shenfun.chebyshev package

Submodules

shenfun.chebyshev.bases module

Module for defining function spaces in the Chebyshev family.

A function is approximated in the Chebyshev basis as

\[u(x) = \sum_{i=0}^{N-1} \hat{u}_i T_i(x)\]

where \(T_i(x)\) is the i’th Chebyshev polynomial of the first kind. The Chebyshev polynomials are orthogonal with weight \(\omega=1/\sqrt{1-x^2}\)

\[\int_{-1}^1 T_i T_k \omega dx = \frac{c_k \pi}{2} \delta_{ki},\]

where \(c_0=2\) and \(c_i=1\) for \(i>0\).

All other bases defined in this module are combinations of \(T_i\)’s. For example, a Dirichlet basis is

\[\phi_i = T_i - T_{i+2}\]

The basis is implemented using a stencil matrix \(K \in \mathbb{R}^{N-2 \times N}\), such that

\[\boldsymbol{\phi} = K \boldsymbol{T},\]

where \(\boldsymbol{\phi}=(\phi_0, \phi_1, \ldots, \phi_{N-3})\) and \(\boldsymbol{T}=(T_0, T_1, \ldots, T_{N-1})\). For the Dirichlet basis \(K = (\delta_{i, j} - \delta_{i+2, j})_{i,j=0}^{N-2, N}\).

All composite bases make use of the fast transforms that exists for \(\boldsymbol{T}\) through fast cosine transforms. The stencil matrix is used to transfer any composite basis back and forth to the orthogonal basis.

class shenfun.chebyshev.bases.BCGeneric(N, bc=None, domain=None, alpha=0, beta=0, **kw)

Bases: CompositeBase

Function space for setting inhomogeneous boundary conditions

Parameters:

• N (int) – Number of quadrature points in the homogeneous space.

• bc (dict) – The boundary conditions in dictionary form, see BoundaryConditions.

• domain (2-tuple of numbers, optional) – The domain of the homogeneous space.

• alpha (number, optional) – Parameter of the Jacobi polynomial

• beta (number, optional) – Parameter of the Jacobi polynomial

basis_function(i=0, x=x)

Return basis function i

Parameters:

• i (int, optional) – The degree of freedom of the basis function

• x (sympy Symbol, optional)

static boundary_condition()

property dim_ortho

eval(x, u, output_array=None)

Evaluate Function u at position x

Parameters:

• x (float or array of floats)

• u (array) – Expansion coefficients or instance of Function

• output_array (array, optional) – Function values at points

Returns:

output_array

Return type:

array

evaluate_basis(x, i=0, output_array=None)

Evaluate basis function i at points x

Parameters:

• x (float or array of floats)

• i (int, optional) – Basis function number

• output_array (array, optional) – Return result in output_array if provided

Returns:

output_array

Return type:

array

evaluate_basis_derivative(x=None, i=0, k=0, output_array=None)

Evaluate k’th derivative of basis function i at x or all quadrature points

Parameters:

• x (float or array of floats, optional) – If not provided use quadrature points of self

• i (int, optional) – Basis function number

• k (int, optional) – k’th derivative

• output_array (array, optional) – return array

Returns:

output_array

Return type:

array

get_orthogonal(**kwargs)

Return orthogonal space (otherwise as self)

Returns:

The orthogonal space in the same family, and otherwise as self.

Return type:

SpectralBase

property is_boundary_basis

shape(forward_output=True)

Return the allocated shape of arrays used for self

Parameters:

forward_output (bool, optional) – If True then return allocated shape of spectral space (the result of a forward transform). If False then return allocated shape of physical space (the input to a forward transform).

static short_name()

slice()

Return index set of current space

stencil_matrix(N=None)

Return stencil matrix in SparseMatrix format

Parameters:

N (int or None, optional) – Shape (N, N) of the stencil matrix. Using self.N if None

to_ortho(input_array, output_array=None)

Project to orthogonal basis

Parameters:

• input_array (array) – Expansion coefficients of input basis

• output_array (array, optional) – Expansion coefficients in orthogonal basis

Returns:

output_array

Return type:

array

vandermonde(x)

Return Vandermonde matrix based on the primary (orthogonal) basis of the family.

Evaluates basis function \(\psi_k(x)\) for all wavenumbers, and all x. Returned Vandermonde matrix is an N x M matrix with N the length of x and M the number of bases.

\[\begin{split}\begin{bmatrix} \psi_0(x_0) & \psi_1(x_0) & \ldots & \psi_{M-1}(x_0)\\ \psi_0(x_1) & \psi_1(x_1) & \ldots & \psi_{M-1}(x_1)\\ \vdots & \ldots \\ \psi_{0}(x_{N-1}) & \psi_1(x_{N-1}) & \ldots & \psi_{M-1}(x_{N-1}) \end{bmatrix}\end{split}\]

Parameters:

x (array of floats) – points for evaluation

Note

This function returns a matrix of evaluated orthogonal basis functions for either family. The true Vandermonde matrix of a basis is obtained through SpectralBase.evaluate_basis_all().

class shenfun.chebyshev.bases.CombinedShenNeumann(N, quad='GC', bc=(0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for Neumann boundary conditions

The basis \(\{\phi_k\}_{k=0}^{N-1}\) is

\[\begin{split}\phi_k = \begin{cases} T_0, \quad &k=0, \\ T_1 - T_3/9, \quad &k=1, \\ T_2/4 - T_4/16, \quad &k=2, \\ -\frac{T_{k-2}}{(k-2)^2} +2\frac{T_k}{k^2} - \frac{T_{k+2}}{(k+2)^2}, &k=3, 4, \ldots, N-3, \\ \frac{1}{8}(4T_1-T_2), \quad &k=N-2 \\ \frac{1}{8}(4T_1+T_2), \quad &k=N-1 \end{cases}\end{split}\]

such that

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u'(-1) &= a \text{ and } u'(1) = b.\end{split}\]

The last two bases are for boundary conditions and only used if a or b are different from 0. In one dimension \(\hat{u}_{N-2}=a\) and \(\hat{u}_{N-1}=b\).

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (2-tuple of floats, optional) – Boundary condition values at, respectively, x=(-1, 1).

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

stencil_matrix(N=None)

Return stencil matrix in SparseMatrix format

Parameters:

N (int or None, optional) – Shape (N, N) of the stencil matrix. Using self.N if None

sympy_stencil(i=i, j=j)

Return stencil matrix as a Sympy matrix

Parameters:

• i, j (Sympy symbols) – indices for row and column

• implicit (bool or str, optional) – Whether to use an unevaluated Sympy function instead of the actual value of the stencil. This makes the matrix prettier, and it can still be evaluated. If implicit is not False, then it must be a string of length one. This string represents the diagonals of the matrix. If implicit=’a’ and the stencil matrix has two diagonals in the main diagonal and the first upper diagonal, then these are called ‘a0’ and ‘a1’.

