void
test_bspline(gsl_bspline_workspace * bw, gsl_bspline_deriv_workspace * dbw)
{
gsl_vector *B;
gsl_matrix *dB;
size_t i, j;
size_t n = 100;
size_t ncoeffs = gsl_bspline_ncoeffs(bw);
size_t order = gsl_bspline_order(bw);
size_t nbreak = gsl_bspline_nbreak(bw);
double a = gsl_bspline_breakpoint(0, bw);
double b = gsl_bspline_breakpoint(nbreak - 1, bw);
B = gsl_vector_alloc(ncoeffs);
dB = gsl_matrix_alloc(ncoeffs, 1);
/* Ensure B-splines form a partition of unity */
for (i = 0; i < n; i++)
{
double xi = a + (b - a) * (i / (n - 1.0));
double sum = 0;
gsl_bspline_eval(xi, B, bw);
for (j = 0; j < ncoeffs; j++)
{
double Bj = gsl_vector_get(B, j);
int s = (Bj < 0 || Bj > 1);
gsl_test(s,
"basis-spline coefficient %u is in range [0,1] for x=%g",
j, xi);
sum += Bj;
}
gsl_test_rel(sum, 1.0, order * GSL_DBL_EPSILON,
"basis-spline order %u is normalized for x=%g", order,
xi);
}
/* Ensure B-splines 0th derivatives agree with regular evaluation */
for (i = 0; i < n; i++)
{
double xi = a + (b - a) * (i / (n - 1.0));
gsl_bspline_eval(xi, B, bw);
gsl_bspline_deriv_eval(xi, 0, dB, bw, dbw);
for (j = 0; j < ncoeffs; j++)
{
gsl_test_abs(gsl_matrix_get(dB, j, 0), gsl_vector_get(B, j),
GSL_DBL_EPSILON,
"b-spline order %d basis #%d evaluation and 0th derivative consistent for x=%g",
order, j, xi);
}
}
gsl_vector_free(B);
gsl_matrix_free(dB);
}
void
test_bspline (gsl_bspline_workspace * bw)
{
gsl_vector *B;
size_t i, j;
size_t n = 100;
size_t ncoeffs = gsl_bspline_ncoeffs (bw);
size_t order = gsl_bspline_order (bw);
size_t nbreak = gsl_bspline_nbreak (bw);
double a = gsl_bspline_breakpoint (0, bw);
double b = gsl_bspline_breakpoint (nbreak - 1, bw);
B = gsl_vector_alloc (ncoeffs);
for (i = 0; i < n; i++)
{
double xi = a + (b - a) * (i / (n - 1.0));
double sum = 0;
gsl_bspline_eval (xi, B, bw);
for (j = 0; j < ncoeffs; j++)
{
double Bj = gsl_vector_get (B, j);
int s = (Bj < 0 || Bj > 1);
gsl_test (s,
"basis-spline coefficient %u is in range [0,1] for x=%g",
j, xi);
sum += Bj;
}
gsl_test_rel (sum, 1.0, order * GSL_DBL_EPSILON,
"basis-spline order %u is normalized for x=%g", order,
xi);
}
gsl_vector_free (B);
}
static VALUE rb_gsl_bspline_order(VALUE obj)
{
gsl_bspline_workspace *w;
Data_Get_Struct(obj, gsl_bspline_workspace, w);
return INT2FIX((int)gsl_bspline_order(w));
}
int
main(int argc, char **argv)
{
size_t order, breakpoints, i;
gsl_ieee_env_setup();
argc = 0; /* prevent warnings about unused parameters */
argv = 0;
for (order = 1; order < 10; order++)
{
for (breakpoints = 2; breakpoints < 100; breakpoints++)
{
double a = -1.23 * order, b = 45.6 * order;
gsl_bspline_workspace *bw = gsl_bspline_alloc(order, breakpoints);
gsl_bspline_deriv_workspace *dbw = gsl_bspline_deriv_alloc(order);
gsl_bspline_knots_uniform(a, b, bw);
test_bspline(bw, dbw);
gsl_bspline_deriv_free(dbw);
