vandermonde.c

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

# include "vandermonde.h"

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

double *bivand1 ( int n, double alpha[], double beta[] )

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

    BIVAND1 returns a bidimensional Vandermonde1 matrix.

  Discussion:

    N = 3, ALPHA = ( 1, 2, 3 ), BETA = ( 10, 20, 30 )

    (x,y)   | (1,10)  (2,10)  (3,10)  (1,20)  (2,20)  (1,30)
    --------+-----------------------------------------------
    1       |     1       1       1       1       1       1  
    x       |     1       2       3       1       2       1
       y    |    10      10      10      20      20      30
    x^2     |     1       4       9       1       4       1
    x  y    |    10      20      30      20      40      30
    x^2y^2  |   100     100     100     400     400     900

  Licensing:

    This code is distributed under the GNU LGPL license.

  Modified:

    23 February 2014

  Author:

    John Burkardt

  Parameters:

    Input, int N, the order of the data vectors.

    Input, double ALPHA[N], BETA[N], the values that define A.

    Output, double BIVAND1[((N+1)*N)/2*((N+1)*N)/2], the matrix.
*/
{
  double *a;
  int e;
  int e1;
  int e2;
  int i1;
  int i2;
  int ii;
  int j1;
  int j2;
  int jj;
  int n2;

  n2 = ( n * ( n + 1 ) ) / 2;
  a = ( double * ) malloc ( n2 * n2 * sizeof ( double ) );

  e1 = 0;
  e2 = 0;
  e = 0;

  for ( ii = 0; ii < n2; ii++ )
  {
    j1 = 0;
    j2 = 0;
    for ( jj = 0; jj < n2; jj++ )
    {
      if ( ii == 0 )
      {
        a[ii+jj*n2] = 1.0;
      }
      else
      {
        a[ii+jj*n2] = pow ( alpha[j1], e1 ) * pow ( beta[j2], e2 );
      }

      if ( j1 + j2 < n - 1 )
      {
        j1 = j1 + 1;
      }
      else
      {
        j1 = 0;
        j2 = j2 + 1;
      }
    }

    if ( e2 < e )
    {
      e1 = e1 - 1;
      e2 = e2 + 1;
    }
    else
    {
      e = e + 1;
      e1 = e;
      e2 = 0;
    }
  }

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

double *bivand2 ( int n, double alpha[], double beta[] )

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

    BIVAND2 returns a bidimensional Vandermonde1 matrix.

  Discussion:

    N = 3, ALPHA = ( 1, 2, 3 ), BETA = ( 10, 20, 30 )

    (x,y)   | (1,10) (2,10) (3,10) (1,20) (2,20) (3,20) (1,30) (2,30) (3,30)
    --------+---------------------------------------------------------------
    1       |     1      1      1      1      1      1      1      1      1  
    x       |     1      2      3      1      2      1      1      2      3
    x^2     |     1      4      9      1      4      1      1      4      9
       y    |    10     10     10     20     20     20     30     30     30
    x  y    |    10     20     30     20     40     60     30     60     90
    x^2y    |    10     40     90     20     80    180     30    120    270
       y^2  |   100    100    100    400    400    400    900    900    900
    x  y^2  |   100    200    300    400    800   1200    900   1800   2700
    x^2y^2  |   100    400    900    400   1600   3600    900   3600   8100

  Licensing:

    This code is distributed under the GNU LGPL license.

  Modified:

    01 May 2014

  Author:

    John Burkardt

  Parameters:

    Input, int N, the order of the data vectors.

    Input, double ALPHA[N], BETA[N], the values that define A.

    Output, double BIVAND2[(N*N)*(N*N)], the matrix.
*/
{
  double *a;
  int i;
  int ix;
  int iy;
  int j;
  int jx;
  int jy;

  a = ( double * ) malloc ( n * n * n * n * sizeof ( double ) );

  i = 0;
  for ( iy = 0; iy < n; iy++ )
  {
    for ( ix = 0; ix < n; ix++ )
    {
      j = 0;
      for ( jy = 0; jy < n; jy++ )
      {
        for ( jx = 0; jx < n; jx++ )
        {
          a[i+j*n*n] = pow ( alpha[jx], ix ) * pow ( beta[jy], iy );
          j = j + 1;
        }
      }
      i = i + 1;
    }
  }

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

double *dvand ( int n, double alpha[], double b[] )

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

    DVAND solves a Vandermonde system A' * x = b.

  Licensing:

    This code is distributed under the GNU LGPL license.

