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

using namespace std;

# include "sparse_grid_open.hpp"

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

int *abscissa_level_open_nd ( int level_max, int dim_num, int test_num, 
  int test_val[] )

//****************************************************************************80
//
//  Purpose:
//
//    ABSCISSA_LEVEL_OPEN_ND: first level at which given abscissa is generated.
//
//  Discussion:
//
//    We assume an underlying product grid.  In each dimension, this product
//    grid has order 2**(LEVEL_MAX+1) - 1.
//
//    We will say a sparse grid has total level LEVEL if each point in the
//    grid has a total level of LEVEL or less.
//
//    The "level" of a point is determined as the sum of the levels of the
//    point in each spatial dimension.
//
//    The level of a point in a single spatial dimension I is determined as
//    the level, between 0 and LEVEL_MAX, at which the point's I'th index
//    would have been generated.
//
//
//    This description is terse and perhaps unenlightening.  Keep in mind
//    that the product grid is the product of 1D grids,
//    that the 1D grids are built up by levels, having
//    orders (total number of points ) 1, 3, 7, 15, 31 and so on,
//    and that these 1D grids are nested, so that each point in a 1D grid
//    has a first level at which it appears.
//
//    Our procedure for generating the points of a sparse grid, then, is
//    to choose a value LEVEL_MAX, to generate the full product grid,
//    but then only to keep those points on the full product grid whose
//    LEVEL is less than or equal to LEVEL_MAX.
//
//
//    Note that this routine is really just testing out the idea of
//    determining the level.  Our true desire is to be able to start
//    with a value LEVEL, and determine, in a straightforward manner,
//    all the points that are generated exactly at that level, or
//    all the points that are generated up to and including that level.
//
//    This allows us to generate the new points to be added to one sparse
//    grid to get the next, or to generate a particular sparse grid at once.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    19 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, int LEVEL_MAX, controls the size of the final sparse grid.
//
//    Input, int DIM_NUM, the spatial dimension.
//
//    Input, int TEST_NUM, the number of points to be tested.
//
//    Input, int TEST_VAL[DIM_NUM*TEST_NUM], the indices of the points 
//    to be tested.  Normally, each index would be between 0 and 2**LEVEL_MAX.
//
//    Output, int ABSCISSA_OPEN_LEVEL_ND[TEST_NUM], the value of LEVEL at which the
//    point would first be generated, assuming that a standard sequence of
//    nested grids is used.
//
{
  int dim;
  int j;
  int level;
  int order;
  int t;
  int *test_level;

  test_level = new int[test_num];

  if ( level_max == 0 )
  {
    for ( j = 0; j < test_num; j++ )
    {
      test_level[j] = 0;
    }
    return test_level;
  }

  order = i4_power ( 2, level_max ) + 1;

  for ( j = 0; j < test_num; j++ )
  {
    test_level[j] = index_to_level_open ( dim_num, test_val+j*dim_num,
      order, level_max );
  }

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

void comp_next ( int n, int k, int a[], bool *more, int *h, int *t )

//****************************************************************************80
//
//  Purpose:
//
//    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.
//    C++ 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, int N, the integer whose compositions are desired.
//
//    Input, int K, the number of parts in the composition.
//
//    Input/output, int A[K], the parts of the composition.
//
//    Input/output, bool *MORE.
//    Set MORE = FALSE on first call.  It will be reset to TRUE on return
//    with a new composition.  Each new call returns another composition until
//    MORE is set to FALSE when the last composition has been computed
//    and returned.
//
//    Input/output, int *H, *T, two internal parameters needed for the
//    computation.  The user should allocate space for these in the calling
//    program, include them in the calling sequence, but never alter them!
//
{
  int i;

  if ( !( *more ) )
  {
    *t = n;
    *h = 0;
    a[0] = n;
    for ( i = 1; i < k; i++ )
    {
       a[i] = 0;
    }
  }
  else
  {
    if ( 1 < *t )
    {
      *h = 0;
    }
    *h = *h + 1;
    *t = a[*h-1];
    a[*h-1] = 0;
    a[0] = *t - 1;
    a[*h] = a[*h] + 1;
  }

  *more = ( a[k-1] != n );

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

double f2_abscissa ( int order, int i )

//****************************************************************************80
//
//  Purpose:
//
//    F2_ABSCISSA returns the I-th abscissa for the Fejer type 2 rule.
//
//  Discussion:
//
//    Our convention is that the abscissas are numbered from left to
//    right.
//
//    This rule is defined on [-1,1].
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    11 April 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int ORDER, the order of the Fejer type 2 rule.
//    1 <= ORDER.
//
//    Input, int I, the index of the desired abscissa.  1 <= I <= ORDER.
//
//    Output, double F2_ABSCISSA, the value of the I-th 
//    abscissa in the Fejer type 2 rule of order ORDER.
//
{
  double pi = 3.141592653589793;
  double value;

  if ( order < 1 )
  {
    value = - r8_huge ( );
    return value;
  }

  if ( i < 1 || order < i )
  {
    cout << "\n";
    cout << "F2_ABSCISSA - Fatal error!\n";
    cout << "  1 <= I <= ORDER is required.\n";
    exit ( 1 );
  }

  if ( order == 1 )
  {
    value = 0.0;
    return value;
  }

  value = cos ( ( double ) ( order + 1 - i ) * pi 
              / ( double ) ( order + 1 ) );

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

void gl_abscissa ( int dim_num, int point_num, int grid_index[], 
  int grid_base[], double grid_point[] )

