Revision 13e45eaa4f4a7f231224f1b8ca4dff03903d250b authored by Mark Clements on 31 August 2017, 20:37:09 UTC, committed by cran-robot on 31 August 2017, 20:37:09 UTC
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bspline.c
/* Note:
* This file includs the original bspline.c of gsl/bspline. We
* merge all other necessary gsl routines to this file, so now we
* can compile and link it without linking to the gsl library.
* The source files are downloaded from http://www.gnu.org/software/gsl/,
* and the current version is 1.14.*
*
*
*
* */
/* bspline/bspline.c
*
* Copyright (C) 2006, 2007, 2008, 2009 Patrick Alken
* Copyright (C) 2008 Rhys Ulerich
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 3 of the License, or (at
* your option) any later version.
*
* This program is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with this program; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
*/
#include "gsl_bspline.h"
/*
* This module contains routines related to calculating B-splines.
* The algorithms used are described in
*
* [1] Carl de Boor, "A Practical Guide to Splines", Springer
* Verlag, 1978.
*
* The bspline_pppack_* internal routines contain code adapted from
*
* [2] "PPPACK - Piecewise Polynomial Package",
* http://www.netlib.org/pppack/
*
*/
/*
gsl_bspline_alloc()
Allocate space for a bspline workspace. The size of the
workspace is O(5k + nbreak)
Inputs: k - spline order (cubic = 4)
nbreak - number of breakpoints
Return: pointer to workspace
*/
/* ***************************** */
/* The following functions only can be called in this file because we define them as static.*/
static inline size_t bspline_find_interval (const double x, int *flag, gsl_bspline_workspace * w);
static inline int bspline_process_interval_for_eval (const double x, size_t * i, int flag,
gsl_bspline_workspace * w);
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);
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);
/* ***************************** */
gsl_bspline_workspace *
gsl_bspline_alloc (const size_t k, const size_t nbreak)
{
if (k == 0)
{
GSL_ERROR ("k must be at least 1", GSL_EINVAL);
}
else if (nbreak < 2)
{
GSL_ERROR ("nbreak must be at least 2", GSL_EINVAL);
}
else
{
gsl_bspline_workspace *w;
w = (gsl_bspline_workspace *) malloc (sizeof (gsl_bspline_workspace));
if (w == 0)
{
GSL_ERROR ("failed to allocate space for workspace",
GSL_ENOMEM);
}
w->k = k;
w->km1 = k - 1;
w->nbreak = nbreak;
w->l = nbreak - 1;
w->n = w->l + k - 1;
w->knots = gsl_vector_alloc (w->n + k);
if (w->knots == 0)
{
free (w);
GSL_ERROR ("failed to allocate space for knots vector",
GSL_ENOMEM);
}
w->deltal = gsl_vector_alloc (k);
if (w->deltal == 0)
{
gsl_vector_free (w->knots);
free (w);
GSL_ERROR ("failed to allocate space for deltal vector",
GSL_ENOMEM);
}
w->deltar = gsl_vector_alloc (k);
if (w->deltar == 0)
{
gsl_vector_free (w->deltal);
gsl_vector_free (w->knots);
free (w);
GSL_ERROR ("failed to allocate space for deltar vector",
GSL_ENOMEM);
}
w->B = gsl_vector_alloc (k);
if (w->B == 0)
{
gsl_vector_free (w->deltar);;
gsl_vector_free (w->deltal);
gsl_vector_free (w->knots);
free (w);
GSL_ERROR
("failed to allocate space for temporary spline vector",
GSL_ENOMEM);
}
return w;
}
return NULL;
} /* gsl_bspline_alloc() */
/*
gsl_bspline_deriv_alloc()
Allocate space for a bspline derivative workspace. The size of the
workspace is O(2k^2)
Inputs: k - spline order (cubic = 4)
Return: pointer to workspace
*/
gsl_bspline_deriv_workspace *
gsl_bspline_deriv_alloc (const size_t k)
{
if (k == 0)
{
GSL_ERROR ("k must be at least 1", GSL_EINVAL);
}
else
{
gsl_bspline_deriv_workspace *dw;
dw =
(gsl_bspline_deriv_workspace *)
malloc (sizeof (gsl_bspline_deriv_workspace));
if (dw == 0)
{
GSL_ERROR ("failed to allocate space for workspace",
GSL_ENOMEM);
}
dw->A = gsl_matrix_alloc (k, k);
if (dw->A == 0)
{
free (dw);
GSL_ERROR
("failed to allocate space for derivative work matrix",
GSL_ENOMEM);
}
dw->dB = gsl_matrix_alloc (k, k + 1);
if (dw->dB == 0)
{
gsl_matrix_free (dw->A);
free (dw);
GSL_ERROR
("failed to allocate space for temporary derivative matrix",
GSL_ENOMEM);
}
dw->k = k;
return dw;
}
return NULL;
} /* gsl_bspline_deriv_alloc() */
/* Return number of coefficients */
size_t
gsl_bspline_ncoeffs (gsl_bspline_workspace * w)
{
return w->n;
}
/* Return order */
size_t
gsl_bspline_order (gsl_bspline_workspace * w)
{
return w->k;
}
/* Return number of breakpoints */
size_t
gsl_bspline_nbreak (gsl_bspline_workspace * w)
{
return w->nbreak;
}
/* Return the location of the i-th breakpoint*/
double
gsl_bspline_breakpoint (size_t i, gsl_bspline_workspace * w)
{
size_t j = i + w->k - 1;
return gsl_vector_get (w->knots, j);
}
/*
gsl_bspline_free()
Free a gsl_bspline_workspace.