  Modified:

    23 February 2014

  Author:

    John Burkardt

  Reference:

    Ake Bjorck, Victor Pereyra,
    Solution of Vandermonde Systems of Equations,
    Mathematics of Computation,
    Volume 24, Number 112, October 1970, pages 893-903.

  Parameters:

    Input, int N, the order of the matrix.

    Input, double ALPHA[N], the parameters that define the matrix.
    The values should be distinct.

    Input, double B[N], the right hand side of the linear system.

    Output, double DVAND[N], the solution of the linear system.
*/
{
  int j;
  int k;
  double *x;

  x = r8vec_copy_new ( n, b );

  for ( k = 0; k < n - 1; k++ )
  {
    for ( j = n - 1; k < j; j-- )
    {
      x[j] = ( x[j] - x[j-1] ) / ( alpha[j] - alpha[j-k-1] );
    }
  }

  for ( k = n - 2; 0 <= k; k-- )
  {
    for ( j = k; j < n - 1; j++ )
    {
      x[j] = x[j] - alpha[k] * x[j+1];
    }
  }

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

void dvandprg ( int n, double alpha[], double b[], double x[], double c[], 
  double m[] )

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

    DVANDPRG solves a Vandermonde system A' * x = f progressively.

  Discussion:

    This function receives the solution to the system of equations A' * x = f
    where A is a Vandermonde matrix for alpha(0) through alpha(n-1),
    and new values alpha(n) and f(n).  It updates the solution.

    To solve a system of Nbig equations, this function may be called 
    repeatedly, with N = 1, 2, ..., Nbig.  Each time, a solution to the 
    current subsystem is returned.

  Licensing:

    This code is distributed under the GNU LGPL license.

  Modified:

    18 April 2014

  Author:

    John Burkardt

  Reference:

    Ake Bjorck, Victor Pereyra,
    Solution of Vandermonde Systems of Equations,
    Mathematics of Computation,
    Volume 24, Number 112, October 1970, pages 893-903.

  Parameters:

    Input, int N, the new order of the matrix, which is 1 
    larger than on the previous call.  For the first call, N must be 1.

    Input, double ALPHA[N], the parameters that define the matrix.
    The values should be distinct.  The value ALPHA(N) has just been
    added to the system.

    Input, double B[N], the right hand side of the linear system.

    Input/output, double X[N].  On input, the first N-1 entries 
    contain the solution of the N-1xN-1 linear system.  On output, the 
    solution to the NxN linear system.

    Input/output, double C[N], M[N].  On input, the first N-1 
    entries contain factorization data for the N-1xN-1 linear system.  On 
    output, factorization data for the NxN linear system.
*/
{
  double cn;
  int j;
 
  c[n-1] = b[n-1];
  for ( j = n - 1; 1 <= j; j-- )
  {
    c[j-1] = ( c[j] - c[j-1] ) / ( alpha[n-1] - alpha[j-1] );
  }

  if ( n == 1 )
  {
    m[n-1] = 1.0;
  }
  else
  {
    m[n-1] = 0.0;
  }

  cn = c[0];
  x[n-1] = c[0];

  for ( j = n - 1; 1 <= j; j-- )
  {
    m[j] = m[j] - alpha[n-2] * m[j-1];
    x[n-j-1] = x[n-j-1] + m[j] * cn;
  }

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

double *pvand ( int n, double alpha[], double b[] )

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

    PVAND solves a Vandermonde system A * x = b.

  Licensing:

    This code is distributed under the GNU LGPL license.

  Modified:

    23 February 2014

  Author:

    John Burkardt

  Reference:

    Ake Bjorck, Victor Pereyra,
    Solution of Vandermonde Systems of Equations,
    Mathematics of Computation,
    Volume 24, Number 112, October 1970, pages 893-903.

  Parameters:

    Input, int N, the order of the matrix.

    Input, double ALPHA[N], the parameters that define the matrix.
    The values should be distinct.

    Input, double B[N], the right hand side of the linear system.

    Output, double PVAND[N], the solution of the linear system.
*/
{
  int j;
  int k;
  double *x;

  x = r8vec_copy_new ( n, b );

  for ( k = 0; k < n - 1; k++ )
  {
    for ( j = n - 1; k < j; j-- )
    {
      x[j] = x[j] - alpha[k] * x[j-1];
    }
  }

  for ( k = n - 2; 0 <= k; k-- )
  {
    for ( j = k + 1; j < n; j++ )
    {
      x[j] = x[j] / ( alpha[j] - alpha[j-k-1] );
    }
    for ( j = k; j < n - 1; j++ )
    {
      x[j] = x[j] - x[j+1];
    }
  }

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

void pvandprg ( int n, double alpha[], double b[], double x[], double d[], 
  double u[] )

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

    PVANDPRG solves a Vandermonde system A * x = f progressively.