//****************************************************************************80
//
//  Purpose:
//
//    GL_ABSCISSA sets abscissas for multidimensional Gauss-Legendre quadrature.
//
//  Discussion:
//
//    The "nesting" as it occurs for Gauss-Legendre sparse grids simply
//    involves the use of a specified set of permissible orders for the
//    rule.  
//
//    The X array lists the (complete) Gauss-Legendre abscissas for rules 
//    of order 1, 3, 7, 15, 31, 63 or 127, in order. 
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    02 October 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int DIM_NUM, the spatial dimension.
//
//    Input, int POINT_NUM, the number of points.
//
//    Input, int GRID_INDEX[DIM_NUM*POINT_NUM], the index of the abscissa
//    from the Gauss-Legendre rule, for each dimension and point.
//
//    Input, int GRID_BASE[DIM_NUM], the number of points used in the 
//    Gauss-Legendre rule for a given dimension.
//
//    Output, double GRID_POINT[DIM_NUM], the grid points of
//    Gauss-Legendre abscissas.
//
{
  int dim;
  int level;
  int point;
  int pointer;
  int skip[8] = { 0, 1, 4, 11, 26, 57, 120, 247 };
  double x[247] = {
       0.0E+00, 
     - 0.774596669241483377035853079956E+00, 
       0.0E+00, 
       0.774596669241483377035853079956E+00, 
     - 0.949107912342758524526189684048E+00, 
     - 0.741531185599394439863864773281E+00, 
     - 0.405845151377397166906606412077E+00, 
       0.0E+00, 
       0.405845151377397166906606412077E+00, 
       0.741531185599394439863864773281E+00, 
       0.949107912342758524526189684048E+00, 
      - 0.987992518020485428489565718587E+00, 
      - 0.937273392400705904307758947710E+00, 
      - 0.848206583410427216200648320774E+00, 
      - 0.724417731360170047416186054614E+00, 
      - 0.570972172608538847537226737254E+00, 
      - 0.394151347077563369897207370981E+00, 
      - 0.201194093997434522300628303395E+00, 
        0.0E+00, 
        0.201194093997434522300628303395E+00, 
       0.394151347077563369897207370981E+00, 
       0.570972172608538847537226737254E+00, 
       0.724417731360170047416186054614E+00, 
       0.848206583410427216200648320774E+00, 
       0.937273392400705904307758947710E+00, 
       0.987992518020485428489565718587E+00, 
      -0.99708748181947707454263838179654,    
      -0.98468590966515248400211329970113,    
      -0.96250392509294966178905249675943,    
      -0.93075699789664816495694576311725,    
      -0.88976002994827104337419200908023,    
      -0.83992032014626734008690453594388,    
      -0.78173314841662494040636002019484,    
      -0.71577678458685328390597086536649,    
      -0.64270672292426034618441820323250,    
      -0.56324916140714926272094492359516,    
      -0.47819378204490248044059403935649,    
      -0.38838590160823294306135146128752,    
      -0.29471806998170161661790389767170,    
      -0.19812119933557062877241299603283,    
      -0.99555312152341520325174790118941E-01,
       0.00000000000000000000000000000000,
       0.99555312152341520325174790118941E-01,
       0.19812119933557062877241299603283,    
       0.29471806998170161661790389767170,    
       0.38838590160823294306135146128752,    
       0.47819378204490248044059403935649,    
       0.56324916140714926272094492359516,
       0.64270672292426034618441820323250,    
       0.71577678458685328390597086536649,    
       0.78173314841662494040636002019484,    
       0.83992032014626734008690453594388,    
       0.88976002994827104337419200908023,    
       0.93075699789664816495694576311725,    
       0.96250392509294966178905249675943,    
       0.98468590966515248400211329970113,   
       0.99708748181947707454263838179654,    
      -0.99928298402912378050701628988630E+00,     
      -0.99622401277797010860209018267357E+00,     
      -0.99072854689218946681089469460884E+00,     
      -0.98280881059372723486251140727639E+00,     
      -0.97248403469757002280196067864927E+00,     
      -0.95977944975894192707035416626398E+00,     
      -0.94472613404100980296637531962798E+00,     
      -0.92736092062184320544703138132518E+00,     
      -0.90772630277853155803695313291596E+00,     
      -0.88587032850785342629029845731337E+00,     
      -0.86184648236412371953961183943106E+00,     
      -0.83571355431950284347180776961571E+00,     
      -0.80753549577345676005146598636324E+00,     
      -0.77738126299037233556333018991104E+00,     
      -0.74532464831784741782932166103759E+00,     
      -0.71144409958484580785143153770401E+00,     
      -0.67582252811498609013110331596954E+00,     
      -0.63854710582136538500030695387338E+00,     
      -0.59970905187762523573900892686880E+00,     
      -0.55940340948628501326769780007005E+00,     
      -0.51772881329003324812447758452632E+00,     
      -0.47478724799480439992221230985149E+00,     
      -0.43068379879511160066208893391863E+00,     
      -0.38552639421224789247761502227440E+00,     
      -0.33942554197458440246883443159432E+00,     
      -0.29249405858625144003615715555067E+00,     
      -0.24484679324595336274840459392483E+00,     
      -0.19660034679150668455762745706572E+00,     
      -0.14787278635787196856983909655297E+00,     
      -0.98783356446945279529703669453922E-01, 
      -0.49452187116159627234233818051808E-01, 
       0.00000000000000000000000000000000E+00,     
       0.49452187116159627234233818051808E-01, 
       0.98783356446945279529703669453922E-01, 
       0.14787278635787196856983909655297E+00,     
       0.19660034679150668455762745706572E+00,     
       0.24484679324595336274840459392483E+00,     
       0.29249405858625144003615715555067E+00,     
       0.33942554197458440246883443159432E+00,     
       0.38552639421224789247761502227440E+00,     
       0.43068379879511160066208893391863E+00,     
       0.47478724799480439992221230985149E+00,     
       0.51772881329003324812447758452632E+00,     
       0.55940340948628501326769780007005E+00,     
       0.59970905187762523573900892686880E+00,     
       0.63854710582136538500030695387338E+00,     
       0.67582252811498609013110331596954E+00,     
       0.71144409958484580785143153770401E+00,     
       0.74532464831784741782932166103759E+00,     
       0.77738126299037233556333018991104E+00,     
       0.80753549577345676005146598636324E+00,     
       0.83571355431950284347180776961571E+00,     
       0.86184648236412371953961183943106E+00,     
       0.88587032850785342629029845731337E+00,     
       0.90772630277853155803695313291596E+00,     
       0.92736092062184320544703138132518E+00,     
       0.94472613404100980296637531962798E+00,     
       0.95977944975894192707035416626398E+00,     
       0.97248403469757002280196067864927E+00,     
       0.98280881059372723486251140727639E+00,     
       0.99072854689218946681089469460884E+00,     
       0.99622401277797010860209018267357E+00,  
       0.99928298402912378050701628988630E+00,
       -0.99982213041530614629963254927125E+00,     
       -0.99906293435531189513828920479421E+00,     
       -0.99769756618980462107441703193392E+00,     
       -0.99572655135202722663543337085008E+00,     
       -0.99315104925451714736113079489080E+00,     
       -0.98997261459148415760778669967548E+00,     
       -0.98619317401693166671043833175407E+00,     
       -0.98181502080381411003346312451200E+00,     
       -0.97684081234307032681744391886221E+00,     
       -0.97127356816152919228894689830512E+00,     
       -0.96511666794529212109082507703391E+00,     
       -0.95837384942523877114910286998060E+00,     
       -0.95104920607788031054790764659636E+00,     
       -0.94314718462481482734544963026201E+00,     
       -0.93467258232473796857363487794906E+00,     
       -0.92563054405623384912746466814259E+00,     
       -0.91602655919146580931308861741716E+00,     
       -0.90586645826182138280246131760282E+00,     
       -0.89515640941708370896904382642451E+00,     
       -0.88390291468002656994525794802849E+00,     
       -0.87211280599856071141963753428864E+00,     
       -0.85979324109774080981203134414483E+00,     
       -0.84695169913409759845333931085437E+00,     
       -0.83359597615489951437955716480123E+00,     
       -0.81973418036507867415511910167470E+00,     
       -0.80537472720468021466656079404644E+00,     
       -0.79052633423981379994544995252740E+00,     
       -0.77519801587020238244496276354566E+00,     
       -0.75939907785653667155666366659810E+00,     
       -0.74313911167095451292056688997595E+00,    
       -0.72642798867407268553569290153270E+00,     
       -0.70927585412210456099944463906757E+00,     
       -0.69169312100770067015644143286666E+00,     
       -0.67369046373825048534668253831602E+00,     
       -0.65527881165548263027676505156852E+00,     
       -0.63646934240029724134760815684175E+00,     
       -0.61727347512685828385763916340822E+00,     
       -0.59770286357006522938441201887478E+00,     
       -0.57776938897061258000325165713764E+00,     
       -0.55748515286193223292186190687872E+00,     
       -0.53686246972339756745816636353452E+00,     
       -0.51591385950424935727727729906662E+00,     
       -0.49465204002278211739494017368636E+00,     
       -0.47308991924540524164509989939699E+00,     
       -0.45124058745026622733189858020729E+00,     
       -0.42911730928019337626254405355418E+00,     
       -0.40673351568978256340867288124339E+00,     
       -0.38410279579151693577907781452239E+00,     
       -0.36123888860586970607092484346723E+00,     
       -0.33815567472039850137600027657095E+00,     
       -0.31486716786289498148601475374890E+00,     
       -0.29138750639370562079451875284568E+00,     
       -0.26773094472238862088834352027938E+00,     
       -0.24391184465391785797071324453138E+00,     
       -0.21994466666968754245452337866940E+00,     
       -0.19584396114861085150428162519610E+00,     
       -0.17162435953364216500834492248954E+00,     
       -0.14730056544908566938932929319807E+00,     
       -0.12288734577408297172603365288567E+00,     
       -0.98399521677698970751091751509101E-01, 
       -0.73851959621048545273440409360569E-01, 
       -0.49259562331926630315379321821927E-01, 
       -0.24637259757420944614897071846088E-01, 
        0.00000000000000000000000000000000E+00,     
        0.24637259757420944614897071846088E-01, 
        0.49259562331926630315379321821927E-01, 
        0.73851959621048545273440409360569E-01, 
        0.98399521677698970751091751509101E-01, 
        0.12288734577408297172603365288567E+00,     
        0.14730056544908566938932929319807E+00,     
        0.17162435953364216500834492248954E+00,     
        0.19584396114861085150428162519610E+00,     
        0.21994466666968754245452337866940E+00,     
        0.24391184465391785797071324453138E+00,     
        0.26773094472238862088834352027938E+00,     
        0.29138750639370562079451875284568E+00,     
        0.31486716786289498148601475374890E+00,     
        0.33815567472039850137600027657095E+00,     
        0.36123888860586970607092484346723E+00,     
        0.38410279579151693577907781452239E+00,     
        0.40673351568978256340867288124339E+00,     
        0.42911730928019337626254405355418E+00,     
        0.45124058745026622733189858020729E+00,     
        0.47308991924540524164509989939699E+00,     
        0.49465204002278211739494017368636E+00,     
        0.51591385950424935727727729906662E+00,     
        0.53686246972339756745816636353452E+00,     
        0.55748515286193223292186190687872E+00,     
        0.57776938897061258000325165713764E+00,     
        0.59770286357006522938441201887478E+00,     
        0.61727347512685828385763916340822E+00,     
        0.63646934240029724134760815684175E+00,     
        0.65527881165548263027676505156852E+00,     
        0.67369046373825048534668253831602E+00,     
        0.69169312100770067015644143286666E+00,    
        0.70927585412210456099944463906757E+00,     
        0.72642798867407268553569290153270E+00,     
        0.74313911167095451292056688997595E+00,     
        0.75939907785653667155666366659810E+00,     
        0.77519801587020238244496276354566E+00,     
        0.79052633423981379994544995252740E+00,     
        0.80537472720468021466656079404644E+00,     
        0.81973418036507867415511910167470E+00,     
        0.83359597615489951437955716480123E+00,     
        0.84695169913409759845333931085437E+00,     
        0.85979324109774080981203134414483E+00,     
        0.87211280599856071141963753428864E+00,     
        0.88390291468002656994525794802849E+00,     
        0.89515640941708370896904382642451E+00,     
        0.90586645826182138280246131760282E+00,     
        0.91602655919146580931308861741716E+00,     
        0.92563054405623384912746466814259E+00,     
        0.93467258232473796857363487794906E+00,     
        0.94314718462481482734544963026201E+00,     
        0.95104920607788031054790764659636E+00,     
        0.95837384942523877114910286998060E+00,     
        0.96511666794529212109082507703391E+00,     
        0.97127356816152919228894689830512E+00,     
        0.97684081234307032681744391886221E+00,     
        0.98181502080381411003346312451200E+00,     
        0.98619317401693166671043833175407E+00,     
        0.98997261459148415760778669967548E+00,     
        0.99315104925451714736113079489080E+00,     
        0.99572655135202722663543337085008E+00,     
        0.99769756618980462107441703193392E+00,     
        0.99906293435531189513828920479421E+00,     
        0.99982213041530614629963254927125E+00
  };

