bspline.c

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      /* find all non-zero d^j/dx^j B_i(x) values */
      error =
	gsl_bspline_deriv_eval_nonzero (x, nderiv, dw->dB, &istart, &iend, w,
					dw);
      if (error)
	{
	  return error;
	}

      /* store values in appropriate part of given matrix */
      for (j = 0; j <= nderiv; j++)
	{
	  for (i = 0; i < istart; i++)
	    gsl_matrix_set (dB, i, j, 0.0);

	  for (i = istart; i <= iend; i++)
	    gsl_matrix_set (dB, i, j, gsl_matrix_get (dw->dB, i - istart, j));

	  for (i = iend + 1; i < w->n; i++)
	    gsl_matrix_set (dB, i, j, 0.0);
	}

      return GSL_SUCCESS;
    }
}				/* gsl_bspline_deriv_eval() */

/*
gsl_bspline_deriv_eval_nonzero()
  At point x evaluate all requested, non-zero B-spline function
derivatives and store them in dB.  These are the
d^j/dx^j B_i(x) with i in [istart, iend] and j in [0, nderiv].
Always d^j/dx^j B_i(x) = 0 for i < istart and for i > iend.

Inputs: x      - point at which to evaluate splines
        nderiv - number of derivatives to request, inclusive
        dB     - (output) where to store dB-spline derivatives
                 (size k by nderiv + 1)
        istart - (output) B-spline function index of
                 first non-zero basis for given x
        iend   - (output) B-spline function index of
                 last non-zero basis for given x.
                 This is also the knot index corresponding to x.
        w      - bspline derivative workspace

Return: success or error

Notes: 1) the w->knots vector must be initialized before calling
          this function

       2) On output, dB contains

            [[B_{istart,  k}, ..., d^nderiv/dx^nderiv B_{istart  ,k}],
             [B_{istart+1,k}, ..., d^nderiv/dx^nderiv B_{istart+1,k}],
             ...
             [B_{iend-1,  k}, ..., d^nderiv/dx^nderiv B_{iend-1,  k}],
             [B_{iend,    k}, ..., d^nderiv/dx^nderiv B_{iend,    k}]]

          evaluated at x.  B_{istart, k} is stored in dB(0,0).
          Each additional column contains an additional derivative.

       3) Note that the zero-th column of the result contains the
          0th derivative, which is simply a function evaluation.

       4) based on PPPACK's bsplvd
*/

int
gsl_bspline_deriv_eval_nonzero (const double x, const size_t nderiv,
				gsl_matrix * dB, size_t * istart,
				size_t * iend, gsl_bspline_workspace * w,
				gsl_bspline_deriv_workspace * dw)
{
  if (dB->size1 != w->k)
    {
      GSL_ERROR ("dB matrix first dimension not of length k", GSL_EBADLEN);
    }
  else if (dB->size2 < nderiv + 1)
    {
      GSL_ERROR
	("dB matrix second dimension must be at least length nderiv+1",
	 GSL_EBADLEN);
    }
  else if (dw->k < w->k) 
    {
      GSL_ERROR ("derivative workspace is too small", GSL_EBADLEN);
    }
  else
    {
      size_t i;			/* spline index */
      size_t j;			/* looping */
      int flag = 0;		/* interval search flag */
      int error = 0;		/* error flag */
      size_t min_nderivk;

      i = bspline_find_interval (x, &flag, w);
      error = bspline_process_interval_for_eval (x, &i, flag, w);
      if (error)
	{
	  return error;
	}

      *istart = i - w->k + 1;
      *iend = i;

      bspline_pppack_bsplvd (w->knots, w->k, x, *iend,
			     w->deltal, w->deltar, dw->A, dB, nderiv);

      /* An order k b-spline has at most k-1 nonzero derivatives
         so we need to zero all requested higher order derivatives */
      min_nderivk = GSL_MIN_INT (nderiv, w->k - 1);
      for (j = min_nderivk + 1; j <= nderiv; j++)
	{
	  for (i = 0; i < w->k; i++)
	    {
	      gsl_matrix_set (dB, i, j, 0.0);
	    }
	}

      return GSL_SUCCESS;
    }
}				/* gsl_bspline_deriv_eval_nonzero() */

/****************************************
 *          INTERNAL ROUTINES           *
 ****************************************/

/*
bspline_find_interval()
  Find knot interval such that t_i <= x < t_{i + 1}
where the t_i are knot values.

