor_value, factor_num, point_num, w )

%*****************************************************************************80
%
%% R8VEC_DIRECT_PRODUCT2 creates a direct product of R8VEC's.
%
%  Discussion:
%
%    To explain what is going on here, suppose we had to construct
%    a multidimensional quadrature rule as the product of K rules
%    for 1D quadrature.
%
%    The product rule will be represented as a list of points and weights.
%
%    The J-th item in the product rule will be associated with
%      item J1 of 1D rule 1,
%      item J2 of 1D rule 2,
%      ...,
%      item JK of 1D rule K.
%
%    In particular,
%      X(J) = ( X(1,J1), X(2,J2), ..., X(K,JK))
%    and
%      W(J) = W(1,J1) * W(2,J2) * ... * W(K,JK)
%
%    So we can construct the quadrature rule if we can properly
%    distribute the information in the 1D quadrature rules.
%
%    This routine carries out that task for the weights W.
%
%    Another way to do this would be to compute, one by one, the
%    set of all possible indices (J1,J2,...,JK), and then index
%    the appropriate information.  An advantage of the method shown
%    here is that you can process the K-th set of information and
%    then discard it.
%
%  Example:
%
%    Rule 1:
%      Order = 4
%      W(1:4) = ( 2, 3, 5, 7 )
%
%    Rule 2:
%      Order = 3
%      W(1:3) = ( 11, 13, 17 )
%
%    Rule 3:
%      Order = 2
%      W(1:2) = ( 19, 23 )
%
%    Product Rule:
%      Order = 24
%      W(1:24) =
%        ( 2 * 11 * 19 )
%        ( 3 * 11 * 19 )
%        ( 4 * 11 * 19 )
%        ( 7 * 11 * 19 )
%        ( 2 * 13 * 19 )
%        ( 3 * 13 * 19 )
%        ( 5 * 13 * 19 )
%        ( 7 * 13 * 19 )
%        ( 2 * 17 * 19 )
%        ( 3 * 17 * 19 )
%        ( 5 * 17 * 19 )
%        ( 7 * 17 * 19 )
%        ( 2 * 11 * 23 )
%        ( 3 * 11 * 23 )
%        ( 5 * 11 * 23 )
%        ( 7 * 11 * 23 )
%        ( 2 * 13 * 23 )
%        ( 3 * 13 * 23 )
%        ( 5 * 13 * 23 )
%        ( 7 * 13 * 23 )
%        ( 2 * 17 * 23 )
%        ( 3 * 17 * 23 )
%        ( 5 * 17 * 23 )
%        ( 7 * 17 * 23 )
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    18 April 2009
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer FACTOR_INDEX, the index of the factor being processed.
%    The first factor processed must be factor 1.
%
%    Input, integer FACTOR_ORDER, the order of the factor.
%
%    Input, real FACTOR_VALUE(FACTOR_ORDER), the factor values for
%    factor FACTOR_INDEX.
%
%    Input, integer FACTOR_NUM, the number of factors.
%
%    Input, integer POINT_NUM, the number of elements in the direct product.
%
%    Input/output, real W(POINT_NUM), the elements of the
%    direct product, updated by the latest factor.
%
%  Local Parameters:
%
%    Local, integer START, the first location of a block of values to set.
%
%    Local, integer CONTIG, the number of consecutive values to set.
%
%    Local, integer SKIP, the distance from the current value of START
%    to the next location of a block of values to set.
%
%    Local, integer REP, the number of blocks of values to set.
%
  persistent contig;
  persistent rep;
  persistent skip;

  if ( factor_index == 1 )
    contig = 1;
    skip = 1;
    rep = point_num;
    w(1:point_num) = 1.0;
  end

  rep = rep / factor_order;
  skip = skip * factor_order;

  for j = 1 : factor_order

    start = 1 + ( j - 1 ) * contig;

    for k = 1 : rep
      w(start:start+contig-1) = w(start:start+contig-1) * factor_value(j);
      start = start + skip;
    end

  end

  contig = contig * factor_order;

  return
end
function r8vec_print_some ( n, a, i_lo, i_hi, title )

%*****************************************************************************80
%
%% R8VEC_PRINT_SOME prints "some" of an R8VEC.
%
%  Discussion:
%
%    An R8VEC is a vector of R8 values.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    16 October 2006
%
%  Author:
%
%    John Burkardt
%
%  Parameters:
%
%    Input, integer N, the dimension of the vector.
%
%    Input, real A(N), the vector to be printed.
%
%    Input, integer MAX_PRINT, the maximum number of lines to print.
%
%    Input, string TITLE, a title.
%
  fprintf ( 1, '\n' );
  fprintf ( 1, '%s\n', title );
  fprintf ( 1, '\n' );

  for i = max ( 1, i_lo ) : min ( n, i_hi )
    fprintf ( 1, '  %8d  %12f\n', i, a(i) );
  end

  return
end
function [ grid_weight, grid_point ] = sparse_grid_herm ( dim_num, ...
  level_max, point_num )