  for ( dim = 0; dim < dim_num; dim++ )
  {
    if ( grid_base[dim] < 0 )
    {
      cout << "\n";
      cout << "GL_ABSCISSA - Fatal error!\n";
      cout << "  Some base values are less than 0.\n";
      exit ( 1 );
    }
  }

  for ( dim = 0; dim < dim_num; dim++ )
  {
    if ( 63 < grid_base[dim] )
    {
      cout << "\n";
      cout << "GL_ABSCISSA - Fatal error!\n";
      cout << "  Some base values are greater than 63.\n";
      exit ( 1 );
    }
  }

  for ( point = 0; point < point_num; point++ )
  {
    for ( dim = 0; dim < dim_num; dim++ )
    {
      level = i4_log_2 ( grid_base[dim] + 1 );

      pointer = skip[level] + ( grid_index[dim+point*dim_num] + grid_base[dim] );

      grid_point[dim+point*dim_num] = x[pointer];
    }
  }

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

double gp_abscissa ( int order, int index )

//****************************************************************************80
//
//  Purpose:
//
//    GP_ABSCISSA returns the I-th abscissa for a Gauss-Patterson rule.
//
//  Discussion:
//
//    The rule is specified by its level.
//
//    The number of points in the rule, known as the order, is
//    related to the level by the formula:
//
//      ORDER = 2^(LEVEL+1)-1.
//
//    Only rules of order 1, 3, 7, 15, 31, 63, 127 and 255 are allowed.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    26 December 2009
//
//  Author:
//
//    John Burkardt
//
//  Reference:
//
//    Prem Kythe, Michael Schaeferkotter,
//    Handbook of Computational Methods for Integration,
//    Chapman and Hall, 2004,
//    ISBN: 1-58488-428-2,
//    LC: QA299.3.K98.
//
//    Thomas Patterson,
//    The Optimal Addition of Points to Quadrature Formulae,
//    Mathematics of Computation,
//    Volume 22, Number 104, October 1968, pages 847-856.
//
//  Parameters:
//
//    Input, int ORDER, the order of the rule.
//
//    Input, int INDEX, the index of the point in the rule.
//
//    Output, double GP_ABSCISSA, the value of the INDEX-th 
//    abscissa in the rule of order ORDER.
//
{
  static double x_001[1] = {
     0.0 };
  static double x_003[3] = {
    -0.77459666924148337704, 
     0.0, 
     0.77459666924148337704 };
  static double x_007[7] = {
    -0.96049126870802028342, 
    -0.77459666924148337704, 
    -0.43424374934680255800, 
     0.0, 
     0.43424374934680255800, 
     0.77459666924148337704, 
     0.96049126870802028342 };
  static double x_015[15] = {
    -0.99383196321275502221, 
    -0.96049126870802028342, 
    -0.88845923287225699889, 
    -0.77459666924148337704, 
    -0.62110294673722640294, 
    -0.43424374934680255800, 
    -0.22338668642896688163, 
     0.0, 
     0.22338668642896688163, 
     0.43424374934680255800, 
     0.62110294673722640294, 
     0.77459666924148337704, 
     0.88845923287225699889, 
     0.96049126870802028342, 
     0.99383196321275502221 };
  static double x_031[31] = {
    -0.99909812496766759766, 
    -0.99383196321275502221, 
    -0.98153114955374010687, 
    -0.96049126870802028342, 
    -0.92965485742974005667, 
    -0.88845923287225699889, 
    -0.83672593816886873550, 
    -0.77459666924148337704, 
    -0.70249620649152707861, 
    -0.62110294673722640294, 
    -0.53131974364437562397, 
    -0.43424374934680255800, 
    -0.33113539325797683309, 
    -0.22338668642896688163, 
    -0.11248894313318662575, 
     0.0, 
     0.11248894313318662575, 
     0.22338668642896688163, 
     0.33113539325797683309, 
     0.43424374934680255800, 
     0.53131974364437562397, 
     0.62110294673722640294, 
     0.70249620649152707861, 
     0.77459666924148337704, 
     0.83672593816886873550, 
     0.88845923287225699889, 
     0.92965485742974005667, 
     0.96049126870802028342, 
     0.98153114955374010687, 
     0.99383196321275502221, 
     0.99909812496766759766 };
  static double x_063[63] = {
    -0.99987288812035761194, 
    -0.99909812496766759766, 
    -0.99720625937222195908, 
    -0.99383196321275502221, 
    -0.98868475754742947994, 
    -0.98153114955374010687, 
    -0.97218287474858179658, 
    -0.96049126870802028342, 
    -0.94634285837340290515, 
    -0.92965485742974005667, 
    -0.91037115695700429250, 
    -0.88845923287225699889, 
    -0.86390793819369047715, 
    -0.83672593816886873550, 
    -0.80694053195021761186, 
    -0.77459666924148337704, 
    -0.73975604435269475868, 
    -0.70249620649152707861, 
    -0.66290966002478059546, 
    -0.62110294673722640294, 
    -0.57719571005204581484, 
    -0.53131974364437562397, 
    -0.48361802694584102756, 
    -0.43424374934680255800, 
    -0.38335932419873034692, 
    -0.33113539325797683309, 
    -0.27774982202182431507, 
    -0.22338668642896688163, 
    -0.16823525155220746498, 
    -0.11248894313318662575, 
    -0.056344313046592789972, 
     0.0, 
     0.056344313046592789972, 
     0.11248894313318662575, 
     0.16823525155220746498, 
     0.22338668642896688163, 
     0.27774982202182431507, 
     0.33113539325797683309, 
     0.38335932419873034692, 
     0.43424374934680255800, 
     0.48361802694584102756, 
     0.53131974364437562397, 
     0.57719571005204581484, 
     0.62110294673722640294, 
     0.66290966002478059546, 
     0.70249620649152707861, 
     0.73975604435269475868, 
     0.77459666924148337704, 
     0.80694053195021761186, 
     0.83672593816886873550, 
     0.86390793819369047715, 
     0.88845923287225699889, 
     0.91037115695700429250, 
     0.92965485742974005667, 
     0.94634285837340290515, 
     0.96049126870802028342, 
     0.97218287474858179658, 
     0.98153114955374010687, 
     0.98868475754742947994, 
     0.99383196321275502221, 
     0.99720625937222195908, 
     0.99909812496766759766, 
     0.99987288812035761194 };
  static double x_127[127] = {
    -0.99998243035489159858, 
    -0.99987288812035761194, 
    -0.99959879967191068325, 
    -0.99909812496766759766, 
    -0.99831663531840739253, 
    -0.99720625937222195908, 
    -0.99572410469840718851, 
    -0.99383196321275502221, 
    -0.99149572117810613240, 
    -0.98868475754742947994, 
    -0.98537149959852037111, 
    -0.98153114955374010687, 
    -0.97714151463970571416, 
    -0.97218287474858179658, 
    -0.96663785155841656709, 
    -0.96049126870802028342, 
    -0.95373000642576113641, 
    -0.94634285837340290515, 
    -0.93832039777959288365, 
    -0.92965485742974005667, 
    -0.92034002547001242073, 
    -0.91037115695700429250, 
    -0.89974489977694003664, 
    -0.88845923287225699889, 
    -0.87651341448470526974, 
    -0.86390793819369047715, 
    -0.85064449476835027976, 
    -0.83672593816886873550, 
    -0.82215625436498040737, 
    -0.80694053195021761186, 
    -0.79108493379984836143, 
    -0.77459666924148337704, 
    -0.75748396638051363793, 
    -0.73975604435269475868, 
    -0.72142308537009891548, 
    -0.70249620649152707861, 
    -0.68298743109107922809, 
    -0.66290966002478059546, 
    -0.64227664250975951377, 
    -0.62110294673722640294, 
    -0.59940393024224289297, 
    -0.57719571005204581484, 
    -0.55449513263193254887, 
    -0.53131974364437562397, 
    -0.50768775753371660215, 
    -0.48361802694584102756, 
    -0.45913001198983233287, 
    -0.43424374934680255800, 
    -0.40897982122988867241, 
    -0.38335932419873034692, 
    -0.35740383783153215238, 
    -0.33113539325797683309, 
    -0.30457644155671404334, 
    -0.27774982202182431507, 
    -0.25067873030348317661, 
    -0.22338668642896688163, 
    -0.19589750271110015392, 
    -0.16823525155220746498, 
    -0.14042423315256017459, 
    -0.11248894313318662575, 
    -0.084454040083710883710, 
    -0.056344313046592789972, 
    -0.028184648949745694339, 
     0.0, 
     0.028184648949745694339, 
     0.056344313046592789972, 
     0.084454040083710883710, 
     0.11248894313318662575, 
     0.14042423315256017459, 
     0.16823525155220746498, 
     0.19589750271110015392, 
     0.22338668642896688163, 
     0.25067873030348317661, 
     0.27774982202182431507, 
     0.30457644155671404334, 
     0.33113539325797683309, 
     0.35740383783153215238, 
     0.38335932419873034692, 
     0.40897982122988867241, 
     0.43424374934680255800, 
     0.45913001198983233287, 
     0.48361802694584102756, 
     0.50768775753371660215, 
     0.53131974364437562397, 
     0.55449513263193254887, 
     0.57719571005204581484, 
     0.59940393024224289297, 
     0.62110294673722640294, 
     0.64227664250975951377, 
     0.66290966002478059546, 
     0.68298743109107922809, 
     0.70249620649152707861, 
     0.72142308537009891548, 
     0.73975604435269475868, 