Inputs: w - workspace to free
Return: none
*/
void
gsl_bspline_free (gsl_bspline_workspace * w)
{
RETURN_IF_NULL (w);
gsl_vector_free (w->knots);
gsl_vector_free (w->deltal);
gsl_vector_free (w->deltar);
gsl_vector_free (w->B);
free (w);
} /* gsl_bspline_free() */
/*
gsl_bspline_deriv_free()
Free a gsl_bspline_deriv_workspace.
Inputs: dw - workspace to free
Return: none
*/
void
gsl_bspline_deriv_free (gsl_bspline_deriv_workspace * dw)
{
RETURN_IF_NULL (dw);
gsl_matrix_free (dw->A);
gsl_matrix_free (dw->dB);
free (dw);
} /* gsl_bspline_deriv_free() */
/*
gsl_bspline_knots()
Compute the knots from the given breakpoints:
knots(1:k) = breakpts(1)
knots(k+1:k+l-1) = breakpts(i), i = 2 .. l
knots(n+1:n+k) = breakpts(l + 1)
where l is the number of polynomial pieces (l = nbreak - 1) and
n = k + l - 1
(using matlab syntax for the arrays)
The repeated knots at the beginning and end of the interval
correspond to the continuity condition there. See pg. 119
of [1].
Inputs: breakpts - breakpoints
w - bspline workspace
Return: success or error
*/
int
gsl_bspline_knots (const gsl_vector * breakpts, gsl_bspline_workspace * w)
{
if (breakpts->size != w->nbreak)
{
GSL_ERROR ("breakpts vector has wrong size", GSL_EBADLEN);
}
else
{
size_t i; /* looping */
for (i = 0; i < w->k; i++)
gsl_vector_set (w->knots, i, gsl_vector_get (breakpts, 0));
for (i = 1; i < w->l; i++)
{
gsl_vector_set (w->knots, w->k - 1 + i,
gsl_vector_get (breakpts, i));
}
for (i = w->n; i < w->n + w->k; i++)
gsl_vector_set (w->knots, i, gsl_vector_get (breakpts, w->l));
return GSL_SUCCESS;
}
return GSL_FAILURE;
} /* gsl_bspline_knots() */
/*
gsl_bspline_knots_uniform()
Construct uniformly spaced knots on the interval [a,b] using
the previously specified number of breakpoints. 'a' is the position
of the first breakpoint and 'b' is the position of the last
breakpoint.
Inputs: a - left side of interval
b - right side of interval
w - bspline workspace
Return: success or error
Notes: 1) w->knots is modified to contain the uniformly spaced
knots
2) The knots vector is set up as follows (using octave syntax):
knots(1:k) = a
knots(k+1:k+l-1) = a + i*delta, i = 1 .. l - 1
knots(n+1:n+k) = b
*/
int
gsl_bspline_knots_uniform (const double a, const double b,
gsl_bspline_workspace * w)
{
size_t i; /* looping */
double delta; /* interval spacing */
double x;
delta = (b - a) / (double) w->l;
for (i = 0; i < w->k; i++)
gsl_vector_set (w->knots, i, a);
x = a + delta;
for (i = 0; i < w->l - 1; i++)
{
gsl_vector_set (w->knots, w->k + i, x);
x += delta;
}
for (i = w->n; i < w->n + w->k; i++)
gsl_vector_set (w->knots, i, b);
return GSL_SUCCESS;
} /* gsl_bspline_knots_uniform() */
/*
gsl_bspline_eval()
Evaluate the basis functions B_i(x) for all i. This is
a wrapper function for gsl_bspline_eval_nonzero() which
formats the output in a nice way.