  Discussion:

    This function receives the solution to the system of equations A * x = f
    where A is a Vandermonde matrix for alpha(0) through alpha(n-1),
    and new values alpha(n) and f(n).  It updates the solution.

    To solve a system of Nbig equations, this function may be called 
    repeatedly, with N = 1, 2, ..., Nbig.  Each time, a solution to the 
    current subsystem is returned.

    Note that the reference, which lists an Algol version of this algorithm, 
    omits a minus sign, writing
      u[j] := u[j] x delta;
    where
      u[j] := - u[j] x delta;
    is actually necessary.  

  Licensing:

    This code is distributed under the GNU LGPL license.

  Modified:

    18 April 2014

  Author:

    John Burkardt

  Reference:

    Ake Bjorck, Victor Pereyra,
    Solution of Vandermonde Systems of Equations,
    Mathematics of Computation,
    Volume 24, Number 112, October 1970, pages 893-903.

  Parameters:

    Input, int N, the new order of the matrix, which is 1 
    larger than on the previous call.  For the first call, N must be 1.

    Input, double ALPHA[N], the parameters that define the matrix.
    The values should be distinct.  The value ALPHA(N) has just been
    added to the system.

    Input, double B[N], the right hand side of the linear system.

    Input/output, double X[N]; on input, the solution of the 
    N-1xN-1 linear system.  On output, the solution of the NxN linear system.

    Input/output, double D[N], U[N]; on input, factorization data 
    for the N-1xN-1 linear system.  On output, factorization data for the
    NxN linear system.
*/
{
  double delta;
  double dn;
  int j;

  d[n-1] = b[n-1];
  for ( j = n - 1; 1 <= j; j-- )
  {
    d[j-1] = d[j] - alpha[n-j-1] * d[j-1];
  }

  dn = d[0];
  u[n-1] = 1.0;

  for ( j = 1; j <= n - 1; j++ )
  {
    delta = alpha[n-1] - alpha[j-1];
    u[j-1] = - u[j-1] * delta;
    u[n-1] = u[n-1] * delta;
    x[j-1] = x[j-1] + dn / u[j-1];
  }

  x[n-1] = dn / u[n-1];

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

double *r8mat_mtv_new ( int m, int n, double a[], double x[] )

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

    R8MAT_MTV_NEW multiplies a transposed matrix times a vector.

  Discussion:

    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
    in column-major order.

    For this routine, the result is returned as the function value.

  Licensing:

    This code is distributed under the GNU LGPL license.

  Modified:

    26 August 2011

  Author:

    John Burkardt

  Parameters:

    Input, int M, N, the number of rows and columns of the matrix.

    Input, double A[M,N], the M by N matrix.

    Input, double X[M], the vector to be multiplied by A.

    Output, double R8MAT_MTV_NEW[N], the product A'*X.
*/
{
  int i;
  int j;
  double *y;

  y = ( double * ) malloc ( n * sizeof ( double ) );

  for ( j = 0; j < n; j++ )
  {
    y[j] = 0.0;
    for ( i = 0; i < m; i++ )
    {
      y[j] = y[j] + a[i+j*m] * x[i];
    }
  }

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

double *r8mat_mv_new ( int m, int n, double a[], double x[] )

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

    R8MAT_MV_NEW multiplies a matrix times a vector.

  Discussion:

    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
    in column-major order.

    For this routine, the result is returned as the function value.

  Licensing:

    This code is distributed under the GNU LGPL license.

  Modified:

    11 April 2007

  Author:

    John Burkardt

  Parameters:

    Input, int M, N, the number of rows and columns of the matrix.

    Input, double A[M,N], the M by N matrix.

    Input, double X[N], the vector to be multiplied by A.

    Output, double R8MAT_MV_NEW[M], the product A*X.
*/
{
  int i;
  int j;
  double *y;

  y = ( double * ) malloc ( m * sizeof ( double ) );

  for ( i = 0; i < m; i++ )
  {
    y[i] = 0.0;
    for ( j = 0; j < n; j++ )
    {
      y[i] = y[i] + a[i+j*m] * x[j];
    }
  }

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

void r8mat_print ( int m, int n, double a[], char *title )

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

    R8MAT_PRINT prints an R8MAT.

  Discussion:

    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
    in column-major order.