     0.75748396638051363793, 
     0.77459666924148337704, 
     0.79108493379984836143, 
     0.80694053195021761186, 
     0.82215625436498040737, 
     0.83672593816886873550, 
     0.85064449476835027976, 
     0.86390793819369047715, 
     0.87651341448470526974, 
     0.88845923287225699889, 
     0.89974489977694003664, 
     0.91037115695700429250, 
     0.92034002547001242073, 
     0.92965485742974005667, 
     0.93832039777959288365, 
     0.94634285837340290515, 
     0.95373000642576113641, 
     0.96049126870802028342, 
     0.96663785155841656709, 
     0.97218287474858179658, 
     0.97714151463970571416, 
     0.98153114955374010687, 
     0.98537149959852037111, 
     0.98868475754742947994, 
     0.99149572117810613240, 
     0.99383196321275502221, 
     0.99572410469840718851, 
     0.99720625937222195908, 
     0.99831663531840739253, 
     0.99909812496766759766, 
     0.99959879967191068325, 
     0.99987288812035761194, 
     0.99998243035489159858 };
  static double x_255[255] = {
    -0.99999759637974846462, 
    -0.99998243035489159858, 
    -0.99994399620705437576, 
    -0.99987288812035761194, 
    -0.99976049092443204733, 
    -0.99959879967191068325, 
    -0.99938033802502358193, 
    -0.99909812496766759766, 
    -0.99874561446809511470, 
    -0.99831663531840739253, 
    -0.99780535449595727456, 
    -0.99720625937222195908, 
    -0.99651414591489027385, 
    -0.99572410469840718851, 
    -0.99483150280062100052, 
    -0.99383196321275502221, 
    -0.99272134428278861533, 
    -0.99149572117810613240, 
    -0.99015137040077015918, 
    -0.98868475754742947994, 
    -0.98709252795403406719, 
    -0.98537149959852037111, 
    -0.98351865757863272876, 
    -0.98153114955374010687, 
    -0.97940628167086268381, 
    -0.97714151463970571416, 
    -0.97473445975240266776, 
    -0.97218287474858179658, 
    -0.96948465950245923177, 
    -0.96663785155841656709, 
    -0.96364062156981213252, 
    -0.96049126870802028342, 
    -0.95718821610986096274, 
    -0.95373000642576113641, 
    -0.95011529752129487656, 
    -0.94634285837340290515, 
    -0.94241156519108305981, 
    -0.93832039777959288365, 
    -0.93406843615772578800, 
    -0.92965485742974005667, 
    -0.92507893290707565236, 
    -0.92034002547001242073, 
    -0.91543758715576504064, 
    -0.91037115695700429250, 
    -0.90514035881326159519, 
    -0.89974489977694003664, 
    -0.89418456833555902286, 
    -0.88845923287225699889, 
    -0.88256884024734190684, 
    -0.87651341448470526974, 
    -0.87029305554811390585, 
    -0.86390793819369047715, 
    -0.85735831088623215653, 
    -0.85064449476835027976, 
    -0.84376688267270860104, 
    -0.83672593816886873550, 
    -0.82952219463740140018, 
    -0.82215625436498040737, 
    -0.81462878765513741344, 
    -0.80694053195021761186, 
    -0.79909229096084140180, 
    -0.79108493379984836143, 
    -0.78291939411828301639, 
    -0.77459666924148337704, 
    -0.76611781930376009072, 
    -0.75748396638051363793, 
    -0.74869629361693660282, 
    -0.73975604435269475868, 
    -0.73066452124218126133, 
    -0.72142308537009891548, 
    -0.71203315536225203459, 
    -0.70249620649152707861, 
    -0.69281376977911470289, 
    -0.68298743109107922809, 
    -0.67301883023041847920, 
    -0.66290966002478059546, 
    -0.65266166541001749610, 
    -0.64227664250975951377, 
    -0.63175643771119423041, 
    -0.62110294673722640294, 
    -0.61031811371518640016, 
    -0.59940393024224289297, 
    -0.58836243444766254143, 
    -0.57719571005204581484, 
    -0.56590588542365442262, 
    -0.55449513263193254887, 
    -0.54296566649831149049, 
    -0.53131974364437562397, 
    -0.51955966153745702199, 
    -0.50768775753371660215, 
    -0.49570640791876146017, 
    -0.48361802694584102756, 
    -0.47142506587165887693, 
    -0.45913001198983233287, 
    -0.44673538766202847374, 
    -0.43424374934680255800, 
    -0.42165768662616330006, 
    -0.40897982122988867241, 
    -0.39621280605761593918, 
    -0.38335932419873034692, 
    -0.37042208795007823014, 
    -0.35740383783153215238, 
    -0.34430734159943802278, 
    -0.33113539325797683309, 
    -0.31789081206847668318, 
    -0.30457644155671404334, 
    -0.29119514851824668196, 
    -0.27774982202182431507, 
    -0.26424337241092676194, 
    -0.25067873030348317661, 
    -0.23705884558982972721, 
    -0.22338668642896688163, 
    -0.20966523824318119477, 
    -0.19589750271110015392, 
    -0.18208649675925219825, 
    -0.16823525155220746498, 
    -0.15434681148137810869, 
    -0.14042423315256017459, 
    -0.12647058437230196685, 
    -0.11248894313318662575, 
    -0.098482396598119202090, 
    -0.084454040083710883710, 
    -0.070406976042855179063, 
    -0.056344313046592789972, 
    -0.042269164765363603212, 
    -0.028184648949745694339, 
    -0.014093886410782462614, 
    0.0, 
    0.014093886410782462614, 
    0.028184648949745694339, 
    0.042269164765363603212, 
    0.056344313046592789972, 
    0.070406976042855179063, 
    0.084454040083710883710, 
    0.098482396598119202090, 
    0.11248894313318662575, 
    0.12647058437230196685, 
    0.14042423315256017459, 
    0.15434681148137810869, 
    0.16823525155220746498, 
    0.18208649675925219825, 
    0.19589750271110015392, 
    0.20966523824318119477, 
    0.22338668642896688163, 
    0.23705884558982972721, 
    0.25067873030348317661, 
    0.26424337241092676194, 
    0.27774982202182431507, 
    0.29119514851824668196, 
    0.30457644155671404334, 
    0.31789081206847668318, 
    0.33113539325797683309, 
    0.34430734159943802278, 
    0.35740383783153215238, 
    0.37042208795007823014, 
    0.38335932419873034692, 
    0.39621280605761593918, 
    0.40897982122988867241, 
    0.42165768662616330006, 
    0.43424374934680255800, 
    0.44673538766202847374, 
    0.45913001198983233287, 
    0.47142506587165887693, 
    0.48361802694584102756, 
    0.49570640791876146017, 
    0.50768775753371660215, 
    0.51955966153745702199, 
    0.53131974364437562397, 
    0.54296566649831149049, 
    0.55449513263193254887, 
    0.56590588542365442262, 
    0.57719571005204581484, 
    0.58836243444766254143, 
    0.59940393024224289297, 
    0.61031811371518640016, 
    0.62110294673722640294, 
    0.63175643771119423041, 
    0.64227664250975951377, 
    0.65266166541001749610, 
    0.66290966002478059546, 
    0.67301883023041847920, 
    0.68298743109107922809, 
    0.69281376977911470289, 
    0.70249620649152707861, 
    0.71203315536225203459, 
    0.72142308537009891548, 
    0.73066452124218126133, 
    0.73975604435269475868, 
    0.74869629361693660282, 
    0.75748396638051363793, 
    0.76611781930376009072, 
    0.77459666924148337704, 
    0.78291939411828301639, 
    0.79108493379984836143, 
    0.79909229096084140180, 
    0.80694053195021761186, 
    0.81462878765513741344, 
    0.82215625436498040737, 
    0.82952219463740140018, 
    0.83672593816886873550, 
    0.84376688267270860104, 
    0.85064449476835027976, 
    0.85735831088623215653, 
    0.86390793819369047715, 
    0.87029305554811390585, 
    0.87651341448470526974, 
    0.88256884024734190684, 
    0.88845923287225699889, 
    0.89418456833555902286, 
    0.89974489977694003664, 
    0.90514035881326159519, 
    0.91037115695700429250, 
    0.91543758715576504064, 
    0.92034002547001242073, 
    0.92507893290707565236, 
    0.92965485742974005667, 
    0.93406843615772578800, 
    0.93832039777959288365, 
    0.94241156519108305981, 
    0.94634285837340290515, 
    0.95011529752129487656, 
    0.95373000642576113641, 
    0.95718821610986096274, 
    0.96049126870802028342, 
    0.96364062156981213252, 
    0.96663785155841656709, 
    0.96948465950245923177, 
    0.97218287474858179658, 
    0.97473445975240266776, 
    0.97714151463970571416, 
    0.97940628167086268381, 
    0.98153114955374010687, 
    0.98351865757863272876, 
    0.98537149959852037111, 
    0.98709252795403406719, 
    0.98868475754742947994, 
    0.99015137040077015918, 
    0.99149572117810613240, 
    0.99272134428278861533, 
    0.99383196321275502221, 
    0.99483150280062100052, 
    0.99572410469840718851, 
    0.99651414591489027385, 
    0.99720625937222195908, 
    0.99780535449595727456, 
    0.99831663531840739253, 
    0.99874561446809511470, 
    0.99909812496766759766, 
    0.99938033802502358193, 
    0.99959879967191068325, 
    0.99976049092443204733, 
    0.99987288812035761194, 
    0.99994399620705437576, 
    0.99998243035489159858, 
    0.99999759637974846462 };
  double value;

