SmoothFunction

The SmoothFunction class and its methods are defined hereafter.

It is used as the base class of two more specific smooth functions:

class siegpy.smoothfunctions.SmoothFunction(x0, lbda, c0=1, cp=1, cm=-1, grid=None)[source]

Bases: object

Note

This is an abstract class.

Smooth functions are used when a smooth complex scaling is applied to the potential. The aim of this class is to easily update the values of the smooth functions (and its derivatives) when the Refection-Free Complex Absorbing Potentials and the Virial operator are defined.

The smooth functions \(q\) used here are of the form:

\[q(x) = c_0 + c_+ r(\lambda (x - x_0)) - c_- r(\lambda (x + x_0))\]

The smooth function should be 0 on a large part of the \([-x_0, x_0]\) range of the grid, while tending to 1 at both infinities.

The initialization of a smooth function requires the value of \(x_0\) and of the parameter \(\lambda\), indicating how smoothly the function goes from 0 to 1 (the larger, the sharper).

Parameters:
  • x0 (float) – Inflection point.
  • lbda (float) – Sharpness parameter (the function is smoother for smaller values).
  • c0 (float) – Constant term \(c_0\) of the smooth function defintion (default to 1).
  • cp (float) – Constant term \(c_+\) of the smooth function defintion (default to 1).
  • cm (float) – Constant term \(c_-\) of the smooth function defintion (default to -1).
  • array (numpy) – Discretization grid of the smooth function (optional).
Raises:

ValueError – If x0 or lbda are not strictly positive.
x0
Returns:Inflection point.
Return type:float
lbda
Returns:Sharpness parameter.
Return type:float
grid
Returns:Discretization grid of the smooth function.
Return type:numpy array
_to_be_updated(new_grid)[source]
Returns:True if the grid has to be updated.
Return type:bool
_update_all_values()[source]

Update the values of the test functions and all of its derivatives.

values
Returns:Values of the smooth function.
Return type:numpy array
dx_values
Returns:Values of the first derivative of the smooth function.
Return type:numpy array
dx2_values
Returns:Values of the second derivative of the smooth function.
Return type:numpy array
dx3_values
Returns:Values of the third derivative of the smooth function.
Return type:numpy array
dxi_values
Returns:Values of the first derivative of the smooth function with respect to both parameters \(x_0\) and \(\lambda\).
Return type:numpy array
dx_dxi_values
Returns:Values of the first derivative with respect to both parameters \(x_0\) and \(\lambda\) of the first derivative of the smooth function.
Return type:numpy array
_get_r_values(grid)[source]

Note

This is an asbtract method.

Parameters:grid (numpy array) – Discretization grid.
Returns:Values of the function \(r\), evaluated on a given set of grid points.
Return type:numpy array
_get_r_dx_values(grid)[source]

Note

This is an asbtract method.

Parameters:grid (numpy array) – Discretization grid.
Returns:
  • Values of first derivative of the function \(r\),
  • evaluated on a given set of grid points.
_get_r_dx2_values(grid)[source]

Note

This is an asbtract method.

Parameters:grid (numpy array) – Discretization grid.
Returns:
  • Values of second derivative of the function \(r\),
  • evaluated on a given set of grid points.
_get_r_dx3_values(grid)[source]

Note

This is an asbtract method.

Parameters:grid (numpy array) – Discretization grid.
Returns:
  • Values of third derivative of the function \(r\),
  • evaluated on a given set of grid points.
class siegpy.smoothfunctions.ErfSmoothFunction(x0, lbda, c0=1, cp=1, cm=-1, grid=None)[source]

Bases: siegpy.smoothfunctions.SmoothFunction

In this case, the function \(r\) corresponds to the error function \(\text{erf}\).

The smooth functions \(q\) used here are of the form:

\[q(x) = c_0 + c_+ r(\lambda (x - x_0)) - c_- r(\lambda (x + x_0))\]

The smooth function should be 0 on a large part of the \([-x_0, x_0]\) range of the grid, while tending to 1 at both infinities.

The initialization of a smooth function requires the value of \(x_0\) and of the parameter \(\lambda\), indicating how smoothly the function goes from 0 to 1 (the larger, the sharper).

Parameters:
  • x0 (float) – Inflection point.
  • lbda (float) – Sharpness parameter (the function is smoother for smaller values).
  • c0 (float) – Constant term \(c_0\) of the smooth function defintion (default to 1).
  • cp (float) – Constant term \(c_+\) of the smooth function defintion (default to 1).
  • cm (float) – Constant term \(c_-\) of the smooth function defintion (default to -1).
  • array (numpy) – Discretization grid of the smooth function (optional).
Raises:

ValueError – If x0 or lbda are not strictly positive.
_get_r_values(grid)[source]
Parameters:grid (numpy array) – Discretization grid.
Returns:
  • Values of the function \(\text{erf}\), evaluated on a
  • given set of grid points.
_get_r_dx_values(grid)[source]
Parameters:grid (numpy array) – Discretization grid.
Returns:
  • Values of first derivative of the function
  • \(\text{erf}\), evaluated on a given set of grid points.
_get_r_dx2_values(grid)[source]
Parameters:grid (numpy array) – Discretization grid.
Returns:
  • Values of second derivative of the function
  • \(\text{erf}\), evaluated on a given set of grid points.
_get_r_dx3_values(grid)[source]
Parameters:grid (numpy array) – Discretization grid.
Returns:
  • Values of third derivative of the function
  • \(\text{erf}\), evaluated on a given set of grid points.
class siegpy.smoothfunctions.TanhSmoothFunction(x0, lbda, c0=1, cp=1, cm=-1, grid=None)[source]

Bases: siegpy.smoothfunctions.SmoothFunction

In this case, the function \(r\) corresponds to the hyperbolic tangent \(\tanh\).

The smooth functions \(q\) used here are of the form:

\[q(x) = c_0 + c_+ r(\lambda (x - x_0)) - c_- r(\lambda (x + x_0))\]

The smooth function should be 0 on a large part of the \([-x_0, x_0]\) range of the grid, while tending to 1 at both infinities.

The initialization of a smooth function requires the value of \(x_0\) and of the parameter \(\lambda\), indicating how smoothly the function goes from 0 to 1 (the larger, the sharper).

Parameters:
  • x0 (float) – Inflection point.
  • lbda (float) – Sharpness parameter (the function is smoother for smaller values).
  • c0 (float) – Constant term \(c_0\) of the smooth function defintion (default to 1).
  • cp (float) – Constant term \(c_+\) of the smooth function defintion (default to 1).
  • cm (float) – Constant term \(c_-\) of the smooth function defintion (default to -1).
  • array (numpy) – Discretization grid of the smooth function (optional).
Raises:

ValueError – If x0 or lbda are not strictly positive.
_get_r_values(grid)[source]
Parameters:grid (numpy array) – Discretization grid.
Returns:
  • Values of the function \(\tanh\), evaluated on a
  • given set of grid points.
_get_r_dx_values(grid)[source]
Parameters:grid (numpy array) – Discretization grid.
Returns:
  • Values of first derivative of the function \(\tanh\),
  • evaluated on a given set of grid points.
_get_r_dx2_values(grid)[source]
Parameters:grid (numpy array) – Discretization grid.
Returns:
  • Values of second derivative of the function \(\tanh\),
  • evaluated on a given set of grid points.
_get_r_dx3_values(grid)[source]
Parameters:grid (numpy array) – Discretization grid.
Returns:
  • Values of third derivative of the function \(\tanh\),
  • evaluated on a given set of grid points.