Inputs: x    - x value
        flag - (output) error flag
        w    - bspline workspace

Return: i (index in w->knots corresponding to left limit of interval)

Notes: The error conditions are reported as follows:

       Condition                        Return value        Flag
       ---------                        ------------        ----
       x < t_0                               0               -1
       t_i <= x < t_{i+1}                    i                0
       t_i < x = t_{i+1} = t_{n+k-1}         i                0
       t_{n+k-1} < x                       l+k-1             +1
*/

static inline size_t
bspline_find_interval (const double x, int *flag, gsl_bspline_workspace * w)
{
  size_t i;

  if (x < gsl_vector_get (w->knots, 0))
    {
      *flag = -1;
      return 0;
    }

  /* find i such that t_i <= x < t_{i+1} */
  for (i = w->k - 1; i < w->k + w->l - 1; i++)
    {
      const double ti = gsl_vector_get (w->knots, i);
      const double tip1 = gsl_vector_get (w->knots, i + 1);

      if (tip1 < ti)
	{
	  GSL_ERROR ("knots vector is not increasing", GSL_EINVAL);
	}

      if (ti <= x && x < tip1)
	break;

      if (ti < x && x == tip1 && tip1 == gsl_vector_get (w->knots, w->k + w->l
							 - 1))
	break;
    }

  if (i == w->k + w->l - 1)
    *flag = 1;
  else
    *flag = 0;

  return i;
}				/* bspline_find_interval() */

/*
bspline_process_interval_for_eval()
  Consumes an x location, left knot from bspline_find_interval, flag
from bspline_find_interval, and a workspace.  Checks that x lies within
the splines' knots, enforces some endpoint continuity requirements, and
avoids divide by zero errors in the underlying bspline_pppack_* functions.
*/
static inline int
bspline_process_interval_for_eval (const double x, size_t * i, const int flag,
				   gsl_bspline_workspace * w)
{
  if (flag == -1)
    {
      GSL_ERROR ("x outside of knot interval", GSL_EINVAL);
    }
  else if (flag == 1)
    {
      if (x <= gsl_vector_get (w->knots, *i) + GSL_DBL_EPSILON)
	{
	  *i -= 1;
	}
      else
	{
	  GSL_ERROR ("x outside of knot interval", GSL_EINVAL);
	}
    }

  if (gsl_vector_get (w->knots, *i) == gsl_vector_get (w->knots, *i + 1))
    {
      GSL_ERROR ("knot(i) = knot(i+1) will result in division by zero",
		 GSL_EINVAL);
    }

  return GSL_SUCCESS;
}				/* bspline_process_interval_for_eval */

/********************************************************************
 * PPPACK ROUTINES
 *
 * The routines herein deliberately avoid using the bspline workspace,
 * choosing instead to pass all work areas explicitly.  This allows
 * others to more easily adapt these routines to low memory or
 * parallel scenarios.
 ********************************************************************/

/*
bspline_pppack_bsplvb()
  calculates the value of all possibly nonzero b-splines at x of order
jout = max( jhigh , (j+1)*(index-1) ) with knot sequence t.

Parameters:
   t      - knot sequence, of length left + jout , assumed to be
            nondecreasing.  assumption t(left).lt.t(left + 1).
            division by zero  will result if t(left) = t(left+1)
   jhigh  -
   index  - integers which determine the order jout = max(jhigh,
            (j+1)*(index-1))  of the b-splines whose values at x
            are to be returned.  index  is used to avoid
            recalculations when several columns of the triangular
            array of b-spline values are needed (e.g., in  bsplpp
            or in  bsplvd ).  precisely,

            if  index = 1 ,
               the calculation starts from scratch and the entire
               triangular array of b-spline values of orders
               1,2,...,jhigh  is generated order by order , i.e.,
               column by column .

            if  index = 2 ,
               only the b-spline values of order j+1, j+2, ..., jout
               are generated, the assumption being that biatx, j,
               deltal, deltar are, on entry, as they were on exit
               at the previous call.

            in particular, if jhigh = 0, then jout = j+1, i.e.,
            just the next column of b-spline values is generated.
   x      - the point at which the b-splines are to be evaluated.
   left   - an integer chosen (usually) so that
            t(left) .le. x .le. t(left+1).
   j      - (output) a working scalar for indexing
   deltal - (output) a working area which must be of length at least jout
   deltar - (output) a working area which must be of length at least jout
   biatx  - (output) array of length jout, with  biatx(i)
            containing the value at  x  of the polynomial of order
            jout which agrees with the b-spline b(left-jout+i,jout,t)
            on the interval (t(left), t(left+1)) .