%*****************************************************************************80
%
%% SPARSE_GRID_HERM computes a sparse grid of Gauss-Hermite points.
%
%  Discussion:
%
%    The quadrature rule is associated with a sparse grid derived from
%    a Smolyak construction using a 1D Gauss-Hermite quadrature rule.
%
%    The user specifies:
%    * the spatial dimension of the quadrature region,
%    * the level that defines the Smolyak grid.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    05 July 2008
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    Fabio Nobile, Raul Tempone, Clayton Webster,
%    A Sparse Grid Stochastic Collocation Method for Partial Differential
%    Equations with Random Input Data,
%    SIAM Journal on Numerical Analysis,
%    Volume 46, Number 5, 2008, pages 2309-2345.
%
%  Parameters:
%
%    Input, integer DIM_NUM, the spatial dimension.
%
%    Input, integer LEVEL_MAX, controls the size of the
%    sparse grid.
%
%    Input, integer POINT_NUM, the number of points in the grid,
%    as determined by SPARSE_GRID_HERM_SIZE.
%
%    Output, real GRID_WEIGHT(1,POINT_NUM), the weights.
%
%    Output, real GRID_POINT(DIM_NUM,POINT_NUM), the points.
%
  grid_point = zeros ( dim_num, point_num );
  grid_weight = zeros ( 1, point_num );
%
%  The outer loop generates LEVELs from LEVEL_MIN to LEVEL_MAX.
%
  point_num2 = 0;

  level_min = max ( 0, level_max + 1 - dim_num );

  for level = level_min : level_max
%
%  The middle loop generates the next partition LEVEL_1D(1:DIM_NUM)
%  that adds up to LEVEL.
%
    level_1d = [];
    more = 0;
    h = 0;
    t = 0;

    while ( 1 )

      [ level_1d, more, h, t ] = comp_next ( level, dim_num, level_1d, more, h, t );
%
%  Transform each 1D level to a corresponding 1D order.
%  The relationship is the same as for other OPEN rules.
%  The Gauss-Hermite rule differs from the other OPEN rules
%  only in the nesting behavior.
%
      order_1d = level_to_order_open ( dim_num, level_1d );

      grid_base2(1:dim_num) = round ( ( order_1d(1:dim_num) - 1 ) / 2 );
%
%  The product of the 1D orders gives us the number of points in this subgrid.
%
      order_nd = prod ( order_1d(1:dim_num) );
%
%  Compute the weights for this product grid.
%
      grid_weight2 = product_weight_herm ( dim_num, order_1d, order_nd );
%
%  Now determine the coefficient of the weight.
%
      coeff = (-1)^( level_max - level ) ...
        * i4_choose ( dim_num - 1, level_max - level );
%
%  The inner (hidden) loop generates all points corresponding to given grid.
%  The grid indices will be between -M to +M, where 2*M + 1 = ORDER_1D(DIM).
%
      grid_index2 = multigrid_index_z ( dim_num, order_1d, order_nd );
%
%  Determine the first level of appearance of each of the points.
%  This allows us to flag certain points as being repeats of points
%  generated on a grid of lower level.
%
%  This is SLIGHTLY tricky.
%
      grid_level = index_level_herm ( level, level_max, dim_num, order_nd, ...
        grid_index2, grid_base2 );
%
%  Only keep those points which first appear on this level.
%
      for point = 1 : order_nd
%
%  Either a "new" point (increase count, create point, create weight)
%
        if ( grid_level(point) == level )

          point_num2 = point_num2 + 1;

          grid_point(1:dim_num,point_num2) = hermite_abscissa ( dim_num, 1, ...
            grid_index2(1:dim_num,point), grid_base2(1:dim_num) );

          grid_weight(point_num2) = coeff * grid_weight2(point);
%
%  or an already existing point (create point temporarily, find match,
%  add weight to matched point's weight).
%
        else

          grid_point_temp(1:dim_num) = hermite_abscissa ( dim_num, 1, ...
            grid_index2(1:dim_num,point), grid_base2(1:dim_num) );

          point3 = -1;

          for point2 = 1 : point_num2
            if ( grid_point(1:dim_num,point2) == grid_point_temp(1:dim_num)' )
              point3 = point2;
              break
            end
          end

          if ( point3 == -1 )
            fprintf ( 1, '\n' );
            fprintf ( 1, 'SPARSE_GRID_HERMITE - Fatal error!\n' );
            fprintf ( 1, '  Could not match point.\n' );
            error ( 'SPARSE_GRID_HERMITE - Fatal error!' );
          end

          grid_weight(point3) = grid_weight(point3) ...
            + coeff * grid_weight2(point);

        end

      end

      if ( ~more )
        break
      end

    end

  end

  return
end
function point_num = sparse_grid_herm_size ( dim_num, level_max )