gsl_bspline_free(bw);
}
}
for (order = 1; order < 10; order++)
{
for (breakpoints = 2; breakpoints < 100; breakpoints++)
{
double a = -1.23 * order, b = 45.6 * order;
gsl_bspline_workspace *bw = gsl_bspline_alloc(order, breakpoints);
gsl_bspline_deriv_workspace *dbw = gsl_bspline_deriv_alloc(order);
gsl_vector *k = gsl_vector_alloc(breakpoints);
for (i = 0; i < breakpoints; i++)
{
double f, x;
f = sqrt(i / (breakpoints - 1.0));
x = (1 - f) * a + f * b;
gsl_vector_set(k, i, x);
};
gsl_bspline_knots(k, bw);
test_bspline(bw, dbw);
gsl_vector_free(k);
gsl_bspline_deriv_free(dbw);
gsl_bspline_free(bw);
}
}
/* Spot check known 0th, 1st, 2nd derivative
evaluations for a particular k = 2 case. */
{
size_t i, j; /* looping */
const double xloc[4] = { 0.0, 1.0, 6.0, 7.0};
const double deriv[4][3] =
{
{ -1.0/2.0, 1.0/2.0, 0.0 },
{ -1.0/2.0, 1.0/2.0, 0.0 },
{ 0.0, -1.0/5.0, 1.0/5.0 },
{ 0.0, -1.0/5.0, 1.0/5.0 }
};
gsl_bspline_workspace *bw = gsl_bspline_alloc(2, 3);
gsl_bspline_deriv_workspace *dbw = gsl_bspline_deriv_alloc(2);
gsl_matrix *dB = gsl_matrix_alloc(gsl_bspline_ncoeffs(bw),
gsl_bspline_order(bw) + 1);
gsl_vector *breakpts = gsl_vector_alloc(3);
gsl_vector_set(breakpts, 0, 0.0);
gsl_vector_set(breakpts, 1, 2.0);
gsl_vector_set(breakpts, 2, 7.0);
gsl_bspline_knots(breakpts, bw);
for (i = 0; i < 4; ++i) /* at each location */
{
/* Initialize dB with poison to ensure we overwrite it */
gsl_matrix_set_all(dB, GSL_NAN);
gsl_bspline_deriv_eval(xloc[i], gsl_bspline_order(bw), dB, bw, dbw);
for (j = 0; j < gsl_bspline_ncoeffs(bw) ; ++j)
{
/* check basis function 1st deriv */
gsl_test_abs(gsl_matrix_get(dB, j, 1), deriv[i][j], GSL_DBL_EPSILON,
"b-spline k=%d basis #%d derivative %d at x = %f",
gsl_bspline_order(bw), j, 1, xloc[i]);
}
for (j = 0; j < gsl_bspline_ncoeffs(bw); ++j)
{
/* check k order basis function has k-th deriv equal to 0 */
gsl_test_abs(gsl_matrix_get(dB, j, gsl_bspline_order(bw)), 0.0,
GSL_DBL_EPSILON,
"b-spline k=%d basis #%d derivative %d at x = %f",
gsl_bspline_order(bw), j, gsl_bspline_order(bw),
xloc[i]);
}
}
gsl_matrix_free(dB);
gsl_bspline_deriv_free(dbw);
gsl_bspline_free(bw);
gsl_vector_free(breakpts);
}
/* Spot check known 0th, 1st, 2nd derivative
evaluations for a particular k = 3 case. */
{
size_t i, j; /* looping */
const double xloc[5] = { 0.0, 5.0, 9.0, 12.0, 15.0 };
const double eval[5][6] =
{
{ 4./25., 69./100., 3./ 20. , 0. , 0. , 0. },
{ 0. , 4./21. , 143./210. , 9./70., 0. , 0. },
{ 0. , 0. , 3./ 10. , 7./10., 0. , 0. },
{ 0. , 0. , 0. , 3./4. , 1./4., 0. },
{ 0. , 0. , 0. , 1./3. , 5./9., 1./9. }
};
const double deriv[5][6] =
{
{ -4./25., 3./50., 1./ 10., 0. , 0. , 0. },
{ 0. , -2./21., 1./105., 3./35., 0. , 0. },
{ 0. , 0. , -1./5. , 1./ 5., 0. , 0. },
{ 0. , 0. , 0. , -1./ 6., 1./6. , 0. },