Example

>>> from shenfun import FunctionSpace
>>> import sympy as sp
>>> i, j = sp.symbols('i,j', integer=True)
>>> D = FunctionSpace(8, 'L', bc=(0, 0), scaled=True)
>>> D._stencil
{0: 1/sqrt(4*n + 6), 2: -1/sqrt(4*n + 6)}
>>> D.sympy_stencil()
KroneckerDelta(i, j)/sqrt(4*i + 6) - KroneckerDelta(j, i + 2)/sqrt(4*i + 6)
>>> D.sympy_stencil(implicit='a')
KroneckerDelta(i, j)*a0(i) + KroneckerDelta(j, i + 2)*a2(i)

Get the main diagonal

>>> D.sympy_stencil(implicit=False).subs(j, i)
1/sqrt(4*i + 6)

class shenfun.chebyshev.bases.Compact3(N, quad='GC', bc=(0, 0, 0, 0, 0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for 6’th order equation

The basis functions \(\phi_k\) for \(k=0, 1, \ldots, N-7\) are

\[\begin{split}\phi_k &= \frac{h_k}{b^{(3)}_{k+3,k}}\frac{(1-x^2)^3}{h^{(3)}_{k+3}} T^{(3)}_{k+3} \\ h^{(3)}_{k+3} &= \frac{\pi (k+3) \Gamma (k+6)}{2k!} = \int_{-1}^1 T^{(3)}_k T^{(3)}_k (1-x^2)^{2.5} dx.\end{split}\]

where \(T^{(3)}_k\) is the 3rd derivative of \(T_k\). This is Phi3 scaled such that the main diagonal of the stencil matrix is unity.

The boundary basis for inhomogeneous boundary conditions is too messy to print, but can be obtained using get_bc_basis(). We have

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u(-1) &= a, u'(-1)=b, u''(-1)=c, u(1)=d u'(1)=e, u''(1)=f.\end{split}\]

The last 6 basis functions are for boundary conditions and only used if there are nonzero boundary conditions.

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (6-tuple of numbers)

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

class shenfun.chebyshev.bases.Compact4(N, quad='GC', bc=(0, 0, 0, 0, 0, 0, 0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for 8’th order equation

The basis functions \(\phi_k\) for \(k=0, 1, \ldots, N-9\) are

\[\begin{split}\phi_k &= \frac{h_k}{b^{(4)}_{k+4,k}}\frac{(1-x^2)^4}{h^{(4)}_{k+4}} T^{(4)}_{k+4} \\ h^{(4)}_{k+4} &= \int_{-1}^1 T^{(4)}_k T^{(4)}_k (1-x^2)^{3.5} dx.\end{split}\]

where \(T^{(4)}_k\) is the 4rd derivative of \(T_k\). This is Phi4 scaled such that the main diagonal of the stencil matrix is unity.

The boundary basis for inhomogeneous boundary conditions is too messy to print, but can be obtained using get_bc_basis(). We have

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u(-1) &= a, u'(-1)=b, u''(-1)=c, u'''(-1)=d, u(1)=e u'(1)=f, u''(1)=g, u'''(1)=h.\end{split}\]

The last 8 basis functions are for boundary conditions and only used if there are nonzero boundary conditions.

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (8-tuple of numbers)

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

class shenfun.chebyshev.bases.DirichletNeumann(N, quad='GC', bc=(0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for mixed Dirichlet/Neumann boundary conditions

The basis \(\{\phi_k\}_{k=0}^{N-1}\) is

\[\begin{split}\phi_k &= T_{k} + \frac{4(k+1)}{2k^2+6k+5}T_{k+1} - \frac{2k^2+2k+1}{2k^2+6k+5}T_{k+2}, \, k=0, 1, \ldots, N-3, \\ \phi_{N-2} &= T_0, \\ \phi_{N-1} &= T_0+T_1,\end{split}\]

such that

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u(-1) &= a, u'(1)=b.\end{split}\]

The last two bases are for boundary conditions and only used if a or b are different from 0. In one dimension \(\hat{u}_{N-2}=a\) and \(\hat{u}_{N-1}=b\).

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (2-tuple of numbers) – Boundary condition values at x=-1 and x=1

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

class shenfun.chebyshev.bases.Generic(N, quad='GC', bc={}, domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for space with any boundary conditions

Any combination of Dirichlet and Neumann is possible.

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (dict, optional) – The dictionary must have keys ‘left’ and ‘right’, to describe boundary conditions on the left and right boundaries, and a list of 2-tuples to specify the condition. Specify Dirichlet on both ends with

  {‘left’: {‘D’: a}, ‘right’: {‘D’: b}}

  for some values a and b, that will be neglected in the current function. Specify mixed Neumann and Dirichlet as

  {‘left’: {‘N’: a}, ‘right’: {‘N’: b}}

  For both conditions on the right do

  {‘right’: {‘N’: a, ‘D’: b}}

  Any combination should be possible, and it should also be possible to use second derivatives N2. See BoundaryConditions.

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

Note

A test function is always using homogeneous boundary conditions.

static boundary_condition()

static short_name()

class shenfun.chebyshev.bases.Heinrichs(N, quad='GC', bc=(0, 0), domain=(-1, 1), dtype=<class 'float'>, scaled=False, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for Dirichlet boundary conditions

The basis \(\{\phi_k\}_{k=0}^{N-1}\) is

\[\begin{split}\phi_k &= \alpha_k(1-x^2) T_{k}, \, k=0, 1, \ldots, N-3, \\ \phi_{N-2} &= \frac{1}{2}(T_0-T_1), \\ \phi_{N-1} &= \frac{1}{2}(T_0+T_1),\end{split}\]

such that

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u(-1) &= a \text{ and } u(1) = b.\end{split}\]

The last two bases are for boundary conditions and only used if a or b are different from 0. In one dimension \(\hat{u}_{N-2}=a\) and \(\hat{u}_{N-1}=b\). If the parameter scaled=True, then \(\alpha_k=1/(k+1)/(k+2)\), otherwise, \(\alpha_k=1\).

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (2-tuple of floats, optional) – Boundary conditions at, respectively, x=(-1, 1).