Inputs: x - point for evaluation
B - (output) where to store B_i(x) values
the length of this vector is
n = nbreak + k - 2 = l + k - 1 = w->n
w - bspline workspace
Return: success or error
Notes: The w->knots vector must be initialized prior to calling
this function (see gsl_bspline_knots())
*/
int
gsl_bspline_eval (const double x, gsl_vector * B, gsl_bspline_workspace * w)
{
if (B->size != w->n)
{
GSL_ERROR ("vector B not of length n", GSL_EBADLEN);
}
else
{
size_t i; /* looping */
size_t istart; /* first non-zero spline for x */
size_t iend; /* last non-zero spline for x, knot for x */
int error; /* error handling */
/* find all non-zero B_i(x) values */
error = gsl_bspline_eval_nonzero (x, w->B, &istart, &iend, w);
if (error)
{
return error;
}
/* store values in appropriate part of given vector */
for (i = 0; i < istart; i++)
gsl_vector_set (B, i, 0.0);
for (i = istart; i <= iend; i++)
gsl_vector_set (B, i, gsl_vector_get (w->B, i - istart));
for (i = iend + 1; i < w->n; i++)
gsl_vector_set (B, i, 0.0);
return GSL_SUCCESS;
}
return GSL_FAILURE;
} /* gsl_bspline_eval() */
/*
gsl_bspline_eval_nonzero()
Evaluate all non-zero B-spline functions at point x.
These are the B_i(x) for i in [istart, iend].
Always B_i(x) = 0 for i < istart and for i > iend.
Inputs: x - point at which to evaluate splines
Bk - (output) where to store B-spline values (length k)
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 workspace
Return: success or error
Notes: 1) the w->knots vector must be initialized before calling
this function
2) On output, B contains
[B_{istart,k}, B_{istart+1,k},
..., B_{iend-1,k}, B_{iend,k}]
evaluated at the given x.
*/
int
gsl_bspline_eval_nonzero (const double x, gsl_vector * Bk, size_t * istart,
size_t * iend, gsl_bspline_workspace * w)
{
if (Bk->size != w->k)
{
GSL_ERROR ("Bk vector length does not match order k", GSL_EBADLEN);
}
else
{
size_t i; /* spline index */
size_t j; /* looping */
int flag = 0; /* interval search flag */
int error = 0; /* error flag */
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_bsplvb (w->knots, w->k, 1, x, *iend, &j, w->deltal,
w->deltar, Bk);
return GSL_SUCCESS;
}
return GSL_FAILURE;
} /* gsl_bspline_eval_nonzero() */
/*
gsl_bspline_deriv_eval()
Evaluate d^j/dx^j B_i(x) for all i, 0 <= j <= nderiv.
This is a wrapper function for gsl_bspline_deriv_eval_nonzero()
which formats the output in a nice way.
Inputs: x - point for evaluation
nderiv - number of derivatives to compute, inclusive.
dB - (output) where to store d^j/dx^j B_i(x)
values. the size of this matrix is
(n = nbreak + k - 2 = l + k - 1 = w->n)
by (nderiv + 1)
w - bspline derivative workspace
Return: success or error
Notes: 1) The w->knots vector must be initialized prior to calling
this function (see gsl_bspline_knots())
2) based on PPPACK's bsplvd
*/
int
gsl_bspline_deriv_eval (const double x, const size_t nderiv, gsl_matrix * dB,
gsl_bspline_workspace * w,
gsl_bspline_deriv_workspace * dw)
{
if (dB->size1 != w->n)
{
GSL_ERROR ("dB matrix first dimension not of length n", 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; /* looping */
size_t j; /* looping */
size_t istart; /* first non-zero spline for x */
size_t iend; /* last non-zero spline for x, knot for x */
int error; /* error handling */
/* 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;
}
return GSL_FAILURE;
} /* 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;
}
return GSL_FAILURE;
} /* 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 */
_gsl_vector_view gsl_matrix_column(gsl_matrix *m, const size_t j)
{
_gsl_vector_view view = NULL_VECTOR_VIEW;
if (j >= m->size2)
{
GSL_ERROR("column index is out of range", GSL_EINVAL);
}
{
gsl_vector v = NULL_VECTOR;
v.data = m->data + j ;
v.size = m->size1;
v.stride = m->tda;
v.block = m->block;
v.owner = 0;
view.vector = v;
return view;
}
}
/* ******************************************************** */
/*
void ErrorMessage(char *msg,int fatal)
{
#ifdef WINDOWS