{ 0. , 0. , 0. , -1./ 9., 1./27., 2./27. }
};
const double deriv2[5][6] =
{
{ 2./25., -17./150., 1.0/30.0 , 0.0 , 0. , 0. },
{ 0. , 1./ 42., -11.0/210.0, 1.0/35.0, 0. , 0. },
{ 0. , 0. , 1.0/15.0 ,-11.0/90.0, 1./18. , 0. },
{ 0. , 0. , 0.0 , 1.0/54.0, -7./162., 2./81. },
{ 0. , 0. , 0.0 , 1.0/54.0, -7./162., 2./81. }
};
gsl_bspline_workspace *bw = gsl_bspline_alloc(3, 5);
gsl_bspline_deriv_workspace *dbw = gsl_bspline_deriv_alloc(3);
gsl_matrix *dB = gsl_matrix_alloc(gsl_bspline_ncoeffs(bw),
gsl_bspline_order(bw) + 1);
gsl_vector *breakpts = gsl_vector_alloc(5);
gsl_vector_set(breakpts, 0, -3.0);
gsl_vector_set(breakpts, 1, 2.0);
gsl_vector_set(breakpts, 2, 9.0);
gsl_vector_set(breakpts, 3, 12.0);
gsl_vector_set(breakpts, 4, 21.0);
gsl_bspline_knots(breakpts, bw);
for (i = 0; i < 5; ++i) /* at each location */
{
/* Initialize dB with poison to ensure we overwrite it */
gsl_matrix_set_all(dB, GSL_NAN);
gsl_bspline_deriv_eval(xloc[i], gsl_bspline_order(bw), dB, bw, dbw);
/* check basis function evaluation */
for (j = 0; j < gsl_bspline_ncoeffs(bw); ++j)
{
gsl_test_abs(gsl_matrix_get(dB, j, 0), eval[i][j], GSL_DBL_EPSILON,
"b-spline k=%d basis #%d derivative %d at x = %f",
gsl_bspline_order(bw), j, 0, xloc[i]);
}
/* check 1st derivative evaluation */
for (j = 0; j < gsl_bspline_ncoeffs(bw); ++j)
{
gsl_test_abs(gsl_matrix_get(dB, j, 1), deriv[i][j], GSL_DBL_EPSILON,
"b-spline k=%d basis #%d derivative %d at x = %f",
gsl_bspline_order(bw), j, 1, xloc[i]);
}
/* check 2nd derivative evaluation */
for (j = 0; j < gsl_bspline_ncoeffs(bw); ++j)
{
gsl_test_abs(gsl_matrix_get(dB, j, 2), deriv2[i][j], GSL_DBL_EPSILON,
"b-spline k=%d basis #%d derivative %d at x = %f",
gsl_bspline_order(bw), j, 2, xloc[i]);
}
}
gsl_matrix_free(dB);
gsl_bspline_deriv_free(dbw);
gsl_bspline_free(bw);
gsl_vector_free(breakpts);
}
/* Check Greville abscissae functionality on a non-uniform k=1 */
{
size_t i; /* looping */
/* Test parameters */
const size_t k = 1;
const double bpoint_data[] = { 0.0, 0.2, 0.5, 0.75, 1.0 };
const size_t nbreak = sizeof(bpoint_data)/sizeof(bpoint_data[0]);
/* Expected results */
const double abscissae_data[] = { 0.1, 0.35, 0.625, 0.875 };
const size_t nabscissae = sizeof(abscissae_data)/sizeof(abscissae_data[0]);
gsl_vector_const_view bpoints = gsl_vector_const_view_array(bpoint_data, nbreak);
gsl_bspline_workspace *w = gsl_bspline_alloc(k, nbreak);
gsl_bspline_knots((const gsl_vector *) &bpoints, w);
gsl_test_int(nabscissae, gsl_bspline_ncoeffs(w),
"b-spline k=%d number of abscissae", k);
for (i = 0; i < nabscissae; ++i)
{
gsl_test_abs(gsl_bspline_greville_abscissa(i, w), abscissae_data[i], 2*k*GSL_DBL_EPSILON,
"b-spline k=%d Greville abscissa #%d at x = %f", k, i, abscissae_data[i]);
}
gsl_bspline_free(w);
}
/* Check Greville abscissae functionality on a non-uniform k=2 */