    Entry A(I,J) is stored as A[I+J*M]

  Licensing:

    This code is distributed under the GNU LGPL license.

  Modified:

    28 May 2008

  Author:

    John Burkardt

  Parameters:

    Input, int M, the number of rows in A.

    Input, int N, the number of columns in A.

    Input, double A[M*N], the M by N matrix.

    Input, char *TITLE, a title.
*/
{
  r8mat_print_some ( m, n, a, 1, 1, m, n, title );

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

void r8mat_print_some ( int m, int n, double a[], int ilo, int jlo, int ihi,
  int jhi, char *title )

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

    R8MAT_PRINT_SOME prints some of an R8MAT.

  Discussion:

    An R8MAT is a doubly dimensioned array of R8 values, stored as a vector
    in column-major order.

  Licensing:

    This code is distributed under the GNU LGPL license.

  Modified:

    26 June 2013

  Author:

    John Burkardt

  Parameters:

    Input, int M, the number of rows of the matrix.
    M must be positive.

    Input, int N, the number of columns of the matrix.
    N must be positive.

    Input, double A[M*N], the matrix.

    Input, int ILO, JLO, IHI, JHI, designate the first row and
    column, and the last row and column to be printed.

    Input, char *TITLE, a title.
*/
{
# define INCX 5

  int i;
  int i2hi;
  int i2lo;
  int j;
  int j2hi;
  int j2lo;

  fprintf ( stdout, "\n" );
  fprintf ( stdout, "%s\n", title );

  if ( m <= 0 || n <= 0 )
  {
    fprintf ( stdout, "\n" );
    fprintf ( stdout, "  (None)\n" );
    return;
  }
/*
  Print the columns of the matrix, in strips of 5.
*/
  for ( j2lo = jlo; j2lo <= jhi; j2lo = j2lo + INCX )
  {
    j2hi = j2lo + INCX - 1;
    if ( n < j2hi )
    {
      j2hi = n;
    }
    if ( jhi < j2hi )
    {
      j2hi = jhi;
    }

    fprintf ( stdout, "\n" );
/*
  For each column J in the current range...

  Write the header.
*/
    fprintf ( stdout, "  Col:  ");
    for ( j = j2lo; j <= j2hi; j++ )
    {
      fprintf ( stdout, "  %7d     ", j - 1 );
    }
    fprintf ( stdout, "\n" );
    fprintf ( stdout, "  Row\n" );
    fprintf ( stdout, "\n" );
/*
  Determine the range of the rows in this strip.
*/
    if ( 1 < ilo )
    {
      i2lo = ilo;
    }
    else
    {
      i2lo = 1;
    }
    if ( m < ihi )
    {
      i2hi = m;
    }
    else
    {
      i2hi = ihi;
    }

    for ( i = i2lo; i <= i2hi; i++ )
    {
/*
  Print out (up to) 5 entries in row I, that lie in the current strip.
*/
      fprintf ( stdout, "%5d:", i - 1 );
      for ( j = j2lo; j <= j2hi; j++ )
      {
        fprintf ( stdout, "  %14g", a[i-1+(j-1)*m] );
      }
      fprintf ( stdout, "\n" );
    }
  }

  return;
# undef INCX
}
/******************************************************************************/

double *r8vec_copy_new ( int n, double a1[] )

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

    R8VEC_COPY_NEW copies an R8VEC.

  Discussion:

    An R8VEC is a vector of R8's.

  Licensing:

    This code is distributed under the GNU LGPL license.

  Modified:

    26 August 2008

  Author:

    John Burkardt

  Parameters:

    Input, int N, the number of entries in the vectors.

    Input, double A1[N], the vector to be copied.

    Output, double R8VEC_COPY_NEW[N], the copy of A1.
*/
{
  double *a2;
  int i;

  a2 = ( double * ) malloc ( n * sizeof ( double ) );

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

void r8vec_print ( int n, double a[], char *title )

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

    R8VEC_PRINT prints an R8VEC.

  Discussion:

    An R8VEC is a vector of R8's.

  Licensing:

    This code is distributed under the GNU LGPL license.

  Modified:

    08 April 2009

  Author:

    John Burkardt

  Parameters:

    Input, int N, the number of components of the vector.

    Input, double A[N], the vector to be printed.

    Input, char *TITLE, a title.
*/
{
  int i;

  fprintf ( stdout, "\n" );
  fprintf ( stdout, "%s\n", title );
  fprintf ( stdout, "\n" );
  for ( i = 0; i < n; i++ )
  {
    fprintf ( stdout, "  %8d: %14g\n", i, a[i] );
  }

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

double *r8vec_uniform_01_new ( int n, int *seed )

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

    R8VEC_UNIFORM_01_NEW returns a unit pseudorandom R8VEC.