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optiona) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• scaled (bool, optional) – Whether or not to use scaled basis

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

stencil_matrix(N=None)

Return stencil matrix in SparseMatrix format

Parameters:

N (int or None, optional) – Shape (N, N) of the stencil matrix. Using self.N if None

sympy_stencil(i=i, j=j)

Return stencil matrix as a Sympy matrix

Parameters:

• i, j (Sympy symbols) – indices for row and column

• implicit (bool or str, optional) – Whether to use an unevaluated Sympy function instead of the actual value of the stencil. This makes the matrix prettier, and it can still be evaluated. If implicit is not False, then it must be a string of length one. This string represents the diagonals of the matrix. If implicit=’a’ and the stencil matrix has two diagonals in the main diagonal and the first upper diagonal, then these are called ‘a0’ and ‘a1’.

Example

>>> from shenfun import FunctionSpace
>>> import sympy as sp
>>> i, j = sp.symbols('i,j', integer=True)
>>> D = FunctionSpace(8, 'L', bc=(0, 0), scaled=True)
>>> D._stencil
{0: 1/sqrt(4*n + 6), 2: -1/sqrt(4*n + 6)}
>>> D.sympy_stencil()
KroneckerDelta(i, j)/sqrt(4*i + 6) - KroneckerDelta(j, i + 2)/sqrt(4*i + 6)
>>> D.sympy_stencil(implicit='a')
KroneckerDelta(i, j)*a0(i) + KroneckerDelta(j, i + 2)*a2(i)

Get the main diagonal

>>> D.sympy_stencil(implicit=False).subs(j, i)
1/sqrt(4*i + 6)

class shenfun.chebyshev.bases.LowerDirichlet(N, quad='GC', bc=(0, None), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space with single Dirichlet boundary condition

The basis \(\{\phi_k\}_{k=0}^{N-1}\) is

\[\begin{split}\phi_k &= T_{k} + T_{k+1}, \, k=0, 1, \ldots, N-2, \\ \phi_{N-1} &= T_0,\end{split}\]

such that

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u(-1) &= a.\end{split}\]

The last basis funciton is for boundary condition and only used if a is different from 0. In one dimension \(\hat{u}_{N-1}=a\).

Parameters:

• N (int) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (tuple of (number, None)) – Boundary conditions at edges of domain.

• domain (2-tuple of numbers, optional) – The computational domain

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

class shenfun.chebyshev.bases.LowerDirichletNeumann(N, quad='GC', bc=(0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for both Dirichlet and Neumann boundary conditions on the left hand side

The basis \(\{\phi_k\}_{k=0}^{N-1}\) is

\[\begin{split}\phi_k &= T_{k} + \frac{4(k+1)}{2k+3}T_{k+1} + \frac{2k+1}{2k+3}T_{k+2}, \, k=0, 1, \ldots, N-3, \\ \phi_{N-2} &= T_0, \\ \phi_{N-1} &= T_0+T_1,\end{split}\]

such that

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u(-1) &= a, u'(-1)=b.\end{split}\]

The last two bases are for boundary conditions and only used if a or b are different from 0. In one dimension \(\hat{u}_{N-2}=a\) and \(\hat{u}_{N-1}=b\).

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (2-tuple of numbers) – Boundary condition values at the left edge of domain

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

class shenfun.chebyshev.bases.MikNeumann(N, quad='GC', bc=(0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for Neumann boundary conditions

The basis functions \(\phi_k\) for \(k=0,1, \ldots, N-3\) are

\[\phi_k &= \frac{2}{k+1}\int (T_{k-1}-T_{k+1}),\]

which (with also boundary functions) leads to the basis

\[\begin{split}\phi_k = \frac{1}{k+1} \begin{cases} T_0, &k=0, \\ 3T_1-T_3/3, &k=1, \\ T_2-T_4/4, &k=2, \\ -\frac{T_{k-2}}{k-2} + 2\frac{T_k}{k} - \frac{T_{k+2}}{k+2} , &k=3, 4, \ldots, N-3, \\ \frac{1}{8}(4T_1-T_2), \quad &k=N-2 \\ \frac{1}{8}(4T_1+T_2), \quad &k=N-1 \end{cases}\end{split}\]

such that

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u'(-1) &= a \text{ and } u'(1) = b.\end{split}\]

The last two bases are for boundary conditions and only used if a or b are different from 0. In one dimension \(\hat{u}_{N-2}=a\) and \(\hat{u}_{N-1}=b\).

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (2-tuple of floats, optional) – Boundary condition values at, respectively, x=(-1, 1).

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

stencil_matrix(N=None)

Return stencil matrix in SparseMatrix format

Parameters:

N (int or None, optional) – Shape (N, N) of the stencil matrix. Using self.N if None

sympy_stencil(i=i, j=j)

Return stencil matrix as a Sympy matrix

Parameters:

• i, j (Sympy symbols) – indices for row and column

• implicit (bool or str, optional) – Whether to use an unevaluated Sympy function instead of the actual value of the stencil. This makes the matrix prettier, and it can still be evaluated. If implicit is not False, then it must be a string of length one. This string represents the diagonals of the matrix. If implicit=’a’ and the stencil matrix has two diagonals in the main diagonal and the first upper diagonal, then these are called ‘a0’ and ‘a1’.

Example

>>> from shenfun import FunctionSpace
>>> import sympy as sp
>>> i, j = sp.symbols('i,j', integer=True)
>>> D = FunctionSpace(8, 'L', bc=(0, 0), scaled=True)
>>> D._stencil
{0: 1/sqrt(4*n + 6), 2: -1/sqrt(4*n + 6)}
>>> D.sympy_stencil()
KroneckerDelta(i, j)/sqrt(4*i + 6) - KroneckerDelta(j, i + 2)/sqrt(4*i + 6)
>>> D.sympy_stencil(implicit='a')
KroneckerDelta(i, j)*a0(i) + KroneckerDelta(j, i + 2)*a2(i)

Get the main diagonal

>>> D.sympy_stencil(implicit=False).subs(j, i)
1/sqrt(4*i + 6)

class shenfun.chebyshev.bases.NeumannDirichlet(N, quad='GC', bc=(0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for mixed Neumann/Dirichlet boundary conditions

The basis \(\{\phi_k\}_{k=0}^{N-1}\) is

\[\begin{split}\phi_k &= T_{k} - \frac{4(k+1)}{2k^2+6k+5}T_{k+1} - \frac{2k^2+2k+1}{2k^2+6k+5}T_{k+2}, \, k=0, 1, \ldots, N-3, \\ \phi_{N-2} &= -T_0+T_1, \\ \phi_{N-1} &= T_0,\end{split}\]

such that

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u'(-1) &= a, u(1)=b.\end{split}\]

The last two bases are for boundary conditions and only used if a or b are different from 0. In one dimension \(\hat{u}_{N-2}=a\) and \(\hat{u}_{N-1}=b\).

