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<center><h1> Multiple uniform splines </h1></center>
<h3> <font color="Blue"> Spline creation </font></h3>
When evaluating many splines at the same point (such as in filling in a
row of a determinant matrix in quantum Monte Carlo calculations), a
signficant speedup can be achieved by using the multiple spline
version of the uniform B-spline routines. The creation routines are
similar to those for the single splines, which the total number of splines
is passed and the raw data is NOT passed in at time of creation.
Rather, we can the <font color="blue">create_multi_UBspline_*</font> routine
to allocate the memory for the splines. We then call the
<font color="blue">set_multi_UBspline_*</font> routine for
each spline, passing the the spline index and the raw data
to be interpolated.
<h4> Base data types </h4>
Each spline creation and evaluation routines have four versions,
corresponding to the four support base data types. Following the
LAPACK naming convention, these types are specified by single-letter
codes:
<ul>
<li> <font color="blue"> s</font>: single-precision real <font color="blue">(float)</font> </li>
<li> <font color="blue"> d</font>: double-precision real <font color="blue">(double)</font> </li>
<li> <font color="blue"> c</font>: single-precision complex <font color="blue">(complex_float)</font></li>
<li> <font color="blue"> z</font>: double-precision complex
<font color="blue">(complex_double)</font> </li>
</ul>
<h4> Grid dimensions </h4>
The uniform grid structure, has the following elements, which must be specified:
<ul>
<li> <font color="blue"> double <i>start</i> </font>:
the first grid point </li>
<li> <font color="blue"> double <i>end</i> </font>:
the last grid point </li>
<li> <font color="blue"> int <i>num</i> </font>:
the number of grid points </li>
</ul>
<h4> Boundary conditions </h4>
The boundary conditions at the first and last grid point must be specified. They are
specifed with BCtype_<i>x</i> structures, where <i>x</i> is one of {s,
d, c, z}, as described above. For the real types (s and d) the
structure contains the following elements
<table CELLPADDING=3 frame=border rules=all bordercolor="gray" >
<tr> <td>Type</td> <td> Name </td> <td> Description </td> </tr>
<tr> <td>bc_code </td> <td> lCode </td> <td> "left" boundary condition code </td> </tr>
<tr> <td>bc_code </td> <td> lCode </td> <td> "right" boundary condition code </td> </tr>
<tr> <td>d_type </td> <td> lVal </td> <td> "left" boundary condition value </td> </tr>
<tr> <td>d_type </td> <td> rVal </td> <td> "right" boundary condition value </td> </tr>
</tr>
<caption> <font color="blue"> BCtype_s and BCtype_d data structure elements</font></caption>
</table>
For the complex types, we must specify both the real and imaginary
parts:
<table CELLPADDING=3 frame=border rules=all bordercolor="gray" >
<tr> <td>Type</td> <td> Name </td> <td> Description </td> </tr>
<tr> <td>bc_code </td> <td> lCode </td> <td> "left" boundary condition code </td> </tr>
<tr> <td>bc_code </td> <td> lCode </td> <td> "right" boundary condition code </td> </tr>
<tr> <td>d_type </td> <td> lVal_r </td> <td> "left" boundary condition real part </td> </tr>
<tr> <td>d_type </td> <td> lVal_i </td> <td> "left" boundary condition imag part </td> </tr>
<tr> <td>d_type </td> <td> rVal_r </td> <td> "right" boundary condition real part </td> </tr>
<tr> <td>d_type </td> <td> rVal_i </td> <td> "right" boundary condition imag part </td> </tr>
</tr>
<caption> <font color="blue"> BCtype_s and BCtype_d data structure elements</font></caption>
</table>
<br/>
<font color="blue">lCode</font> and <font color="blue">lVal</font> specify the boundary conditions
at the first grid point (the "left" boundary), while <font color="blue">rCode</font> and
<font color="blue">rVal</font> specify the boundary conditions on the last grid point (the "right"
boundary).<br>
<font color="blue">bc_code</font> is one of the enumerated value,
{<font color="blue">PERIODIC</font>, <font color="blue">DERIV1</font>,
<font color="blue">DERIV2</font>, <font color="blue">FLAT</font>,
<font color="blue">NATURAL</font>, <font color="blue">ANTIPERIODIC</font> }. <br/>
<font color="blue">d_type</font> is the the C type corresponding to
{s, d, c, z}, i.e. {float, double, complex_float, complex_double}
<br/> <br/>
The codes have the following meaning
<table CELLPADDING=3 frame=border rules=all bordercolor="gray" width="50%">
<tr> <td> Code </td> <td> Meaning </td></tr>