Method:
   the recurrence relation

                      x - t(i)              t(i+j+1) - x
      b(i,j+1)(x) = -----------b(i,j)(x) + ---------------b(i+1,j)(x)
                    t(i+j)-t(i)            t(i+j+1)-t(i+1)

   is used (repeatedly) to generate the (j+1)-vector  b(left-j,j+1)(x),
   ...,b(left,j+1)(x)  from the j-vector  b(left-j+1,j)(x),...,
   b(left,j)(x), storing the new values in  biatx  over the old. the
   facts that

      b(i,1) = 1  if  t(i) .le. x .lt. t(i+1)

   and that

      b(i,j)(x) = 0  unless  t(i) .le. x .lt. t(i+j)

   are used. the particular organization of the calculations follows
   algorithm (8) in chapter x of [1].

Notes:

   (1) This is a direct translation of PPPACK's bsplvb routine with
       j, deltal, deltar rewritten as input parameters and
       utilizing zero-based indexing.

   (2) This routine contains no error checking.  Please use routines
       like gsl_bspline_eval().
*/

static void
bspline_pppack_bsplvb (const gsl_vector * t,
		       const size_t jhigh,
		       const size_t index,
		       const double x,
		       const size_t left,
		       size_t * j,
		       gsl_vector * deltal,
		       gsl_vector * deltar, gsl_vector * biatx)
{
  size_t i;			/* looping */
  double saved;
  double term;

  if (index == 1)
    {
      *j = 0;
      gsl_vector_set (biatx, 0, 1.0);
    }

  for ( /* NOP */ ; *j < jhigh - 1; *j += 1)
    {
      gsl_vector_set (deltar, *j, gsl_vector_get (t, left + *j + 1) - x);
      gsl_vector_set (deltal, *j, x - gsl_vector_get (t, left - *j));

      saved = 0.0;

      for (i = 0; i <= *j; i++)
	{
	  term = gsl_vector_get (biatx, i) / (gsl_vector_get (deltar, i)
					      + gsl_vector_get (deltal,
								*j - i));

	  gsl_vector_set (biatx, i,
			  saved + gsl_vector_get (deltar, i) * term);

	  saved = gsl_vector_get (deltal, *j - i) * term;
	}

      gsl_vector_set (biatx, *j + 1, saved);
    }

  return;
}				/* gsl_bspline_pppack_bsplvb */

/*
bspline_pppack_bsplvd()
  calculates value and derivs of all b-splines which do not vanish at x

Parameters:
   t      - the knot array, of length left+k (at least)
   k      - the order of the b-splines to be evaluated
   x      - the point at which these values are sought
   left   - an integer indicating the left endpoint of the interval
            of interest. the k b-splines whose support contains the
            interval (t(left), t(left+1)) are to be considered.
            it is assumed that t(left) .lt. t(left+1)
            division by zero will result otherwise (in  bsplvb).
            also, the output is as advertised only if
            t(left) .le. x .le. t(left+1) .
   deltal - a working area which must be of length at least k
   deltar - a working area which must be of length at least k
   a      - an array of order (k,k), to contain b-coeffs of the
            derivatives of a certain order of the k b-splines
            of interest.
   dbiatx - an array of order (k,nderiv). its entry (i,m) contains
            value of (m)th derivative of (left-k+i)-th b-spline
            of order k for knot sequence  t, i=1,...,k, m=0,...,nderiv.
   nderiv - an integer indicating that values of b-splines and
            their derivatives up to AND INCLUDING the nderiv-th
            are asked for. (nderiv is replaced internally by the
            integer mhigh in (1,k) closest to it.)

Method:
   values at x of all the relevant b-splines of order k,k-1,..., k+1-nderiv
   are generated via bsplvb and stored temporarily in dbiatx.  then, the
   b-coeffs of the required derivatives of the b-splines of interest are
   generated by differencing, each from the preceeding one of lower order,
   and combined with the values of b-splines of corresponding order in
   dbiatx  to produce the desired values .