  if ( order < 1 )
  {
    value = - r8_huge ( );
  }
  else if ( index < 1 || order < index )
  {
    value = - r8_huge ( );
  }
  else if ( order == 1 )
  {
    value = x_001[index-1];
  }
  else if ( order == 3 )
  {
    value = x_003[index-1];
  }
  else if ( order == 7 )
  {
    value = x_007[index-1];
  }
  else if ( order == 15 )
  {
    value = x_015[index-1];
  }
  else if ( order == 31 )
  {
    value = x_031[index-1];
  }
  else if ( order == 63 )
  {
    value = x_063[index-1];
  }
  else if ( order == 127 )
  {
    value = x_127[index-1];
  }
  else if ( order == 255 )
  {
    value = x_255[index-1];
  }
  else
  {
    value = - r8_huge ( );
  }
  return value;
}
//****************************************************************************80

int i4_choose ( int n, int k )

//****************************************************************************80
//
//  Purpose:
//
//    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, int N, K, are the values of N and K.
//
//    Output, int I4_CHOOSE, the number of combinations of N
//    things taken K at a time.
//
{
  int i;
  int mn;
  int mx;
  int value;

  mn = i4_min ( k, n - k );

  if ( mn < 0 )
  {
    value = 0;
  }
  else if ( mn == 0 )
  {
    value = 1;
  }
  else
  {
    mx = i4_max ( k, n - k );
    value = mx + 1;

    for ( i = 2; i <= mn; i++ )
    {
      value = ( value * ( mx + i ) ) / i;
    }
  }
  return value;
}
//****************************************************************************80

int i4_log_2 ( int i )

//****************************************************************************80
//
//  Purpose:
//
//    I4_LOG_2 returns the integer part of the logarithm base 2 of an I4.
//
//  Example:
//
//        I  I4_LOG_10
//    -----  --------
//        0    0
//        1    0
//        2    1
//        3    1
//        4    2
//        5    2
//        7    2
//        8    3
//        9    3
//     1000    9
//     1024   10
//
//  Discussion:
//
//    I4_LOG_2 ( I ) + 1 is the number of binary digits in I.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    04 January 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I, the number whose logarithm base 2 is desired.
//
//    Output, int I4_LOG_2, the integer part of the logarithm base 2 of
//    the absolute value of X.
//
{
  int i_abs;
  int two_pow;
  int value;

  if ( i == 0 )
  {
    value = 0;
  }
  else
  {
    value = 0;
    two_pow = 2;

    i_abs = abs ( i );

    while ( two_pow <= i_abs )
    {
      value = value + 1;
      two_pow = two_pow * 2;
    }
  }

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

int i4_max ( int i1, int i2 )

//****************************************************************************80
//
//  Purpose:
//
//    I4_MAX returns the maximum of two I4's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    13 October 1998
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I1, I2, are two integers to be compared.
//
//    Output, int I4_MAX, the larger of I1 and I2.
//
{
  int value;

  if ( i2 < i1 )
  {
    value = i1;
  }
  else
  {
    value = i2;
  }
  return value;
}
//****************************************************************************80

int i4_min ( int i1, int i2 )

//****************************************************************************80
//
//  Purpose:
//
//    I4_MIN returns the minimum of two I4's.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    13 October 1998
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I1, I2, two integers to be compared.
//
//    Output, int I4_MIN, the smaller of I1 and I2.
//
{
  int value;

  if ( i1 < i2 )
  {
    value = i1;
  }
  else
  {
    value = i2;
  }
  return value;
}
//****************************************************************************80

int i4_modp ( int i, int j )

//****************************************************************************80
//
//  Purpose:
//
//    I4_MODP returns the nonnegative remainder of I4 division.
//
//  Formula:
//
//    If 
//      NREM = I4_MODP ( I, J ) 
//      NMULT = ( I - NREM ) / J
//    then
//      I = J * NMULT + NREM
//    where NREM is always nonnegative.
//
//    The MOD function computes a result with the same sign as the
//    quantity being divided.  Thus, suppose you had an angle A,
//    and you wanted to ensure that it was between 0 and 360.
//    Then mod(A,360) would do, if A was positive, but if A
//    was negative, your result would be between -360 and 0.
//
//    On the other hand, I4_MODP(A,360) is between 0 and 360, always.
//
//  Example:
//
//        I         J     MOD  I4_MODP   I4_MODP Factorization
// 
//      107        50       7       7    107 =  2 *  50 + 7
//      107       -50       7       7    107 = -2 * -50 + 7
//     -107        50      -7      43   -107 = -3 *  50 + 43
//     -107       -50      -7      43   -107 =  3 * -50 + 43
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    26 May 1999
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I, the number to be divided.
//
//    Input, int J, the number that divides I.
//
//    Output, int I4_MODP, the nonnegative remainder when I is 
//    divided by J.
//
{
  int value;

  if ( j == 0 )
  {
    cout << "\n";
    cout << "I4_MODP - Fatal error!\n";
    cout << "  I4_MODP ( I, J ) called with J = " << j << "\n";
    exit ( 1 );
  }

  value = i % j;

  if ( value < 0 )
  {
    value = value + abs ( j );
  }

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

int i4_power ( int i, int j )

//****************************************************************************80
//
//  Purpose:
//
//    I4_POWER returns the value of I^J.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    01 April 2004
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I, J, the base and the power.  J should be nonnegative.
//
//    Output, int I4_POWER, the value of I^J.
//
{
  int k;
  int value;

  if ( j < 0 )
  {
    if ( i == 1 )
    {
      value = 1;
    }
    else if ( i == 0 )
    {
      cout << "\n";
      cout << "I4_POWER - Fatal error!\n";
      cout << "  I^J requested, with I = 0 and J negative.\n";
      exit ( 1 );
    }
    else
    {
      value = 0;
    }
  }
  else if ( j == 0 )
  {
    if ( i == 0 )
    {
      cout << "\n";
      cout << "I4_POWER - Fatal error!\n";
      cout << "  I^J requested, with I = 0 and J = 0.\n";
      exit ( 1 );
    }
    else
    {
      value = 1;
    }
  }
  else if ( j == 1 )
  {
    value = i;
  }
  else
  {
    value = 1;
    for ( k = 1; k <= j; k++ )
    {
      value = value * i;
    }
  }
  return value;
}
//****************************************************************************80

string i4_to_string ( int i4, string format )

//****************************************************************************80
//
//  Purpose:
//
//    I4_TO_STRING converts an I4 to a C++ string.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    09 July 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int I4, an integer.
//
//    Input, string FORMAT, the format string.
//
//    Output, string I4_TO_STRING, the string.
//
{
  char i4_char[80];
  string i4_string;

  sprintf ( i4_char, format.c_str ( ), i4 );

  i4_string = string ( i4_char );

  return i4_string;
}
//****************************************************************************80

int i4vec_product ( int n, int a[] )