{
size_t i; /* looping */
/* Test parameters */
const size_t k = 2;
const double bpoint_data[] = { 0.0, 0.2, 0.5, 0.75, 1.0 };
const size_t nbreak = sizeof(bpoint_data)/sizeof(bpoint_data[0]);
/* Expected results */
const double abscissae_data[] = { 0.0, 0.2, 0.5, 0.75, 1.0 };
const size_t nabscissae = sizeof(abscissae_data)/sizeof(abscissae_data[0]);
gsl_vector_const_view bpoints = gsl_vector_const_view_array(bpoint_data, nbreak);
gsl_bspline_workspace *w = gsl_bspline_alloc(k, nbreak);
gsl_bspline_knots((const gsl_vector *) &bpoints, w);
gsl_test_int(nabscissae, gsl_bspline_ncoeffs(w),
"b-spline k=%d number of abscissae", k);
for (i = 0; i < nabscissae; ++i)
{
gsl_test_abs(gsl_bspline_greville_abscissa(i, w), abscissae_data[i], 2*k*GSL_DBL_EPSILON,
"b-spline k=%d Greville abscissa #%d at x = %f", k, i, abscissae_data[i]);
}
gsl_bspline_free(w);
}
/* Check Greville abscissae functionality on non-uniform k=3 */
{
size_t i; /* looping */
/* Test parameters */
const size_t k = 3;
const double bpoint_data[] = { 0.0, 0.2, 0.5, 0.75, 1.0 };
const size_t nbreak = sizeof(bpoint_data)/sizeof(bpoint_data[0]);
/* Expected results */
const double abscissae_data[] = { 0.0, 1.0/10.0, 7.0/20.0,
5.0/ 8.0, 7.0/ 8.0, 1.0 };
const size_t nabscissae = sizeof(abscissae_data)/sizeof(abscissae_data[0]);
gsl_vector_const_view bpoints = gsl_vector_const_view_array(bpoint_data, nbreak);
gsl_bspline_workspace *w = gsl_bspline_alloc(k, nbreak);
gsl_bspline_knots((const gsl_vector *) &bpoints, w);
gsl_test_int(nabscissae, gsl_bspline_ncoeffs(w),
"b-spline k=%d number of abscissae", k);
for (i = 0; i < nabscissae; ++i)
{
gsl_test_abs(gsl_bspline_greville_abscissa(i, w), abscissae_data[i], 2*k*GSL_DBL_EPSILON,
"b-spline k=%d Greville abscissa #%d at x = %f", k, i, abscissae_data[i]);
}
gsl_bspline_free(w);
}
/* Check Greville abscissae functionality on non-uniform k=4 */
{
size_t i; /* looping */
/* Test parameters */
const size_t k = 4;
const double bpoint_data[] = { 0.0, 0.2, 0.5, 0.75, 1.0 };
const size_t nbreak = sizeof(bpoint_data)/sizeof(bpoint_data[0]);
/* Expected results */
const double abscissae_data[] = { 0.0, 1.0/15.0, 7.0/30.0, 29.0/60.0,
3.0/ 4.0, 11.0/12.0, 1.0 };
const size_t nabscissae = sizeof(abscissae_data)/sizeof(abscissae_data[0]);
gsl_vector_const_view bpoints = gsl_vector_const_view_array(bpoint_data, nbreak);
gsl_bspline_workspace *w = gsl_bspline_alloc(k, nbreak);
gsl_bspline_knots((const gsl_vector *) &bpoints, w);
gsl_test_int(nabscissae, gsl_bspline_ncoeffs(w),
"b-spline k=%d number of abscissae", k);
for (i = 0; i < nabscissae; ++i)
{
gsl_test_abs(gsl_bspline_greville_abscissa(i, w), abscissae_data[i], 2*k*GSL_DBL_EPSILON,
"b-spline k=%d Greville abscissa #%d at x = %f", k, i, abscissae_data[i]);
}
gsl_bspline_free(w);
}
exit(gsl_test_summary());
}