MessageBox(HWND_DESKTOP,msg,"Info!",MB_ICONEXCLAMATION|MB_OK);
#else
if (fatal) error("%s",msg);else warning("%s",msg);
#endif
}
*/
/* *******************vecotr & block************** */
double gsl_vector_get (const gsl_vector * v, const size_t i)
{
#if GSL_RANGE_CHECK
if (GSL_RANGE_COND(i >= v->size))
{
GSL_ERROR("index out of range", GSL_EINVAL);
}
#endif
return v->data[i * v->stride];
}
void gsl_vector_set (gsl_vector * v, const size_t i, double x)
{
#if GSL_RANGE_CHECK
if (GSL_RANGE_COND(i >= v->size))
{
GSL_ERROR ("index out of range", GSL_EINVAL);
}
#endif
v->data[i * v->stride] = x;
}
gsl_vector *gsl_vector_alloc(const size_t n)
{
gsl_block * block;
gsl_vector * v;
if (n == 0)
{
GSL_ERROR ("vector length n must be positive integer",
GSL_EINVAL);
}
v = (gsl_vector *) malloc (sizeof (gsl_vector));
if (v == 0)
{
GSL_ERROR ("failed to allocate space for vector struct",
GSL_ENOMEM);
}
block = gsl_block_alloc(n);
if (block == 0)
{
free (v) ;
GSL_ERROR ("failed to allocate space for block",
GSL_ENOMEM);
}
v->data = block->data ;
v->size = n;
v->stride = 1;
v->block = block;
v->owner = 1;
return v;
}
gsl_block *gsl_block_alloc (const size_t n)
{
gsl_block * b;
if (n == 0)
{
GSL_ERROR ("block length n must be positive integer",
GSL_EINVAL);
}
b = (gsl_block *) malloc (sizeof (gsl_block));
if (b == 0)
{
GSL_ERROR ("failed to allocate space for block struct",
GSL_ENOMEM);
}
b->data = (double *) malloc ( n * sizeof(double));
if (b->data == 0)
{
free (b); /* exception in constructor, avoid memory leak */
GSL_ERROR ("failed to allocate space for block data",
GSL_ENOMEM);
}
b->size = n;
return b;
}
void gsl_vector_free (gsl_vector * v)
{
RETURN_IF_NULL (v);
if (v->owner)
{
gsl_block_free (v->block) ;
}
free (v);
}
void gsl_block_free(gsl_block * b)
{
RETURN_IF_NULL (b);
free (b->data);
free (b);
}
/* ***************matrix********** */
gsl_matrix *gsl_matrix_alloc(const size_t n1, const size_t n2)
{
gsl_block * block;
gsl_matrix * m;
if (n1 == 0)
{
GSL_ERROR ("matrix dimension n1 must be positive integer",
GSL_EINVAL);
}
else if (n2 == 0)
{
GSL_ERROR("matrix dimension n2 must be positive integer",
GSL_EINVAL);
}
m = (gsl_matrix *) malloc (sizeof (gsl_matrix));
if (m == 0)
{
GSL_ERROR("failed to allocate space for matrix struct",
GSL_ENOMEM);
}
/* FIXME: n1*n2 could overflow for large dimensions */
block = gsl_block_alloc (n1 * n2) ;
if (block == 0)
{
GSL_ERROR("failed to allocate space for block",
GSL_ENOMEM);
}
m->data = block->data;
m->size1 = n1;
m->size2 = n2;
m->tda = n2;
m->block = block;
m->owner = 1;
return m;
}
void gsl_matrix_free (gsl_matrix * m)
{
RETURN_IF_NULL (m);
if (m->owner)
{
gsl_block_free (m->block);
}
free (m);
}
int gsl_matrix_get_col(gsl_vector * v,
const gsl_matrix * m,
const size_t j)
{
const size_t M = m->size1;
const size_t N = m->size2;
const size_t tda = m->tda;
if (j >= N)
{
GSL_ERROR ("column index is out of range", GSL_EINVAL);
}
if (v->size != M)
{
GSL_ERROR ("matrix column size and vector length are not equal",
GSL_EBADLEN);
}
{
double *v_data = v->data;
const double *column_data = m->data + j;
const size_t stride = v->stride ;
size_t i;
for (i = 0; i < M; i++)
{
unsigned int k;
for (k = 0; k < 1; k++)
{
v_data[stride * 1 * i + k] =
column_data[1 * i * tda + k];
}
}
}
return GSL_SUCCESS;
}
double gsl_matrix_get(const gsl_matrix * m, const size_t i, const size_t j)
{
#if GSL_RANGE_CHECK
if (GSL_RANGE_COND(1))
{
if (i >= m->size1)
{
GSL_ERROR_VAL("first index out of range", GSL_EINVAL, 0) ;
}
else if (j >= m->size2)
{
GSL_ERROR_VAL("second index out of range", GSL_EINVAL, 0) ;
}
}
#endif
return m->data[i * m->tda + j] ;
}
void gsl_matrix_set(gsl_matrix * m, const size_t i, const size_t j, const double x)
{
#if GSL_RANGE_CHECK
if (GSL_RANGE_COND(1))
{
if (i >= m->size1)
{
GSL_ERROR_VOID("first index out of range", GSL_EINVAL) ;
}
else if (j >= m->size2)
{
GSL_ERROR_VOID("second index out of range", GSL_EINVAL) ;
}
}
#endif
m->data[i * m->tda + j] = x ;
}
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