WallGo.polynomial

Class that stores and perfom operation on the coefficients of a polynomial series.

Classes

`Polynomial`(coefficients, grid[, basis, ...])	Class that stores and perfoms operations on the coefficients of a polynomial series.
`SpectralConvergenceInfo`(coefficients[, ...])	Carries information about the convergence of a polynomial expansion.

class Polynomial(coefficients, grid, basis='Cardinal', direction='z', endpoints=False)[source]

Class that stores and perfoms operations on the coefficients of a polynomial series.

Initialization of Polynomial object.

Parameters:

coefficients (array-like) – Array of rank N containing the coefficients of a polynomial defined by the object grid.
grid (Grid) – An object of the Grid class defining the polynomial.
basis (string or tuple of strings, optional) – Tuple of length N specifying in what basis each dimension of coefficients is defined. Each component can either be ‘Cardinal’, ‘Chebyshev’ or ‘Array’. The latter is to be used if the axis does not corresponds to a polynomial series. Can also be a single string, in which case all the dimensions are assumed to be in that basis. The default is ‘Cardinal’.
direction (string or tuple of strings, optional) – Tuple of length N specifying what direction each dimension of coefficients represents. Each component can either be ‘z’, ‘pz’, ‘pp’ or ‘Array’. The latter can only be used if basis is ‘Array’ for that axis. Can also be a single string, in which case all the dimensions are assumed to be in that direction. The default is ‘z’.
endpoints (bool or tuple of bool, optional) – Tuple of length N specifying wheither each dimension includes the endpoints. Can also be a single bool, in which case all the dimensions are assumed to be the same. If False, the polynomial is assumed to be 0 at the endpoints. The default is False.

Return type:

None.

cardinal(compactCoord, n, direction)[source]

Computes the cardinal polynomials \(C_n(x)\) defined by grid.

Parameters:

compactCoord (array_like) – Compact coordinate at which to evaluate the Chebyshev polynomial. Must be broadcastable with n.
n (array_like) – Order of the cardinal polynomial to evaluate. Must be broadcastable with x.
direction (string) – Select the direction in which to compute the matrix. Can either be ‘z’, ‘pz’ or ‘pp’.

Returns:

cn – Values of the cardinal functions.

Return type:

array_like

changeBasis(newBasis, inverseTranspose=False)[source]

Change the basis of the polynomial. Will change self.coefficients accordingly.

Parameters:

newBasis (string or tuple of strings, optional) – Tuple of length N specifying in what basis each dimension of self.coefficients is defined. Each component can either be ‘Cardinal’ or ‘Chebyshev’. Can also be a single string, in which case all the dimensions are assumed to be in that basis.
inverseTranspose (bool, optional) – If True, take the inverse-transpose of the transformation matrix. This is useful, for example, when changing the basis of the collision array.

Return type:

None.

chebyshev(compactCoord, n, restriction=None)[source]

Computes the Chebyshev polynomial \(T_n(x)\).

Parameters:

compactCoord (array_like) – Compact coordinate at which to evaluate the Chebyshev polynomial. Must be broadcastable with n.
n (array_like) – Order of the Chebyshev polynomial to evaluate. Must be broadcastable with x.
restriction (None or string, optional) – Select the restriction on the Chebyshev basis. If None, evaluates the unrestricted basis. If ‘full’, the polynomials are 0 at \(x=\pm 1\). If ‘partial’, the polynomials are 0 at \(x=+1\).

Returns:

chen – Values of the polynomial

Return type:

array_like

derivMatrix(basis, direction, endpoints=False)[source]

Computes the derivative matrix of either the Chebyshev or cardinal polynomials in some direction.

Parameters:

basis (string) – Select the basis of polynomials. Can be ‘Cardinal’ or ‘Chebyshev’
direction (string) – Select the direction in which to compute the matrix. Can be ‘z’, ‘pz’ or ‘pp’.
endpoints (Bool, optional) – If True, include endpoints of grid. Default is False.

Returns:

deriv – Derivative matrix.

Return type:

array_like

derivative(axis)[source]

Computes the derivative of the polynomial and returns it in a Polynomial object.

Parameters:: axis (int or tuple) – axis along which the derivative is taken. Can either be a int or a tuple of int.
Returns:: Object of the class Polynomial containing the coefficients of the derivative polynomial (in the compact coordinates). The axis along which the derivative is taken is always returned with the endpoints in the cardinal basis.
Return type:: Polynomial

evaluate(compactCoord, axes=None)[source]

Evaluates the polynomial at the compact coordinates x.

Parameters:

compactCoord (array-like) – Compact coordinates at which to evaluate the polynomial. Must have a shape (len(axes),:) or (len(axes),).
axes (tuple or None, optional) – Axes along which to be evaluated. If None, evaluate the polynomial along all the axes. Default is None.

Returns:

Values of the polynomial at x.

Return type:

array-like

integrate(axis=None, weight=1)[source]

Computes the integral of the polynomial \(\int_{-1}^1 dx P(x)w(x)\) along some axis using Gauss-Chebyshev-Lobatto quadrature.

Parameters:

axis (None, int or tuple) – axis along which the integral is taken. Can either be None, a int or a tuple of int. If None, integrate along all the axes.
weight (array-like, optional) – Integration weight. Must be an object broadcastable with self.coefficients. Default is 1.

Returns:

If axis=None, returns a float. Otherwise, returns an object of the class Polynomial containing the coefficients of the integrated polynomial along the remaining axes.

Return type:

Polynomial or float

matrix(basis, direction, endpoints=False)[source]

Returns the matrix \(M_{ij}=T_j(x_i)\) or \(M_{ij}=C_j(x_i)\) computed in a specific direction.

Parameters:

basis (string) – Select the basis of polynomials. Can be ‘Cardinal’ or ‘Chebyshev’
direction (string) – Select the direction in which to compute the matrix. Can either be ‘z’, ‘pz’ or ‘pp’
endpoints (Bool, optional) – If True, include endpoints of grid. Default is False.

Return type:

ndarray

class SpectralConvergenceInfo(coefficients, weightPower=0, offset=0)[source]

Carries information about the convergence of a polynomial expansion.

Assumes input is a 1d array of coefficients in the Chebyshev basis. Fits a slope to the logarithm of the absolute value of these coefficients. Also, finds the average value of the index, treating the coefficients as a histogram.

Initialise given an array.

Parameters:

coefficients (ndarray)
weightPower (int)
offset (int)

apparentConvergence: bool = False: True if spectral expansion appears to be converging, False otherwise.

coefficients: ndarray: Coefficients of expansion in the Chebyshev basis, must be 1d.

offset: int = 0: Offest in \(n\). Default is zero.

spectralExponent: float = 0.0: Exponent \(\sigma\) of \(A e^{\sigma n}\) fit to spectral expansion.

spectralPeak: int = 0: Positions of the peak of the spectral expansion.

weightPower: int = 0: Additional powers of \(n\) to weight by in assessing convergence. Default is zero.