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (2-tuple of numbers) – Boundary condition values at x=-1 and x=1

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

class shenfun.chebyshev.bases.Orthogonal(N, quad='GC', domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: JacobiBase

Function space for regular Chebyshev series

The orthogonal basis is

\[T_k, \quad k = 0, 1, \ldots, N-1,\]

where \(T_k\) is the \(k\)’th Chebyshev polynomial of the first kind.

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

L2_norm_sq(i)

Return square of L2-norm

\[\| \phi_i \|^2_{\omega} = (\phi_i, \phi_i)_{\omega} = \int_{I} \phi_i \overline{\phi}_i \omega dx\]

where \(\phi_i\) is the i’th orthogonal basis function for the orthogonal basis in the given family.

Parameters:

i (int) – The number of the orthogonal basis function

eval(x, u, output_array=None)

Evaluate Function u at position x

Parameters:

• x (float or array of floats)

• u (array) – Expansion coefficients or instance of Function

• output_array (array, optional) – Function values at points

Returns:

output_array

Return type:

array

evaluate_basis(x, i=0, output_array=None)

Evaluate basis function i at points x

Parameters:

• x (float or array of floats)

• i (int, optional) – Basis function number

• output_array (array, optional) – Return result in output_array if provided

Returns:

output_array

Return type:

array

evaluate_basis_derivative(x=None, i=0, k=0, output_array=None)

Evaluate k’th derivative of basis function i at x or all quadrature points

Parameters:

• x (float or array of floats, optional) – If not provided use quadrature points of self

• i (int, optional) – Basis function number

• k (int, optional) – k’th derivative

• output_array (array, optional) – return array

Returns:

output_array

Return type:

array

evaluate_basis_derivative_all(x=None, k=0, argument=0)

Return k’th derivative of basis evaluated at x or all quadrature points as a Vandermonde matrix.

Parameters:

• x (float or array of floats, optional) – If not provided use quadrature points of self

• k (int, optional) – k’th derivative

• argument (int) – Zero for test and 1 for trialfunction

Returns:

Vandermonde matrix

Return type:

array

static family()

get_bc_space()

get_orthogonal(**kwargs)

Return orthogonal space (otherwise as self)

Returns:

The orthogonal space in the same family, and otherwise as self.

Return type:

SpectralBase

property is_orthogonal

l2_norm_sq(i=None)

Return square of l2-norm

\[\| u \|^2_{N,\omega} = (u, u)_{N,\omega} = \sun_{j=0}^{N-1} u(x_j) \overline{u}(x_j) \omega_j\]

where \(u=\{\phi_i\}_{i=0}^{N-1}\) and \(\phi_i\) is the i’th orthogonal basis function in the given family.

Parameters:

i (None or int) – If None then return the square of the l2-norm for all i=0, 1, …, N-1. Else, return for given i.

orthogonal_basis_function(i=0, x=x)

Return the orthogonal basis function i

Parameters:

• i (int, optional) – The degree of freedom of the basis function

• x (sympy Symbol, optional)

plan(shape, axis, dtype, options)

Plan transform

Allocate work arrays for transforms and set up methods forward, backward and scalar_product with or without padding

Parameters:

• shape (array) – Local shape of global array

• axis (int) – This base’s axis in global TensorProductSpace

• dtype (numpy.dtype) – Type of array

• options (dict) – Options for planning transforms

points_and_weights(N=None, map_true_domain=False, weighted=True, **kw)

Return points and weights of quadrature for weighted integral

\[\int_{\Omega} f(x) w(x) dx \approx \sum_{i} f(x_i) w_i\]

Parameters:

• N (int, optional) – Number of quadrature points

• map_true_domain (bool, optional) – Whether or not to map points to true domain

• weighted (bool, optional) – Whether to use quadrature weights for a weighted inner product (default), or a regular, non-weighted inner product.

Note

The weight of the space is included in the returned quadrature weights.

static short_name()

stencil_matrix(N=None)

sympy_stencil(i=i, j=j)

to_ortho(input_array, output_array=None)

Project to orthogonal basis

Parameters:

• input_array (array) – Expansion coefficients of input basis

• output_array (array, optional) – Expansion coefficients in orthogonal basis

Returns:

output_array

Return type:

array

vandermonde(x)

Return Vandermonde matrix based on the primary (orthogonal) basis of the family.

Evaluates basis function \(\psi_k(x)\) for all wavenumbers, and all x. Returned Vandermonde matrix is an N x M matrix with N the length of x and M the number of bases.

\[\begin{split}\begin{bmatrix} \psi_0(x_0) & \psi_1(x_0) & \ldots & \psi_{M-1}(x_0)\\ \psi_0(x_1) & \psi_1(x_1) & \ldots & \psi_{M-1}(x_1)\\ \vdots & \ldots \\ \psi_{0}(x_{N-1}) & \psi_1(x_{N-1}) & \ldots & \psi_{M-1}(x_{N-1}) \end{bmatrix}\end{split}\]

Parameters:

x (array of floats) – points for evaluation

Note

This function returns a matrix of evaluated orthogonal basis functions for either family. The true Vandermonde matrix of a basis is obtained through SpectralBase.evaluate_basis_all().

weight(x=x)

Weight of inner product space

Parameters:

x (coordinate)

class shenfun.chebyshev.bases.Phi1(N, quad='GC', bc=(0, 0), domain=(-1.0, 1.0), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for Dirichlet boundary conditions.

The basis \(\{\phi_k\}_{k=0}^{N-1}\) is

\[\begin{split}\phi_k &= \frac{T_k - T_{k+2}}{\pi (k+1)} = \frac{2(1-x^2)}{\pi k(k+1)} T'_{k+1}, \, k=0, 1, \ldots, N-3, \\ \phi_{N-2} &= \frac{1}{2}(T_0-T_1), \\ \phi_{N-1} &= \frac{1}{2}(T_0+T_1),\end{split}\]

such that

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u(-1) &= a \text{ and } u(1) = b.\end{split}\]

The last two bases are for boundary conditions and only used if a or b are different from 0. In one dimension \(\hat{u}_{N-2}=a\) and \(\hat{u}_{N-1}=b\).

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (2-tuple of floats, optional) – Boundary conditions at, respectively, x=(-1, 1).

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

class shenfun.chebyshev.bases.Phi2(N, quad='GC', bc=(0, 0, 0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for biharmonic equation.