<tr> <td> <font color="blue">PERIODIC </font> </td> <td> Use periodic boundary conditions. The value, first derivative and second derivative at the left
boundary match those at the right boundary. </td> </tr>
<tr> <td> <font color="blue">DERIV1 </font> </td> <td> The value of the first derivative is specified in lVal or rVal. </td> </tr>
<tr> <td> <font color="blue">DERIV2 </font> </td> <td> The value of the second derivative is specified in lVal or rVal. </td> </tr>
<tr> <td> <font color="blue">FLAT </font> </td> <td> The value of the first derivative is set to zero at the boundary. </td> </tr>
<tr> <td> <font color="blue">NATURAL </font> </td> <td> The value of the second derivative is set to zero at the boundary. </td> </tr>
<tr> <td> <font color="blue">ANTIPERIODIC </font> </td> <td> Use anti-periodic boundary conditions. The value, first derivative and second derivative at the left
boundary are the negative of those at the right boundary. </td> </tr>
</table>
<h4> Data to be interpolated </h4>
The data to be interpolated should have
<i>N<sub>x</sub></i> <i>N<sub>y</sub></i> <i>N<sub>z</sub></i> contiguous
elements, arranged is row-order (C-style) format. That is, the offset
of the (ix,iy,iz) element is (ix*(Ny+iy)*Nz+iz). Complex numbers are
stored in the standard format of (real,imaginary) pairs, which the
real element first.
<h4> Function prototypes: </h4>
<u>Single-precision real: </u><br>
<PRE>
multi_UBspline_1d_s * create_multi_UBspline_1d_s (Ugrid x_grid, BCtype_s xBC, int num_splines);
multi_UBspline_2d_s * create_multi_UBspline_2d_s (Ugrid x_grid, Ugrid y_grid,
BCtype_s xBC, BCtype_s yBC, int num_splines);
multi_UBspline_3d_s * create_multi_UBspline_3d_s (Ugrid x_grid, Ugrid y_grid, Ugrid z_grid,
BCtype_s xBC, BCtype_s yBC, BCtype_s zBC, int num_splines);
void set_multi_UBspline_1d_s (multi_UBspline_1d_s* spline, int spline_num, float *data);
void set_multi_UBspline_2d_s (multi_UBspline_2d_s* spline, int spline_num, float *data);
void set_multi_UBspline_3d_s (multi_UBspline_3d_s* spline, int spline_num, float *data);
</PRE>
<u>Single-precision complex: </u><br>
<PRE>
multi_UBspline_1d_c * create_multi_UBspline_1d_c (Ugrid x_grid, BCtype_c xBC, int num_splines);
multi_UBspline_2d_c * create_multi_UBspline_2d_c (Ugrid x_grid, Ugrid y_grid,
BCtype_c xBC, BCtype_c yBC, int num_splines);
multi_UBspline_3d_c * create_multi_UBspline_3d_c (Ugrid x_grid, Ugrid y_grid, Ugrid z_grid,
BCtype_c xBC, BCtype_c yBC, BCtype_c zBC,
int num_splines);
void set_multi_UBspline_1d_c (multi_UBspline_1d_c* spline, int spline_num, complex_float *data);
void set_multi_UBspline_2d_c (multi_UBspline_2d_c* spline, int spline_num, complex_float *data);
void set_multi_UBspline_3d_c (multi_UBspline_3d_c* spline, int spline_num, complex_float *data);
</PRE>
<u>Double-precision real: </u><br>
<PRE>
multi_UBspline_1d_d * create_multi_UBspline_1d_d (Ugrid x_grid, BCtype_d xBC, int num_splines);
multi_UBspline_2d_d * create_multi_UBspline_2d_d (Ugrid x_grid, Ugrid y_grid,
BCtype_d xBC, BCtype_d yBC, int num_splines);
multi_UBspline_3d_d * create_multi_UBspline_3d_d (Ugrid x_grid, Ugrid y_grid, Ugrid z_grid,
BCtype_d xBC, BCtype_d yBC, BCtype_d zBC, int num_splines);
void set_multi_UBspline_1d_d (multi_UBspline_1d_d* spline, int spline_num, double *data);
void set_multi_UBspline_2d_d (multi_UBspline_2d_d* spline, int spline_num, double *data);
void set_multi_UBspline_3d_d (multi_UBspline_3d_d* spline, int spline_num, double *data);
</PRE>
<u> Double-precision complex: </u><br>
<PRE>
multi_UBspline_1d_z * create_multi_UBspline_1d_z (Ugrid x_grid, BCtype_z xBC, int num_splines);
multi_UBspline_2d_z * create_multi_UBspline_2d_z (Ugrid x_grid, Ugrid y_grid,
BCtype_z xBC, BCtype_z yBC, int num_splines);
multi_UBspline_3d_z * create_multi_UBspline_3d_z (Ugrid x_grid, Ugrid y_grid, Ugrid z_grid,
BCtype_z xBC, BCtype_z yBC, BCtype_z zBC,
int num_splines);
void set_multi_UBspline_1d_z (multi_UBspline_1d_z* spline, int spline_num, complex_double *data);
void set_multi_UBspline_2d_z (multi_UBspline_2d_z* spline, int spline_num, complex_double *data);
void set_multi_UBspline_3d_z (multi_UBspline_3d_z* spline, int spline_num, complex_double *data);
</PRE>
<hr/>
<h3> <font color="Blue"> Spline destruction </font></h3>
The memory used for spline storage can be freed simply by a call to
<pre>
void
destroy_Bspline (void *<i>spline</i>);
</pre>
The <i>spline</i> parameter can be a spline of any type and dimension,
uniform or nonuniform.