//****************************************************************************80
//
//  Purpose:
//
//    I4VEC_PRODUCT multiplies the entries of an I4VEC.
//
//  Example:
//
//    Input:
//
//      A = ( 1, 2, 3, 4 )
//
//    Output:
//
//      I4VEC_PRODUCT = 24
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    17 May 2003
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int N, the number of entries in the vector.
//
//    Input, int A[N], the vector
//
//    Output, int I4VEC_PRODUCT, the product of the entries of A.
//
{
  int i;
  int product;

  product = 1;
  for ( i = 0; i < n; i++ )
  {
    product = product * a[i];
  }

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

int index_to_level_open ( int dim_num, int t[], int order, int level_max )

//****************************************************************************80
//
//  Purpose:
//
//    INDEX_TO_LEVEL_OPEN determines the level of a point given its index.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    19 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, int DIM_NUM, the spatial dimension.
//
//    Input, int T[DIM_NUM], the grid index of a point.
//
//    Input, int ORDER, the order of the rule.
//
//    Input, int LEVEL_MAX, the level with respect to which the
//    index applies.
//
//    Output, int INDEX_TO_LEVEL_OPEN, the first level on which
//    the point associated with the given index will appear.
//
{
  int dim;
  int level;
  int s;
  int value;

  value = 0;

  for ( dim = 0; dim < dim_num; dim++ )
  {
    s = t[dim];

    s = i4_modp ( s, order );

    if ( s == 0 )
    {
      level = 0;
    }
    else
    {
      level = level_max;

      while ( ( s % 2 ) == 0 )
      {
        s = s / 2;
        level = level - 1;
      }
    }

    if ( level == 0 )
    {
      level = 1;
    }
    else if ( level == 1 )
    {
      level = 0;
    }

    value = value + level;
  }

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

void level_to_order_open ( int dim_num, int level[], int order[] )

//****************************************************************************80
//
//  Purpose:
//
//    LEVEL_TO_ORDER_OPEN 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 2, Newton Cotes 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 2, Newton Cotes Open, 
//    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:
//
//    19 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, int DIM_NUM, the spatial dimension.
//
//    Input, int LEVEL[DIM_NUM], the nesting level.
//
//    Output, int ORDER[DIM_NUM], the order (number of points) 
//    of the rule.
//
{
  int dim;

  for ( dim = 0; dim < dim_num; dim++ )
  {
    if ( level[dim] < 0 )
    {
      order[dim] = -1;
    }
    else if ( level[dim] == 0 )
    {
      order[dim] = 1;
    }
    else
    {
      order[dim] = i4_power ( 2, level[dim] + 1 ) - 1 ;
    }
  }
  return;
}
//****************************************************************************80

int *levels_open_index ( int dim_num, int level_max, int point_num )

//****************************************************************************80
//
//  Purpose:
//
//    LEVELS_OPEN_INDEX computes grids with 0 <= LEVEL <= LEVEL_MAX.
//
//  Discussion:
//
//    The necessary dimensions of GRID_INDEX can be
//    determined by calling LEVELS_OPEN_INDEX_SIZE first.
//
//  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, int DIM_NUM, the spatial dimension.
//
//    Input, int LEVEL_MAX, the maximum value of LEVEL.
//
//    Input, int POINT_NUM, the total number of points in the grids.
//
//    Output, int LEVELS_MAX_INDEX[DIM_NUM*POINT_NUM], a list of point indices,
//    representing a subset of the product grid of level LEVEL_MAX,
//    representing (exactly once) each point that will show up in a
//    sparse grid of level LEVEL_MAX.
//
{
  int dim;
  int factor;
  int *grid_index;
  int *grid_index2;
  int h;
  int level;
  int *level_1d;
  bool more;
  int *order_1d;
  int order_nd;
  int point;
  int point_num2;
  int t;
  bool test;
//
//  The outer loop generates LEVELs from 0 to LEVEL_MAX.
//
  grid_index = new int[dim_num*point_num];
  level_1d = new int[dim_num];
  order_1d = new int[dim_num];

  point_num2 = 0;

  for ( level = 0; level <= level_max; level++ )
  {
//
//  The middle loop generates the next partition that adds up to LEVEL.
//
    more = false;
    h = 0;
    t = 0;

    for ( ; ; )
    {
      comp_next ( level, dim_num, level_1d, &more, &h, &t );
//
//  Transform each 1D level to a corresponding 1D order.
//
      level_to_order_open ( dim_num, level_1d, order_1d );
//
//  The product of the 1D orders gives us the number of points in this grid.
//
      order_nd = i4vec_product ( dim_num, order_1d );
//
//  The inner (hidden) loop generates all points corresponding to given grid.
//
      grid_index2 = multigrid_index1 ( dim_num, order_1d, order_nd );
//
//  Only keep those points which first appear on this level.
//  If you keep a point, it is necessary to rescale each of its components
//  so that we save the coordinates as they apply on the final grid.
//
      for ( point = 0; point < order_nd; point++ )
      {
        test = true;
        for ( dim = 0; dim < dim_num; dim++ )
        {
          if ( grid_index2[dim+point*dim_num] % 2 == 0 )
          {
            test = false;
          }
        }

        if ( test )
        {
          for ( dim = 0; dim < dim_num; dim++ )
          {
            grid_index[dim+point_num2*dim_num] =
              i4_power ( 2, level_max - level_1d[dim] )
              * grid_index2[dim+point*dim_num];
          }
          point_num2 = point_num2 + 1;
        }
      }

      delete [] grid_index2;

      if ( !more )
      {
        break;
      }
    }
  }
  delete [] level_1d;
  delete [] order_1d;

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

int *multigrid_index1 ( int dim_num, int order_1d[], int order_nd )

//****************************************************************************80
//
//  Purpose:
//
//    MULTIGRID_INDEX1 returns an indexed multidimensional grid.
//
//  Discussion:
//
//    For dimension DIM, the second index of INDX may vary from 
//    1 to ORDER_1D[DIM].
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    23 May 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, int DIM_NUM, the spatial dimension of the points.
//
//    Input, int ORDER_1D[DIM_NUM], the order of the
//    rule in each dimension.
//
//    Input, int ORDER_ND, the product of the entries of ORDER_1D.
//
//    Output, int 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.
//
{
  int *a;
  int dim;
  bool more;
  int p;
  int *indx;

  indx = new int[dim_num*order_nd];
  a = new int[dim_num];
  more = false;
  p = 0;

  for ( ; ; )
  {
    vec_colex_next2 ( dim_num, order_1d, a, &more );

    if ( !more )
    {
      break;
    }

    for ( dim = 0; dim < dim_num; dim++ )
    {
      indx[dim+p*dim_num] = a[dim] + 1;
    }
    p = p + 1;
  }

  delete [] a;

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

void multigrid_scale_open ( int dim_num, int order_nd, int level_max, 
  int level_1d[], int grid_index[] )

//****************************************************************************80
//
//  Purpose:
//
//    MULTIGRID_SCALE_OPEN renumbers a grid as a subgrid on a higher level.
//
//  Discussion:
//
//    This routine takes a grid associated with a given value of
//    LEVEL, and multiplies all the indices by a power of 2, so that
//    the indices reflect the position of the same points, but in
//    a grid of level LEVEL_MAX.
//
//    For an open grid, going from one level to the next, a set of indices
//    will be rescaled by 2*INDEX-1.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    08 June 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, int DIM_NUM, the spatial dimension.
//
//    Input, int ORDER_ND, the number of points in the grid.
//
//    Input, int LEVEL_MAX, the maximum value of LEVEL.
//
//    Input, int LEVEL_1D[DIM_NUM], the level in each dimension.
//
//    Input/output, int GRID_INDEX[DIM_NUM*POINT_NUM], the index
//    values for each grid point.  On input, these indices are based in
//    the level for which the grid was generated; on output, the
//    indices are appropriate for the grid as a subgrid of a grid
//    of level LEVEL_MAX.
//
{
  int dim;
  int factor;
  int order;

  for ( dim = 0; dim < dim_num; dim++ )
  {
    factor = i4_power ( 2, level_max - level_1d[dim] );

    for ( order = 0; order < order_nd; order++ )
    {
      grid_index[dim+order*dim_num] = grid_index[dim+order*dim_num] * factor;
    }
  }

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

double nco_abscissa ( int order, int i )

//****************************************************************************80
//
//  Purpose:
//
//    NCO_ABSCISSA returns the I-th abscissa for the Newton Cotes open rule.
//
//  Discussion:
//
//    Our convention is that the abscissas are numbered from left to
//    right.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    25 May 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int ORDER, the order of the rule.
//    1 <= ORDER.
//
//    Input, int I, the index of the desired abscissa.  
//    1 <= I <= ORDER.
//
//    Output, double NCO_ABSCISSA, the value of the I-th 
//    abscissa in the Newton Cotes open rule of order ORDER.
//
{
  double value;
  double x_max = +1.0;
  double x_min = -1.0;

  if ( order < 1 )
  {
    value = - r8_huge ( );
    return value;
  }

  if ( i < 1 || order < i )
  {
    cout << "\n";
    cout << "NCO_ABSCISSA - Fatal error!\n";
    cout << "  1 <= I <= ORDER is required.\n";
    exit ( 1 );
  }

  if ( order == 1 )
  {
    value = ( x_min + x_max ) / 2.0;
    return value;
  }

  value = ( ( double ) ( order - i + 1 ) * x_min
          + ( double ) (         i     ) * x_max )
          / ( double ) ( order     + 1 );