The basis functions \(\phi_k\) for \(k=0, 1, \ldots, N-5\) are

\[\phi_k = \frac{2(1-x^2)^2 T''_{k+2}}{\pi (k+1)(k+2)^2(k+3)} ,\]

which (along with boundary functions) gives the basis

\[\begin{split}\phi_k &= \frac{1}{2 \pi (k+1)(k+2)}(T_k - \frac{2(k+2)}{k+3}T_{k+2} + \frac{k+1}{k+3}T_{k+4}), \, k=0, 1, \ldots, N-5, \\ \phi_{N-4} &= \frac{1}{16}(8T_0-9T_1+T_3), \\ \phi_{N-3} &= \frac{1}{16}(2T_0-T_1-2T_2+T_3), \\ \phi_{N-2} &= \frac{1}{16}(8T_0+9T_1-T_3), \\ \phi_{N-1} &= \frac{1}{16}(2T_0-T_1+2T_2+T_3),\end{split}\]

such that

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u(-1)&=a, u'(-1) = b, u(1)=c, u'(1) = d.\end{split}\]

The last four bases are for boundary conditions and only used if a, b, c or d are different from 0. In one dimension \(\hat{u}_{N-4}=a\), \(\hat{u}_{N-3}=b\), \(\hat{u}_{N-2}=c\) and \(\hat{u}_{N-1}=d\).

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (4-tuple of numbers) – The values of the 4 boundary conditions at x=(-1, 1). The two conditions at x=-1 first and then x=1. With (a, b, c, d) corresponding to bc = {‘left’: {‘D’: a, ‘N’: b}, ‘right’: {‘D’: c, ‘N’: d}}

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

class shenfun.chebyshev.bases.Phi3(N, quad='GC', bc=(0, 0, 0, 0, 0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for 6’th order equation

The basis functions \(\phi_k\) for \(k=0, 1, \ldots, N-7\) are

\[\begin{split}\phi_k &= \frac{(1-x^2)^3}{h^{(3)}_{k+3}} T^{(3)}_{k+3} \\ h^{(3)}_{k+3} &= \frac{\pi (k+3) \Gamma (k+6)}{2k!} = \int_{-1}^1 T^{(3)}_k T^{(3)}_k (1-x^2)^{2.5} dx.\end{split}\]

where \(T^{(3)}_k\) is the 3rd derivative of \(T_k\). The boundary basis for inhomogeneous boundary conditions is too messy to print, but can be obtained using get_bc_basis(). We have

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u(-1) &= a, u'(-1)=b, u''(-1)=c, u(1)=d u'(1)=e, u''(1)=f.\end{split}\]

The last 6 basis functions are for boundary conditions and only used if there are nonzero boundary conditions.

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (6-tuple of numbers)

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

class shenfun.chebyshev.bases.Phi4(N, quad='GC', bc=(0, 0, 0, 0, 0, 0, 0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space with 4 Dirichlet and 4 Neumann boundary conditions

The basis functions \(\phi_k\) for \(k=0, 1, \ldots, N-9\) are

\[\begin{split}\phi_k &= \frac{(1-x^2)^4}{h^{(4)}_{k+4}} T^{(4)}_{k+4} \\ h^{(4)}_k &= \frac{\pi k \Gamma (k+4)}{2(k-4)!} = \int_{-1}^1 T^{(4)}_k T^{(4)}_k (1-x^2)^{3.5} dx,\end{split}\]

where \(T^{(4)}_k\) is the 4th derivative of \(T_k\). The boundary basis for inhomogeneous boundary conditions is too messy to print, but can be obtained using get_bc_basis().

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (8-tuple of numbers)

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

class shenfun.chebyshev.bases.Phi6(N, quad='GC', bc=(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for 12th order equation

The basis functions \(\phi_k\) for \(k=0, 1, \ldots, N-13\) are

\[\begin{split}\phi_k &= \frac{(1-x^2)^6}{h^{(6)}_{k+6}} T^{(6)}_{k+6} \\ h^{(6)}_k &= \frac{\pi k (k+5)!}{2(k-6)!} = \int_{-1}^1 T^{(6)}_k T^{(6)}_k (1-x^2)^{5.5} dx,\end{split}\]

where \(T^{(6)}_k\) is the 6th derivative of \(T_k\). The boundary basis for inhomogeneous boundary conditions is too messy to print, but can be obtained using get_bc_basis().

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (12-tuple of numbers)

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

class shenfun.chebyshev.bases.PolarDirichlet(N, quad='GC', bc=(0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for polar coordinates.

The basis \(\{\phi_k\}_{k=0}^{N-1}\) is

\[\begin{split}\phi_k &= T_k - T_{k+2}, \, k=0, 1 \phi_k &= T_{k-2} - T_{k+2}, \, k=2, 3, \ldots, N-3, \\ \phi_{N-2} &= \frac{1}{2}(T_0-T_1), \\ \phi_{N-1} &= \frac{1}{2}(T_0+T_1),\end{split}\]

such that

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u(-1)&=a, u(1)=b\end{split}\]

The last four bases are for boundary conditions and only used if a, b, c or d are different from 0. In one dimension \(\hat{u}_{N-4}=a\), \(\hat{u}_{N-3}=b\), \(\hat{u}_{N-2}=c\) and \(\hat{u}_{N-1}=d\).

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (4-tuple of numbers) – The values of the 4 boundary conditions at x=(-1, 1). The two conditions on x=-1 first, and then x=1. With (a, b, c, d) corresponding to bc = {‘left’: [(‘D’, a), (‘N’, b)], ‘right’: [(‘D’, c), (‘N’, d)]}

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

stencil_matrix(N=None)

Return stencil matrix in SparseMatrix format

Parameters:

N (int or None, optional) – Shape (N, N) of the stencil matrix. Using self.N if None

sympy_stencil(i=i, j=j)

Return stencil matrix as a Sympy matrix

Parameters:

• i, j (Sympy symbols) – indices for row and column

• implicit (bool or str, optional) – Whether to use an unevaluated Sympy function instead of the actual value of the stencil. This makes the matrix prettier, and it can still be evaluated. If implicit is not False, then it must be a string of length one. This string represents the diagonals of the matrix. If implicit=’a’ and the stencil matrix has two diagonals in the main diagonal and the first upper diagonal, then these are called ‘a0’ and ‘a1’.