<hr/>
<h3> <font color="Blue"> Spline evaluation </font> </h3>
In contrast to the single-spline routines, <font color="blue">eval_multi_UBspline_*</font> routines
evaluate all <font color="blue">num_splines</font> splines at a given point with a single call.
The prototypes for the functions are very similar to the single-spline prototypes, but the
value, gradient, hessian, and laplacian arguments are now arrays which much be
of sufficicient size to hold the values for all the splines. If the multi-spline object
was created with <i>N</i> splines, the value parameter must be an array of length <i>N</i>.
Similarly, the gradient array must have dimension <i>N*D</i>, where <i>D</i> is the
spline dimensionality (i.e. 1, 2, or 3). Finally the hessian argument must have
dimension <i>N*D*D</i>.
For each of the four datatypes, there are four evaluation routines, depending on
which quantities need to be computed:
<ul>
<li> Value only
<li> Value and gradient
<li> Value, gradient, and Laplacian
<li> Value, gradient, and Hessian (matrix of 2nd derivatives)
</ul>
For consistency, all results are returned through pointers passed to the evaluation
routines. Currently, no bounds checking is done for the sake of speed. The user is
responsible for ensuring that the points passed to the evaluation functions fall within
the grids specified at the time of spline creation.<br>
<h4> Function prototypes: </h4>
<table CELLPADDING=5 frame=border rules=all>
<tr>
<td> <font color="Blue">Quick Jump Table</font> </td>
<td> 1D </td>
<td> 2D </td>
<td> 3D </td>
</tr>
<tr>
<td> Single-precision real </td>
<td> <a href="#1Ds"> 1ds </a>
<td> <a href="#2Ds"> 2ds </a>
<td> <a href="#3Ds"> 3ds </a>
</tr>
<tr>
<td> Single-precision complex </td>
<td> <a href="#1Dc"> 1dc </a>
<td> <a href="#2Dc"> 2dc </a>
<td> <a href="#3Dc"> 3dc </a>
</tr>
<tr>
<td> Double-precision real </td>
<td> <a href="#1Dd"> 1dd </a>
<td> <a href="#2Dd"> 2dd </a>
<td> <a href="#3Dd"> 3dd </a>
</tr>
<tr>
<td> Double-precision complex </td>
<td> <a href="#1Dz"> 1dz </a>
<td> <a href="#2Dz"> 2dz </a>
<td> <a href="#3Dz"> 3dz </a>
</tr>
</table>
<br>
<h4> <font color="Blue"> Single-precision real: </font></h4>
<u> <a name="1Ds"> 1D </a></u>
<PRE>
inline void
eval_multi_UBspline_1d_s (multi_UBspline_1d_s * restrict spline,
double x, float* restrict val);
inline void
eval_multi_UBspline_1d_s_vg (multi_UBspline_1d_s * restrict spline, double x,
float* restrict val, float* restrict grad);
inline void
eval_multi_UBspline_1d_s_vgl (multi_UBspline_1d_s * restrict spline, double x,
float* restrict val, float* restrict grad, float* restrict lapl);
inline void /* identical to above routine in 1D */
eval_multi_UBspline_1d_s_vgh (multi_UBspline_1d_s * restrict spline, double x,
float* restrict val, float* restrict grad, float* restrict hess);
</PRE>
<u> <a name="2Ds"> 2D </a> </u>
<PRE>
inline void
eval_multi_UBspline_2d_s (multi_UBspline_2d_s * restrict spline, double x, double y,
float* restrict val);
inline void
eval_multi_UBspline_2d_s_vg (multi_UBspline_2d_s * restrict spline, double x, double y,
float* restrict val, float* restrict grad);
inline void
eval_multi_UBspline_2d_s_vgl (multi_UBspline_2d_s * restrict spline, double x, double y,
float* restrict val, float* restrict grad, float* restrict lapl);
inline void
eval_multi_UBspline_2d_s_vgh (multi_UBspline_2d_s * restrict spline, double x, double y,
float* restrict val, float* restrict grad, float* restrict hess);
</PRE>
<u><a name="3Ds"> 3D </a></u>
<PRE>
inline void
eval_multi_UBspline_3d_s (multi_UBspline_3d_s * restrict spline, double x, double y, double z,