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

double r8_epsilon ( )

//****************************************************************************80
//
//  Purpose:
//
//    R8_EPSILON returns the R8 roundoff unit.
//
//  Discussion:
//
//    The roundoff unit is a number R which is a power of 2 with the
//    property that, to the precision of the computer's arithmetic,
//      1 < 1 + R
//    but
//      1 = ( 1 + R / 2 )
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license.
//
//  Modified:
//
//    01 September 2012
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, double R8_EPSILON, the R8 round-off unit.
//
{
  const double value = 2.220446049250313E-016;

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

double r8_huge ( )

//****************************************************************************80
//
//  Purpose:
//
//    R8_HUGE returns a "huge" R8.
//
//  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.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    06 October 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Output, double R8_HUGE, a "huge" R8 value.
//
{
  double value;

  value = 1.0E+30;

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

void r8mat_write ( string output_filename, int m, int n, double table[] )

//****************************************************************************80
//
//  Purpose:
//
//    R8MAT_WRITE writes an R8MAT file.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    29 June 2009
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, string OUTPUT_FILENAME, the output filename.
//
//    Input, int M, the spatial dimension.
//
//    Input, int N, the number of points.
//
//    Input, double TABLE[M*N], the table data.
//
{
  int i;
  int j;
  ofstream output;
//
//  Open the file.
//
  output.open ( output_filename.c_str ( ) );

  if ( !output )
  {
    cerr << "\n";
    cerr << "R8MAT_WRITE - Fatal error!\n";
    cerr << "  Could not open the output file.\n";
    return;
  }
//
//  Write the data.
//
  for ( j = 0; j < n; j++ )
  {
    for ( i = 0; i < m; i++ )
    {
      output << "  " << setw(24) << setprecision(16) << table[i+j*m];
    }
    output << "\n";
  }
//
//  Close the file.
//
  output.close ( );

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

int sparse_grid_f2s_size ( int dim_num, int level_max )

//****************************************************************************80
//
//  Purpose:
//
//    SPARSE_GRID_F2S_SIZE sizes a sparse grid using Fejer Type 2 Slow rules.
//
//  Discussion:
//
//    The grid is defined as the sum of the product rules whose LEVEL
//    satisfies:
//
//      0 <= LEVEL <= LEVEL_MAX.
//
//    This calculation is much faster than a previous method.  It simply
//    computes the number of new points that are added at each level in the
//    1D rule, and then counts the new points at a given DIM_NUM dimensional
//    level vector as the product of the new points added in each dimension.
//
//    This approach will work for nested families, and may be extensible
//    to other families, and to mixed rules.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    26 December 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.
//
//  Parameters:
//
//    Input, int DIM_NUM, the spatial dimension.
//
//    Input, int LEVEL_MAX, the maximum value of LEVEL.
//
//    Output, int SPARSE_GRID_F2S_SIZE, the number of points in the grid.
//
{
  int dim;
  int h;
  int l;
  int level;
  int *level_1d;
  bool more;
  int *new_1d;
  int o;
  int p;
  int point_num;
  int t;
  int v;
//
//  Special case.
//
  if ( level_max < 0 )
  {
    point_num = 0;
    return point_num;
  }

  if ( level_max == 0 )
  {
    point_num = 1;
    return point_num;
  }
//
//  Construct the vector that counts the new points in the 1D rule.
//
  new_1d = new int[level_max+1];

  new_1d[0] = 1;

  p = 1;
  o = 1;

  for ( l = 1; l <= level_max; l++ )
  {
    p = 2 * l + 1;
    if ( o < p )
    {
      new_1d[l] = o + 1;
      o = 2 * o + 1;
    }
    else
    {
      new_1d[l] = 0;
    }
  }
//
//  Count the number of points by counting the number of new points 
//  associated with each level vector.
//
  level_1d = new int[dim_num];

  point_num = 0;

  for ( level = 0; level <= level_max; level++ )
  {
    more = false;
    h = 0;
    t = 0;

    for ( ; ;)
    {
      comp_next ( level, dim_num, level_1d, &more, &h, &t );

      v = 1;
      for ( dim = 0; dim < dim_num; dim++ )
      {
        v = v * new_1d[level_1d[dim]];
      }

      point_num = point_num + v;

      if ( !more )
      {
        break;
      }
    }
  }
  delete [] level_1d;
  delete [] new_1d;

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

int sparse_grid_gps_size ( int dim_num, int level_max )

//****************************************************************************80
//
//  Purpose:
//
//    SPARSE_GRID_GPS_SIZE sizes a sparse grid using Gauss-Patterson-Slow rules.
//
//  Discussion:
//
//    The Gauss-Patterson-Slow family assumes that, for the underlying 1D
//    rules, a precision of 2*L+1 is needed at level L.  Therefore, the
//    lowest possible order Gauss-Patterson rule is chosen that will achieve
//    that precision.  This retains a combination of the advantages of
//    nestedness and high accuracy.
//
//    The grid is defined as the sum of the product rules whose LEVEL
//    satisfies:
//
//      0 <= LEVEL <= LEVEL_MAX.
//
//    This calculation is much faster than a previous method.  It simply
//    computes the number of new points that are added at each level in the
//    1D rule, and then counts the new points at a given DIM_NUM dimensional
//    level vector as the product of the new points added in each dimension.
//
//    This approach will work for nested families, and may be extensible
//    to other families, and to mixed rules.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    25 December 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.
//
//  Parameters:
//
//    Input, int DIM_NUM, the spatial dimension.
//
//    Input, int LEVEL_MAX, the maximum value of LEVEL.
//
//    Output, int SPARSE_GRID_CC_SIZE, the number of points in the grid.
//
{
  int dim;
  int h;
  int j;
  int level;
  int *level_1d;
  bool more;
  int *new_1d;
  int o;
  int *order_1d;
  int p;
  int point_num;
  int t;
  int v;
//
//  Special case.
//
  if ( level_max < 0 )
  {
    point_num = 0;
    return point_num;
  }

  if ( level_max == 0 )
  {
    point_num = 1;
    return point_num;
  }
//
//  Count the points in the 1D rule.
//
  order_1d = new int[level_max+1];
  order_1d[0] = 1;
  for ( level = 1; level <= level_max; level++ )
  {
    p = 5;
    o = 3;
    while ( p < 2 * level + 1 )
    {
      p = 2 * p + 1;
      o = 2 * o + 1;
    }
    order_1d[level] = o;
  }
//
//  Count the new points in the 1D rule.
//
  new_1d = new int[level_max+1];

  new_1d[0] = 1;
  for ( level = 1; level <= level_max; level++ )
  {
    new_1d[level] = order_1d[level] - order_1d[level-1];
  }
//
//  Count the number of points by counting the number of new points 
//  associated with each level vector.
//
  level_1d = new int[dim_num];

  point_num = 0;

  for ( level = 0; level <= level_max; level++ )
  {
    more = false;
    h = 0;
    t = 0;

    for ( ; ;)
    {
      comp_next ( level, dim_num, level_1d, &more, &h, &t );

      v = 1;
      for ( dim = 0; dim < dim_num; dim++ )
      {
        v = v * new_1d[level_1d[dim]];
      }

      point_num = point_num + v;

      if ( !more )
      {
        break;
      }
    }
  }
  delete [] level_1d;
  delete [] new_1d;
  delete [] order_1d;

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

int sparse_grid_ofn_size ( int dim_num, int level_max )