%*****************************************************************************80
%
%% SPARSE_GRID_HERM_SIZE sizes a sparse grid of Gauss-Hermite points.
%
%  Discussion:
%
%    The grid is defined as the sum of the product rules whose LEVEL
%    satisfies:
%
%      LEVEL_MIN <= LEVEL <= LEVEL_MAX.
%
%    where LEVEL_MAX is user specified, and
%
%      LEVEL_MIN = max ( 0, LEVEL_MAX + 1 - DIM_NUM ).
%
%    The grids are only very weakly nested, since Gauss-Hermite rules
%    only have the origin in common.
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    05 July 2008
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    Fabio Nobile, Raul Tempone, Clayton Webster,
%    A Sparse Grid Stochastic Collocation Method for Partial Differential
%    Equations with Random Input Data,
%    SIAM Journal on Numerical Analysis,
%    Volume 46, Number 5, 2008, pages 2309-2345.
%
%  Parameters:
%
%    Input, integer DIM_NUM, the spatial dimension.
%
%    Input, integer LEVEL_MAX, the maximum value of LEVEL.
%
%    Output, integer POINT_NUM, the number of points in the grid.
%

%
%  Special case.
%
  if ( level_max == 0 )
    point_num = 1;
    return;
  end
%
%  The outer loop generates LEVELs from LEVEL_MIN to LEVEL_MAX.
%
  point_num = 0;

  level_min = max ( 0, level_max + 1 - dim_num );

  for level = level_min : level_max
%
%  The middle loop generates the next partition that adds up to LEVEL.
%
    level_1d = [];
    more = false;
    h = 0;
    t = 0;

    while ( 1 )

      [ level_1d, more, h, t ] = comp_next ( level, dim_num, level_1d, more, h, t );
%
%  Transform each 1D level to a corresponding 1D order.
%
      order_1d = level_to_order_open ( dim_num, level_1d );

      for dim = 1 : dim_num
%
%  If we can reduce the level in this dimension by 1 and
%  still not go below LEVEL_MIN.
%
        if ( level_min < level && 1 < order_1d(dim) )
          order_1d(dim) = order_1d(dim) - 1;
        end

      end

      point_num = point_num + prod ( order_1d(1:dim_num) );

      if ( ~more )
        break
      end

    end
  end

  return
end
function timestamp ( )

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

  return
end
function [ a, more ] = vec_colex_next2 ( dim_num, base, a, more )

%*****************************************************************************80
%
%% VEC_COLEX_NEXT2 generates vectors in colex order.
%
%  Discussion:
%
%    The vectors are produced in colexical order, starting with
%    (0,0,...,0),
%    (1,0,...,0),
%    ...
%    (BASE(1)-1,BASE(2)-1,...,BASE(DIM_NUM)-1).
%
%  Example:
%
%    DIM_NUM = 2,
%    BASE = [ 3, 3]
%
%    0   0
%    1   0
%    2   0
%    0   1
%    1   1
%    2   1
%    0   2
%    1   2
%    2   2
%
%  Licensing:
%
%    This code is distributed under the GNU LGPL license.
%
%  Modified:
%
%    11 November 2007
%
%  Author:
%
%    John Burkardt
%
%  Reference:
%
%    Dennis Stanton, Dennis White,
%    Constructive Combinatorics,
%    Springer, 1986,
%    ISBN: 0387963472,
%    LC: QA164.S79.
%
%  Parameters:
%
%    Input, integer DIM_NUM, the spatial dimension.
%
%    Input, integer BASE(DIM_NUM), the base to be used in each dimension.
%
%    Input, integer A(DIM_NUM), except on the first call, this should
%    be the output value of A on the last call.
%
%    Input, logical MORE, should be FALSE on the first call, and
%    thereafter should be the output value of MORE from the previous call.
%
%    Output, integer A(DIM_NUM), the next vector.
%
%    Output, logical MORE, is TRUE if another vector was computed.
%    If MORE is FALSE on return, then ignore the output value A, and
%    stop calling the routine.
%
  if ( ~more )

    a(1:dim_num) = 0;
    more = 1;

  else

    for i = 1 : dim_num

      a(i) = a(i) + 1;

      if ( a(i) < base(i) )
        return
      end

      a(i) = 0;

    end

    more = 0;

  end

  return
end