Example

>>> from shenfun import FunctionSpace
>>> import sympy as sp
>>> i, j = sp.symbols('i,j', integer=True)
>>> D = FunctionSpace(8, 'L', bc=(0, 0), scaled=True)
>>> D._stencil
{0: 1/sqrt(4*n + 6), 2: -1/sqrt(4*n + 6)}
>>> D.sympy_stencil()
KroneckerDelta(i, j)/sqrt(4*i + 6) - KroneckerDelta(j, i + 2)/sqrt(4*i + 6)
>>> D.sympy_stencil(implicit='a')
KroneckerDelta(i, j)*a0(i) + KroneckerDelta(j, i + 2)*a2(i)

Get the main diagonal

>>> D.sympy_stencil(implicit=False).subs(j, i)
1/sqrt(4*i + 6)

class shenfun.chebyshev.bases.ShenBiPolar(N, quad='GC', bc=(0, 0, 0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for the Biharmonic equation in polar coordinates

u(-1)=a, u(1)=c, u’(-1)=b and u’(1)=d

Parameters:

• N (int) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (4-tuple of numbers)

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

stencil_matrix(N=None)

Return stencil matrix in SparseMatrix format

Parameters:

N (int or None, optional) – Shape (N, N) of the stencil matrix. Using self.N if None

sympy_stencil(i=i, j=j)

Return stencil matrix as a Sympy matrix

Parameters:

• i, j (Sympy symbols) – indices for row and column

• implicit (bool or str, optional) – Whether to use an unevaluated Sympy function instead of the actual value of the stencil. This makes the matrix prettier, and it can still be evaluated. If implicit is not False, then it must be a string of length one. This string represents the diagonals of the matrix. If implicit=’a’ and the stencil matrix has two diagonals in the main diagonal and the first upper diagonal, then these are called ‘a0’ and ‘a1’.

Example

>>> from shenfun import FunctionSpace
>>> import sympy as sp
>>> i, j = sp.symbols('i,j', integer=True)
>>> D = FunctionSpace(8, 'L', bc=(0, 0), scaled=True)
>>> D._stencil
{0: 1/sqrt(4*n + 6), 2: -1/sqrt(4*n + 6)}
>>> D.sympy_stencil()
KroneckerDelta(i, j)/sqrt(4*i + 6) - KroneckerDelta(j, i + 2)/sqrt(4*i + 6)
>>> D.sympy_stencil(implicit='a')
KroneckerDelta(i, j)*a0(i) + KroneckerDelta(j, i + 2)*a2(i)

Get the main diagonal

>>> D.sympy_stencil(implicit=False).subs(j, i)
1/sqrt(4*i + 6)

class shenfun.chebyshev.bases.ShenBiharmonic(N, quad='GC', bc=(0, 0, 0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for biharmonic equation.

The basis \(\{\phi_k\}_{k=0}^{N-1}\) is

\[\begin{split}\phi_k &= T_n - \frac{2(k+2)}{k+3}T_{k+2}+\frac{k+1}{k+3}T_{k+4}, \, k=0, 1, \ldots, N-5, \\ \phi_{N-4} &= \frac{1}{16}(8T_0-9T_1+T_3), \\ \phi_{N-3} &= \frac{1}{16}(2T_0-T_1-2T_2+T_3), \\ \phi_{N-2} &= \frac{1}{16}(8T_0+9T_1-T_3), \\ \phi_{N-1} &= \frac{1}{16}(2T_0-T_1+2T_2+T_3),\end{split}\]

such that

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u(-1)&=a, u'(-1) = b, u(1)=c, u'(1) = d.\end{split}\]

The last four bases are for boundary conditions and only used if a, b, c or d are different from 0. In one dimension \(\hat{u}_{N-4}=a\), \(\hat{u}_{N-3}=b\), \(\hat{u}_{N-2}=c\) and \(\hat{u}_{N-1}=d\).

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (4-tuple of numbers) – The values of the 4 boundary conditions at x=(-1, 1). The two conditions on x=-1 first, and then x=1. With (a, b, c, d) corresponding to bc = {‘left’: [(‘D’, a), (‘N’, b)], ‘right’: [(‘D’, c), (‘N’, d)]}

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

class shenfun.chebyshev.bases.ShenDirichlet(N, quad='GC', bc=(0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for Dirichlet boundary conditions.

The basis \(\{\phi_k\}_{k=0}^{N-1}\) is

\[\begin{split}\phi_k &= T_k - T_{k+2}, \, k=0, 1, \ldots, N-3, \\ \phi_{N-2} &= \frac{1}{2}(T_0-T_1), \\ \phi_{N-1} &= \frac{1}{2}(T_0+T_1),\end{split}\]

such that

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u(-1) &= a \text{ and } u(1) = b.\end{split}\]

The last two bases are for boundary conditions and only used if a or b are different from 0. In one dimension \(\hat{u}_{N-2}=a\) and \(\hat{u}_{N-1}=b\).

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (2-tuple of floats, optional) – Boundary conditions at, respectively, x=(-1, 1).

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

class shenfun.chebyshev.bases.ShenNeumann(N, quad='GC', bc=(0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for Neumann boundary conditions

The basis \(\{\phi_k\}_{k=0}^{N-1}\) is

\[\begin{split}\phi_k &= T_{k} - \left(\frac{k}{k+2}\right)^2 T_{k+2}, \, k=0, 1, \ldots, N-3, \\ \phi_{N-2} &= \frac{1}{8}(4T_1-T_2), \\ \phi_{N-1} &= \frac{1}{8}(4T_1+T_2),\end{split}\]

such that

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u'(-1) &= a \text{ and } u'(1) = b.\end{split}\]

The last two bases are for boundary conditions and only used if a or b are different from 0. In one dimension \(\hat{u}_{N-2}=a\) and \(\hat{u}_{N-1}=b\).

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (2-tuple of floats, optional) – Boundary condition values at, respectively, x=(-1, 1).

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

class shenfun.chebyshev.bases.UpperDirichlet(N, quad='GC', bc=(None, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space with single Dirichlet on upper edge

The basis \(\{\phi_k\}_{k=0}^{N-1}\) is

\[\begin{split}\phi_k &= T_{k} - T_{k+1}, \, k=0, 1, \ldots, N-2, \\ \phi_{N-1} &= T_0,\end{split}\]

such that

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u(1) &= a.\end{split}\]

The last basis function is for boundary condition and only used if a is different from 0. In one dimension \(\hat{u}_{N-1}=a\).

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (2-tuple of (None, number), optional) – The number is the boundary condition value

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

class shenfun.chebyshev.bases.UpperDirichletNeumann(N, quad='GC', bc=(0, 0), domain=(-1, 1), dtype=<class 'float'>, padding_factor=1, dealias_direct=False, coordinates=None, **kw)

Bases: CompositeBase

Function space for both Dirichlet and Neumann boundary conditions on the right hand side.