float* restrict val);
inline void
eval_multi_UBspline_3d_s_vg (multi_UBspline_3d_s * restrict spline, double x, double y, double z,
float* restrict val, float* restrict grad);
inline void
eval_multi_UBspline_3d_s_vgl (multi_UBspline_3d_s * restrict spline, double x, double y, double z,
float* restrict val, float* restrict grad, float* restrict lapl);
inline void
eval_multi_UBspline_3d_s_vgh (multi_UBspline_3d_s * restrict spline, double x, double y,
float* restrict val, float* restrict grad, float* restrict hess);
</PRE>
<h4><font color="Blue">Single-precision complex: </font></h4>
<u> <a name="1Dc"> 1D </a></u>
<PRE>
inline void
eval_multi_UBspline_1d_c (multi_UBspline_1d_c * restrict spline,
double x, complex_float* restrict val);
inline void
eval_multi_UBspline_1d_c_vg (multi_UBspline_1d_c * restrict spline, double x,
complex_float* restrict val, complex_float* restrict grad);
inline void
eval_multi_UBspline_1d_c_vgl (multi_UBspline_1d_c * restrict spline, double x,
complex_float* restrict val, complex_float* restrict grad, complex_float* restrict lapl);
inline void /* identical to above routine in 1D */
eval_multi_UBspline_1d_c_vgh (multi_UBspline_1d_c * restrict spline, double x,
complex_float* restrict val, complex_float* restrict grad, complex_float* restrict hess);
</PRE>
<u> <a name="2Dc"> 2D </a> </u>
<PRE>
inline void
eval_multi_UBspline_2d_c (multi_UBspline_2d_c * restrict spline, double x, double y,
complex_float* restrict val);
inline void
eval_multi_UBspline_2d_c_vg (multi_UBspline_2d_c * restrict spline, double x, double y,
complex_float* restrict val, complex_float* restrict grad);
inline void
eval_multi_UBspline_2d_c_vgl (multi_UBspline_2d_c * restrict spline, double x, double y,
complex_float* restrict val, complex_float* restrict grad, complex_float* restrict lapl);
inline void
eval_multi_UBspline_2d_c_vgh (multi_UBspline_2d_c * restrict spline, double x, double y,
complex_float* restrict val, complex_float* restrict grad, complex_float* restrict hess);
</PRE>
<u> <a name="3Dc"> 3D </a> </u>
<PRE>
inline void
eval_multi_UBspline_3d_c (multi_UBspline_3d_c * restrict spline, double x, double y, double z,
complex_float* restrict val);
inline void
eval_multi_UBspline_3d_c_vg (multi_UBspline_3d_c * restrict spline, double x, double y, double z,
complex_float* restrict val, complex_float* restrict grad);
inline void
eval_multi_UBspline_3d_c_vgl (multi_UBspline_3d_c * restrict spline, double x, double y, double z,
complex_float* restrict val, complex_float* restrict grad, complex_float* restrict lapl);
inline void
eval_multi_UBspline_3d_c_vgh (multi_UBspline_3d_c * restrict spline, double x, double y,
complex_float* restrict val, complex_float* restrict grad, complex_float* restrict hess);
</PRE>
<h4> <font color="Blue">Double-precision real:</font></h4>
<u> <a name="1Dd"> 1D </a></u>
<PRE>
inline void
eval_multi_UBspline_1d_d (multi_UBspline_1d_d * restrict spline,
double x, double* restrict val);
inline void
eval_multi_UBspline_1d_d_vg (multi_UBspline_1d_d * restrict spline, double x,
double* restrict val, double* restrict grad);
inline void
eval_multi_UBspline_1d_d_vgl (multi_UBspline_1d_d * restrict spline, double x,
double* restrict val, double* restrict grad, double* restrict lapl);
inline void /* identical to above routine in 1D */
eval_multi_UBspline_1d_d_vgh (multi_UBspline_1d_d * restrict spline, double x,
double* restrict val, double* restrict grad, double* restrict hess);
</PRE>
<u> <a name="2Dd"> 2D </a> </u>
<PRE>
inline void