  Discussion:

    This routine implements the recursion

      seed = 16807 * seed mod ( 2^31 - 1 )
      unif = seed / ( 2^31 - 1 )

    The integer arithmetic never requires more than 32 bits,
    including a sign bit.

  Licensing:

    This code is distributed under the GNU LGPL license.

  Modified:

    19 August 2004

  Author:

    John Burkardt

  Reference:

    Paul Bratley, Bennett Fox, Linus Schrage,
    A Guide to Simulation,
    Second Edition,
    Springer, 1987,
    ISBN: 0387964673,
    LC: QA76.9.C65.B73.

    Bennett Fox,
    Algorithm 647:
    Implementation and Relative Efficiency of Quasirandom
    Sequence Generators,
    ACM Transactions on Mathematical Software,
    Volume 12, Number 4, December 1986, pages 362-376.

    Pierre L'Ecuyer,
    Random Number Generation,
    in Handbook of Simulation,
    edited by Jerry Banks,
    Wiley, 1998,
    ISBN: 0471134031,
    LC: T57.62.H37.

    Peter Lewis, Allen Goodman, James Miller,
    A Pseudo-Random Number Generator for the System/360,
    IBM Systems Journal,
    Volume 8, Number 2, 1969, pages 136-143.

  Parameters:

    Input, int N, the number of entries in the vector.

    Input/output, int *SEED, a seed for the random number generator.

    Output, double R8VEC_UNIFORM_01_NEW[N], the vector of pseudorandom values.
*/
{
  int i;
  const int i4_huge = 2147483647;
  int k;
  double *r;

  if ( *seed == 0 )
  {
    fprintf ( stderr, "\n" );
    fprintf ( stderr, "R8VEC_UNIFORM_01_NEW - Fatal error!\n" );
    fprintf ( stderr, "  Input value of SEED = 0.\n" );
    exit ( 1 );
  }

  r = ( double * ) malloc ( n * sizeof ( double ) );

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

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

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

    r[i] = ( double ) ( *seed ) * 4.656612875E-10;
  }

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

void timestamp ( )

/******************************************************************************/
/*
  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:

    24 September 2003

  Author:

    John Burkardt

  Parameters:

    None
*/
{
# define TIME_SIZE 40

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

  now = time ( NULL );
  tm = localtime ( &now );

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

  fprintf ( stdout, "%s\n", time_buffer );

  return;
# undef TIME_SIZE
}
/******************************************************************************/

double *vand1 ( int n, double x[] )

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

    VAND1 returns the Vandermonde1 matrix A with 1's on the first row.

  Formula:

    A(I,J) = X(J)^(I-1)

  Example:

    N = 5, X = ( 2, 3, 4, 5, 6 )

    1  1   1   1   1
    2  3   4   5   6
    4  9  16  25  36
    8 27  64 125  216
   16 81 256 625 1296

  Properties:

    A is generally not symmetric: A' /= A.

    A is nonsingular if, and only if, the X values are distinct.

    det ( A ) = product ( 1 <= I <= N ) ( 1 <= J .lt. I ) ( X(I) - X(J) ).
             = product ( 1 <= J <= N ) X(J)
             * product ( 1 <= I .lt. J ) ( X(J) - X(I) ).

    A is generally ill-conditioned.

  Licensing:

    This code is distributed under the GNU LGPL license.

  Modified:

    23 February 2014

  Author:

    John Burkardt

  Reference:

    Robert Gregory, David Karney,
    A Collection of Matrices for Testing Computational Algorithms,
    Wiley, 1969, page 27,
    LC: QA263.G68.

    Nicholas Higham,
    Stability analysis of algorithms for solving confluent
    Vandermonde-like systems,
    SIAM Journal on Matrix Analysis and Applications,
    Volume 11, 1990, pages 23-41.

  Parameters:

    Input, int N, the order of the matrix desired.

    Input, double X[N], the values that define A.

    Output, double VAND1[N*N], the matrix.
*/
{
  double *a;
  int i;
  int j;

  a = ( double * ) malloc ( n * n * sizeof ( double ) );

  for ( i = 0; i < n; i++ )
  {
    for ( j = 0; j < n; j++ )
    {
      if ( i == 0 && x[j] == 0.0 )
      {
        a[i+j*n] = 1.0;
      }
      else
      {
        a[i+j*n] = pow ( x[j], i );
      }
    }
  }

  return a;
}