The basis \(\{\phi_k\}_{k=0}^{N-1}\) is

\[\begin{split}\phi_k &= T_{k} - \frac{4(k+1)}{2k+3}T_{k+1} + \frac{2k+1}{2k+3}T_{k+2}, \, k=0, 1, \ldots, N-3, \\ \phi_{N-2} &= T_0, \\ \phi_{N-1} &= -T_0+T_1,\end{split}\]

such that

\[\begin{split}u(x) &= \sum_{k=0}^{N-1} \hat{u}_k \phi_k(x), \\ u(1) &= a, u'(1)=b.\end{split}\]

The last two bases are for boundary conditions and only used if a or b are different from 0. In one dimension \(\hat{u}_{N-2}=a\) and \(\hat{u}_{N-1}=b\).

Parameters:

• N (int, optional) – Number of quadrature points

• quad (str, optional) – Type of quadrature

  • GL - Chebyshev-Gauss-Lobatto

  • GC - Chebyshev-Gauss

• bc (2-tuple of numbers) – Boundary condition values at the right edge of domain

• domain (2-tuple of numbers, optional) – The computational domain

• dtype (data-type, optional) – Type of input data in real physical space. Will be overloaded when basis is part of a TensorProductSpace.

• padding_factor (float, optional) – Factor for padding backward transforms.

• dealias_direct (bool, optional) – Set upper 1/3 of coefficients to zero before backward transform

• coordinates (2- or 3-tuple (coordinate, position vector (, sympy assumptions)), optional) – Map for curvilinear coordinatesystem, and parameters to Coordinates

static boundary_condition()

static short_name()

shenfun.chebyshev.la module

class shenfun.chebyshev.la.ADDSolver(mats)

Bases: SparseMatrixSolver

Poisson_Solve_ADD = <shenfun.optimization.runtimeoptimizer object>

perform_lu()

solve(b, u, axis, lu)

Solve Au=b

Solve along axis if b and u are multidimensional arrays.

Parameters:

• b, u (arrays of rhs and output) – Both can be multidimensional

• axis (int) – The axis we are solving over

• lu (LU-decomposition) – Can be either the output from splu, or a dia-matrix containing the L and U matrices. The latter is used in subclasses.

class shenfun.chebyshev.la.ANNSolver(mats)

Bases: SparseMatrixSolver

__call__(b, u=None, axis=0, constraints=((0, 0),))

Solve matrix problem Au = b along axis

This routine also applies boundary conditions and constraints, and performes LU-decomposition on the fully assembled matrix.

Parameters:

• b (array) – Array of right hand side on entry and solution on exit unless u is provided.

• u (array, optional) – Output array

• axis (int, optional) – The axis over which to solve for if b and u are multidimensional

• constraints (tuple of 2-tuples) – The 2-tuples represent (row, val) The constraint indents the matrix row and sets b[row] = val

Note

If u is not provided, then b is overwritten with the solution and returned

class shenfun.chebyshev.la.Biharmonic(*args)

Bases: object

Multidimensional Biharmonic solver for

\[a_0 u'''' + \alpha u'' + \beta u = b\]

where \(u\) is the solution, \(b\) is the right hand side and \(a_0, \alpha\) and \(\beta\) are scalars, or arrays of scalars for a multidimensional problem.

The user must provide mass, stiffness and biharmonic matrices with associated scale arrays \((a_0/\alpha/\beta)\). The matrices and scales can be provided in any order

Parameters:

• S (TPMatrix or SpectralMatrix)

• A (TPMatrix or SpectralMatrix)

• B (TPMatrix or SpectralMatrix)

• scale_S (array, optional)

• scale_A (array, optional)

• scale_B (array, optional)

If only three arguments are passed, then we decide which matrix is which through inspection. The three scale arrays must then be available as S.scale, A.scale, B.scale. If six arguments are provided they must be in order S, A, B, scale S, scale A, scale B.

Variables are extracted from the matrices

The solver can be used along any axis of a multidimensional problem. For example, if the Chebyshev basis (Biharmonic) is the last in a 3-dimensional TensorProductSpace, where the first two dimensions use Fourier, then the 1D equation listed above arises when one is solving the 3D biharmonic equation

\[\nabla^4 u = b\]

With the spectral Galerkin method we multiply this equation with a test function (\(v\)) and integrate (weighted inner product \((\cdot, \cdot)_w\)) over the domain

\[(v, \nabla^4 u)_w = (v, b)_w\]

See the Poisson problem for details, since it is actually quite involved. But basically, one obtains a linear algebra system to be solved along the \(z\)-axis for all combinations of the two Fourier indices \(k\) and \(l\)

\[(S_{mj} - 2(k^2 + l^2) A_{mj}) + (k^2 + l^2)^2 B_{mj}) \hat{u}[k, l, j] = (v, b)_w[k, l, m]\]

Note that \(k\) only varies along \(x\)-direction, whereas \(l\) varies along \(y\). To allow for Numpy broadcasting these two variables are stored as arrays of shape

\[ \begin{align}\begin{aligned}k : (N, 1, 1)\\l : (1, M, 1)\end{aligned}\end{align} \]

Here it is assumed that the solution array \(\hat{u}\) has shape (N, M, P). Now, multiplying \(k\) array with \(\hat{u}\) is achieved as

\[k \cdot \hat{u}\]

Numpy will then take care of broadcasting \(k\) to an array of shape (N, M, P) before performing the elementwise multiplication. Likewise, the constant scale \(1\) in front of the \(A_{mj}\) matrix is stored with shape (1, 1, 1), and multiplying with \(\hat{u}\) is performed as if it was a scalar (as it here happens to be).

This is where the scale arrays in the signature to the Helmholt solver comes from. \(a_0\) is here \(1\), whereas \(\alpha\) and \(\beta\) are \(-2(k^2+l^2)\) and \((k^2+l^2)^2\), respectively. Note that \(k+l\) is an array of shape (N, M, 1).

Biharmonic_Solve = <shenfun.optimization.runtimeoptimizer object>

Biharmonic_factor_pr = <shenfun.optimization.runtimeoptimizer object>

Biharmonic_matvec = <shenfun.optimization.runtimeoptimizer object>

LU_Biharmonic = <shenfun.optimization.runtimeoptimizer object>

__call__(b, u=None, **kw)

Solve matrix problem

Parameters:

• b (array) – Array of right hand side on entry and solution on exit unless u is provided.

• u (array) – Output array

If b and u are multidimensional, then the axis over which to solve for is determined on creation of the class.

matvec(v, c)

class shenfun.chebyshev.la.Helmholtz(*args)

Bases: object

Helmholtz solver

\[\alpha u'' + \beta u = b\]

where \(u\) is the solution, \(b\) is the right hand side and \(\alpha\) and \(\beta\) are scalars, or arrays of scalars for a multidimensional problem.

The user must provide mass and stiffness matrices with scale arrays \((\alpha/\beta)\) to each matrix. The matrices and scales can be provided as instances of TPMatrix, or SpectralMatrix.