Notes:

   (1) This is a direct translation of PPPACK's bsplvd routine with
       deltal, deltar rewritten as input parameters (to later feed them
       to bspline_pppack_bsplvb) and utilizing zero-based indexing.

   (2) This routine contains no error checking.
*/

static void
bspline_pppack_bsplvd (const gsl_vector * t,
		       const size_t k,
		       const double x,
		       const size_t left,
		       gsl_vector * deltal,
		       gsl_vector * deltar,
		       gsl_matrix * a,
		       gsl_matrix * dbiatx, const size_t nderiv)
{
  int i, ideriv, il, j, jlow, jp1mid, kmm, ldummy, m, mhigh;
  double factor, fkmm, sum;

  size_t bsplvb_j;
  gsl_vector_view dbcol = gsl_matrix_column (dbiatx, 0);

  mhigh = GSL_MIN_INT (nderiv, k - 1);
  bspline_pppack_bsplvb (t, k - mhigh, 1, x, left, &bsplvb_j, deltal, deltar,
			 &dbcol.vector);
  if (mhigh > 0)
    {
      /* the first column of dbiatx always contains the b-spline
         values for the current order. these are stored in column
         k-current order before bsplvb is called to put values
         for the next higher order on top of it.  */
      ideriv = mhigh;
      for (m = 1; m <= mhigh; m++)
	{
	  for (j = ideriv, jp1mid = 0; j < (int) k; j++, jp1mid++)
	    {
	      gsl_matrix_set (dbiatx, j, ideriv,
			      gsl_matrix_get (dbiatx, jp1mid, 0));
	    }
	  ideriv--;
	  bspline_pppack_bsplvb (t, k - ideriv, 2, x, left, &bsplvb_j, deltal,
				 deltar, &dbcol.vector);
	}

      /* at this point,  b(left-k+i, k+1-j)(x) is in  dbiatx(i,j)
         for i=j,...,k-1 and j=0,...,mhigh. in particular, the
         first column of dbiatx is already in final form. to obtain
         corresponding derivatives of b-splines in subsequent columns,
         generate their b-repr. by differencing, then evaluate at x. */
      jlow = 0;
      for (i = 0; i < (int) k; i++)
	{
	  for (j = jlow; j < (int) k; j++)
	    {
	      gsl_matrix_set (a, j, i, 0.0);
	    }
	  jlow = i;
	  gsl_matrix_set (a, i, i, 1.0);
	}

      /* at this point, a(.,j) contains the b-coeffs for the j-th of the
         k b-splines of interest here. */
      for (m = 1; m <= mhigh; m++)
	{
	  kmm = k - m;
	  fkmm = (float) kmm;
	  il = left;
	  i = k - 1;

	  /* for j=1,...,k, construct b-coeffs of (m)th  derivative
	     of b-splines from those for preceding derivative by
	     differencing and store again in  a(.,j) . the fact that
	     a(i,j) = 0  for i .lt. j  is used. */
	  for (ldummy = 0; ldummy < kmm; ldummy++)
	    {
	      factor =
		fkmm / (gsl_vector_get (t, il + kmm) -
			gsl_vector_get (t, il));
	      /* the assumption that t(left).lt.t(left+1) makes
	         denominator in factor nonzero. */
	      for (j = 0; j <= i; j++)
		{
		  gsl_matrix_set (a, i, j, factor * (gsl_matrix_get (a, i, j)
						     - gsl_matrix_get (a,
								       i - 1,
								       j)));
		}
	      il--;
	      i--;
	    }

	  /* for i=1,...,k, combine b-coeffs a(.,i) with b-spline values
	     stored in dbiatx(.,m) to get value of (m)th  derivative
	     of i-th b-spline (of interest here) at x, and store in
	     dbiatx(i,m). storage of this value over the value of a
	     b-spline of order m there is safe since the remaining
	     b-spline derivatives of the same order do not use this
	     value due to the fact that a(j,i) = 0 for j .lt. i . */
	  for (i = 0; i < (int) k; i++)
	    {
	      sum = 0;
	      jlow = GSL_MAX_INT (i, m);
	      for (j = jlow; j < (int) k; j++)
		{
		  sum +=
		    gsl_matrix_get (a, j, i) * gsl_matrix_get (dbiatx, j, m);
		}
	      gsl_matrix_set (dbiatx, i, m, sum);
	    }
	}
    }

  return;

}				/* bspline_pppack_bsplvd */