eval_multi_UBspline_2d_d (multi_UBspline_2d_d * restrict spline, double x, double y,
double* restrict val);
inline void
eval_multi_UBspline_2d_d_vg (multi_UBspline_2d_d * restrict spline, double x, double y,
double* restrict val, double* restrict grad);
inline void
eval_multi_UBspline_2d_d_vgl (multi_UBspline_2d_d * restrict spline, double x, double y,
double* restrict val, double* restrict grad, double* restrict lapl);
inline void
eval_multi_UBspline_2d_d_vgh (multi_UBspline_2d_d * restrict spline, double x, double y,
double* restrict val, double* restrict grad, double* restrict hess);
</PRE>
<u> <a name="3Dd"> 3D </a></u>
<PRE>
inline void
eval_multi_UBspline_3d_d (multi_UBspline_3d_d * restrict spline, double x, double y, double z,
double* restrict val);
inline void
eval_multi_UBspline_3d_d_vg (multi_UBspline_3d_d * restrict spline, double x, double y, double z,
double* restrict val, double* restrict grad);
inline void
eval_multi_UBspline_3d_d_vgl (multi_UBspline_3d_d * restrict spline, double x, double y, double z,
double* restrict val, double* restrict grad, double* restrict lapl);
inline void
eval_multi_UBspline_3d_d_vgh (multi_UBspline_3d_d * restrict spline, double x, double y,
double* restrict val, double* restrict grad, double* restrict hess);
</PRE>
<h4> <font color="Blue">Double-precision complex: </font></h4>
<u> <a name="1Dz"> 1D </a></u>
<PRE>
inline void
eval_multi_UBspline_1d_z (multi_UBspline_1d_z * restrict spline,
double x, complex_double* restrict val);
inline void
eval_multi_UBspline_1d_z_vg (multi_UBspline_1d_z * restrict spline, double x,
complex_double* restrict val, complex_double* restrict grad);
inline void
eval_multi_UBspline_1d_z_vgl (multi_UBspline_1d_z * restrict spline, double x,
complex_double* restrict val, complex_double* restrict grad, complex_double* restrict lapl);
inline void /* identical to above routine in 1D */
eval_multi_UBspline_1d_z_vgh (multi_UBspline_1d_z * restrict spline, double x,
complex_double* restrict val, complex_double* restrict grad, complex_double* restrict hess);
</PRE>
<u> <a name="2Dz"> 2D </a> </u>
<PRE>
inline void
eval_multi_UBspline_2d_z (multi_UBspline_2d_z * restrict spline, double x, double y,
complex_double* restrict val);
inline void
eval_multi_UBspline_2d_z_vg (multi_UBspline_2d_z * restrict spline, double x, double y,
complex_double* restrict val, complex_double* restrict grad);
inline void
eval_multi_UBspline_2d_z_vgl (multi_UBspline_2d_z * restrict spline, double x, double y,
complex_double* restrict val, complex_double* restrict grad, complex_double* restrict lapl);
inline void
eval_multi_UBspline_2d_z_vgh (multi_UBspline_2d_z * restrict spline, double x, double y,
complex_double* restrict val, complex_double* restrict grad, complex_double* restrict hess);
</PRE>
<u> <a name="3Dz"> 3D </a> </u>
<PRE>
inline void
eval_multi_UBspline_3d_z (multi_UBspline_3d_z * restrict spline, double x, double y, double z,
complex_double* restrict val);
inline void
eval_multi_UBspline_3d_z_vg (multi_UBspline_3d_z * restrict spline, double x, double y, double z,
complex_double* restrict val, complex_double* restrict grad);
inline void
eval_multi_UBspline_3d_z_vgl (multi_UBspline_3d_z * restrict spline, double x, double y, double z,
complex_double* restrict val, complex_double* restrict grad, complex_double* restrict lapl);
inline void
eval_multi_UBspline_3d_z_vgh (multi_UBspline_3d_z * restrict spline, double x, double y,
complex_double* restrict val, complex_double* restrict grad, complex_double* restrict hess);
</PRE>
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