//****************************************************************************80
//
//  Purpose:
//
//    SPARSE_GRID_OFN_SIZE sizes a sparse grid using Open Fully Nested rules.
//
//  Discussion:
//
//    The grid is defined as the sum of the product rules whose LEVEL
//    satisfies:
//
//      0 <= LEVEL <= LEVEL_MAX.
//
//    This calculation is much faster than a previous method.  It simply
//    computes the number of new points that are added at each level in the
//    1D rule, and then counts the new points at a given DIM_NUM dimensional
//    level vector as the product of the new points added in each dimension.
//
//    This approach will work for nested families, and may be extensible
//    to other families, and to mixed rules.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    23 December 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.
//
//  Parameters:
//
//    Input, int DIM_NUM, the spatial dimension.
//
//    Input, int LEVEL_MAX, the maximum value of LEVEL.
//
//    Output, int SPARSE_GRID_OFN_SIZE, the number of points in the grid.
//
{
  int dim;
  int h;
  int j;
  int l;
  int level;
  int *level_1d;
  bool more;
  int *new_1d;
  int point_num;
  int t;
  int v;
//
//  Special case.
//
  if ( level_max < 0 )
  {
    point_num = 0;
    return point_num;
  }

  if ( level_max == 0 )
  {
    point_num = 1;
    return point_num;
  }
//
//  Construct the vector that counts the new points in the 1D rule.
//
  new_1d = new int[level_max+1];

  new_1d[0] = 1;
  for ( l = 1; l <= level_max; l++ )
  {
    new_1d[l] = 2 * new_1d[l-1];
  }
//
//  Count the number of points by counting the number of new points 
//  associated with each level vector.
//
  level_1d = new int[dim_num];

  point_num = 0;

  for ( level = 0; level <= level_max; level++ )
  {
    more = false;
    h = 0;
    t = 0;

    for ( ; ;)
    {
      comp_next ( level, dim_num, level_1d, &more, &h, &t );

      v = 1;
      for ( dim = 0; dim < dim_num; dim++ )
      {
        v = v * new_1d[level_1d[dim]];
      }

      point_num = point_num + v;

      if ( !more )
      {
        break;
      }
    }
  }
  delete [] level_1d;
  delete [] new_1d;

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

int sparse_grid_onn_size ( int dim_num, int level_max )

//****************************************************************************80
//
//  Purpose:
//
//    SPARSE_GRID_ONN_SIZE sizes a sparse grid using Open Non-Nested rules.
//
//  Discussion:
//
//    The grid is defined as the sum of the product rules whose LEVEL
//    satisfies:
//
//      0 <= LEVEL <= LEVEL_MAX.
//
//    This calculation is much faster than a previous method.  It simply
//    computes the number of new points that are added at each level in the
//    1D rule, and then counts the new points at a given DIM_NUM dimensional
//    level vector as the product of the new points added in each dimension.
//
//    This approach will work for nested families, and may be extensible
//    to other families, and to mixed rules.
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    15 January 2010
//
//  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, int DIM_NUM, the spatial dimension.
//
//    Input, int LEVEL_MAX, the maximum value of LEVEL.
//
//    Output, int SPARSE_GRID_ONN_SIZE, the number of points in the grid.
//
{
  int dim;
  int h;
  int j;
  int l;
  int level;
  int *level_1d;
  int level_min;
  bool more;
  int *order_1d;
  int point_num;
  int t;
  int v;
//
//  Special case.
//
  if ( level_max < 0 )
  {
    point_num = 0;
    return point_num;
  }

  if ( level_max == 0 )
  {
    point_num = 1;
    return point_num;
  }
//
//  Construct the 1D order vector.
//
  order_1d = new int[level_max+1];

  for ( l = 0; l <= level_max; l++ )
  {
    order_1d[l] = 2 * l + 1;
  }

  level_1d = new int[dim_num];

  level_min = i4_max ( 0, level_max + 1 - dim_num );

  point_num = 0;

  for ( level = level_min; level <= level_max; level++ )
  {
    more = false;
    h = 0;
    t = 0;

    for ( ; ;)
    {
      comp_next ( level, dim_num, level_1d, &more, &h, &t );

      v = 1;
      for ( dim = 0; dim < dim_num; dim++ )
      {
        v = v * order_1d[level_1d[dim]];
      }

      point_num = point_num + v;

      if ( !more )
      {
        break;
      }
    }
  }
  delete [] level_1d;
  delete [] order_1d;

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

int sparse_grid_own_size ( int dim_num, int level_max )

//****************************************************************************80
//
//  Purpose:
//
//    SPARSE_GRID_OWN_SIZE sizes a sparse grid using Open Weakly Nested rules.
//
//  Discussion:
//
//    This calculation is much faster than a previous method.  
//
//    This calculation assumes that a linear growth rule is being used,
//    that is, that the 1D rules have orders 1, 3, 5, 7, 9, and so on.
//
//    This calculation assumes that the 1D family of quadrature rules 
//    contains only one repeated point, presumably the value 0.0.
//    This assumption holds for Gauss-Legendre, Gauss-Hermite and 
//    Generalized Gauss-Hermite rules.
//
//    The routine then counts the number of unique abscissas that will
//    be generated for a sparse grid of given dimension and level.
//
//    The computation is complicated.  It starts by counting just those
//    abscissas which have no 0.0 in them.  This is relatively easy, since
//    it is like counting the points in a sparse grid that uses open 
//    non-nested rules, but for which the order of each rule is reduced by 1.
//
//    Then we have to count the abscissas with one 0.0, two 0.0's and so
//    on to DIM_NUM zeros.  We are assuming this is an isotropic grid,
//    so for a particular number K of zeroes we only need to count the case
//    where the first K entries are zero, and multiply by C(DIM_NUM,K).
//
//    To count the number of entries with K zeroes, (and assuming 0 < K),
//    then, we essentially count the number of abscissas in an open 
//    non-nested rule as before, but modifed so that the minimum level is 0,
//    rather than LEVEL_MAX - DIM_NUM + 1.
//
//    I will mention that this was a rather difficult computation to
//    figure out!
//
//  Licensing:
//
//    This code is distributed under the GNU LGPL license. 
//
//  Modified:
//
//    12 January 2010
//
//  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, int DIM_NUM, the spatial dimension.
//
//    Input, int LEVEL_MAX, the maximum value of LEVEL.
//
//    Output, int SPARSE_GRID_OWN_SIZE, the number of points in the grid.
//
{
  int dim;
  int dim_num2;
  int h;
  int j;
  int l;
  int level;
  int level_min;
  int *level_1d;
  bool more;
  int *new_1d;
  int point_num;
  int point_num2;
  int t;
  int v;
//
//  Special case.
//
  if ( level_max < 0 )
  {
    point_num = 0;
    return point_num;
  }

  if ( level_max == 0 )
  {
    point_num = 1;
    return point_num;
  }
//
//  Construct the vector that counts the new points in the 1D rule.
//
  new_1d = new int[level_max+1];

  new_1d[0] = 0;
  for ( l = 1; l <= level_max; l++ )
  {
    new_1d[l] = 2 * l;
  }
//
//  Count the nonzero points in the full dimensional table with the usual
//  LEVEL_MIN restriction.
//
//  Then count the points with 1, 2, 3, ... DIM_NUM zeroes, by counting
//  the nonzero points in a DIM_NUM2 table, with LEVEL_MIN set to 0, and
//  multiplying by the appropriate combinatorial coefficient.
//
  point_num = 0;

  for ( dim_num2 = dim_num; 0 <= dim_num2; dim_num2-- )
  {
    if ( dim_num2 == dim_num )
    {
      level_min = i4_max ( 0, level_max - dim_num + 1 );
    }
    else
    {
      level_min = 0;
    }

    if ( dim_num2 == 0 )
    {
      point_num2 = 1;
    }
    else
    {
      level_1d = new int[dim_num2];

      point_num2 = 0;

      for ( level = level_min; level <= level_max; level++ )
      {
        more = false;
        h = 0;
        t = 0;

        for ( ; ; )
        {
          comp_next ( level, dim_num2, level_1d, &more, &h, &t );

          v = 1;
          for ( dim = 0; dim < dim_num2; dim++ )
          {
            v = v * new_1d[level_1d[dim]];
          }

          point_num2 = point_num2 + v;

          if ( !more )
          {
            break;
          }
        }
      }
      delete [] level_1d;
    }
    point_num = point_num + i4_choose ( dim_num, dim_num2 ) * point_num2;
  }
  delete [] new_1d;

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

void vec_colex_next2 ( int dim_num, int base[], int a[], bool *more )

//****************************************************************************80
//
//  Purpose:
//
//    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)-1,0,        ...,0)
//
//    (0,        1,        ...,0)
//    (1,        1,        ...,0)
//    ...
//    (BASE(1)-1,1,        ...,0)
//
//    (0,        2,        ...,0)
//    (1,        2,        ...,0)
//    ...
//    (BASE(1)-1,BASE(2)-1,...,BASE(DIM_NUM)-1).
//
//  Examples:
//
//    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:
//
//    25 May 2007
//
//  Author:
//
//    John Burkardt
//
//  Parameters:
//
//    Input, int DIM_NUM, the spatial dimension.
//
//    Input, int BASE[DIM_NUM], the bases to be used in each dimension.
//    In dimension I, entries will range from 0 to BASE[I]-1.
//
//    Output, int A[DIM_NUM], the next vector.
//
//    Input/output, bool *MORE.  Set this variable false before
//    the first call.  On return, MORE is TRUE if another vector has
//    been computed.  If MORE is returned FALSE, ignore the output 
//    vector and stop calling the routine.
//
{
  int i;

  if ( !( *more ) )
  {
    for ( i = 0; i < dim_num; i++ )
    {
      a[i] = 0;
    }
    *more = true;
  }
  else
  {
    for ( i = 0; i < dim_num; i++ )
    {
      a[i] = a[i] + 1;

      if ( a[i] < base[i] )
      {
        return;
      }
      a[i] = 0;
    }
    *more = false;
  }

  return;
}