Parameters:

• A (SpectralMatrix or TPMatrix) – mass or stiffness matrix

• B (SpectralMatrix or TPMatrix) – mass or stiffness matrix

• scale_A (array, optional) – Scale array to stiffness matrix

• scale_B (array, optional) – Scale array to mass matrix

The two matrices must be one stiffness and one mass matrix. Which is which will be found by inspection if only two arguments are provided. The scales \(\alpha\) and \(\beta\) must then be available as A.scale and B.scale. If four arguments are provided they must be in the order

• stiffness matrix, mass matrix, scale stiffness, scale mass

The solver can be used along any axis of a multidimensional problem. For example, if the Chebyshev basis (Dirichlet or Neumann) is the last in a 3-dimensional TensorProductSpace, where the first two dimensions use Fourier, then the 1D Helmholtz equation arises when one is solving the 3D Poisson equation

\[\nabla^2 u = b\]

With the spectral Galerkin method we multiply this equation with a test function (\(v\)) and integrate (weighted inner product \((\cdot, \cdot)_w\)) over the domain

\[(v, \nabla^2 u)_w = (v, b)_w\]

See Poisson’s equation for details, since it is actually quite involved. But basically, one obtains a linear algebra system to be solved along the \(z\)-axis for all combinations of the two Fourier indices \(k\) and \(l\)

\[(A_{mj} - (k^2 + l^2) B_{mj}) \hat{u}[k, l, j] = (v, b)_w[k, l, m]\]

Note that \(k\) only varies along \(x\)-direction, whereas \(l\) varies along \(y\). To allow for Numpy broadcasting these two variables are stored as arrays of shape

\[ \begin{align}\begin{aligned}k : (N, 1, 1)\\l : (1, M, 1)\end{aligned}\end{align} \]

Here it is assumed that the solution array \(\hat{u}\) has shape (N, M, P). Now, multiplying k array with \(\hat{u}\) is achieved as an elementwise multiplication

\[k \cdot \hat{u}\]

Numpy will then take care of broadcasting \(k\) to an array of shape (N, M, P) before performing the elementwise multiplication. Likewise, the constant scale \(1\) in front of the \(A_{mj}\) matrix is stored with shape (1, 1, 1), and multiplying with \(\hat{u}\) is performed as if it was a scalar (as it here happens to be).

This is where the scale arrays come from. \(\alpha\) is here \(1\), whereas \(\beta\) is \((k^2+l^2)\). Note that \(k+l\) is an array of shape (N, M, 1).

Helmholtz_Neumann_matvec = <shenfun.optimization.runtimeoptimizer object>

Helmholtz_matvec = <shenfun.optimization.runtimeoptimizer object>

LU_Helmholtz = <shenfun.optimization.runtimeoptimizer object>

Solve_Helmholtz = <shenfun.optimization.runtimeoptimizer object>

__call__(b, u=None, constraints=())

Solve matrix problem

Parameters:

• b (array) – Array of right hand side on entry and solution on exit unless u is provided.

• u (array) – Output array

If b and u are multidimensional, then the axis over which to solve for is determined on creation of the class.

matvec(v, c)

Matrix vector product c = dot(self, v)

Parameters:

• v (array)

• c (array)

Returns:

c

Return type:

array

shenfun.chebyshev.matrices module

This module contains specific inner product matrices for the different bases in the Chebyshev family.

A naming convention is used for the first capital letter for all matrices. The first letter refers to type of matrix.

• Mass matrices start with B

• One derivative start with C

• Two derivatives (Laplace) start with A

• Four derivatives (Biharmonic) start with S

A matrix may consist of different types of test and trialfunctions. The next letters in the matrix name uses the ‘short_name’ method for all these different bases, see chebyshev.bases.py.

So a mass matrix using ShenDirichlet test and ShenNeumann trial is named BSDSNmat.

All matrices in this module may be looked up using the ‘mat’ dictionary, which takes test and trialfunctions along with the number of derivatives to be applied to each. As such the mass matrix BSDSDmat may be looked up as

>>> from shenfun.chebyshev.matrices import mat
>>> from shenfun.chebyshev.bases import ShenDirichlet as SD
>>> B = mat[((SD, 0), (SD, 0))]

and an instance of the matrix can be created as

>>> B0 = SD(10)
>>> BM = B((B0, 0), (B0, 0))

Check that this matrix corresponds to the matrix ‘d’ hardcoded below:

>>> import numpy as np
>>> d = {-2: -np.pi/2,
...       0: np.array([ 1.5*np.pi, np.pi, np.pi, np.pi, np.pi, np.pi, np.pi, np.pi]),
...       2: -np.pi/2}
>>> [np.all(BM[k] == v) for k, v in d.items()]
[True, True, True]

However, this way of creating matrices is not reccommended use. It is far more elegant to use the TrialFunction/TestFunction interface, and to generate the matrix as an inner product:

>>> from shenfun import TrialFunction, TestFunction, inner
>>> u = TrialFunction(B0)
>>> v = TestFunction(B0)
>>> BM = inner(u, v)
>>> [np.all(BM[k] == v) for k, v in d.items()]
[True, True, True]

To see that this is in fact the BSDSDmat:

>>> print(BM.__class__)
<class 'shenfun.chebyshev.matrices.BSDSDmat'>

class shenfun.chebyshev.matrices.ACNCNmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Stiffness matrix \(A=(a_{kj}) \in \mathbb{R}^{M \times N}\), where

\[a_{kj} = (\phi''_j, \phi_k)_w\]

where \(\phi_k \in\) chebyshev.bases.CombinedShenNeumann, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

get_solver()

Return appropriate solver for self

Note

Fall back on generic Solve, which is using Scipy sparse matrices with splu/spsolve. This is still pretty fast.

class shenfun.chebyshev.matrices.AHHHHmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Stiffness matrix \(A=(a_{kj}) \in \mathbb{R}^{M \times N}\), where

\[a_{kj} = (\phi''_j, \phi_k)_w\]

where \(\phi_k \in\) chebyshev.bases.Heinrichs, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

class shenfun.chebyshev.matrices.ASBSBmat(test, trial, scale=1, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Stiffness matrix \(A=(a_{kj}) \in \mathbb{R}^{M \times N}\), where

\[a_{kj} = (\phi''_j, \phi_k)_w\]

where \(\phi_k \in\) chebyshev.bases.ShenBiharmonic, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

class shenfun.chebyshev.matrices.ASBSDmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Stiffness matrix \(A=(a_{kj}) \in \mathbb{R}^{M \times N}\), where

\[a_{kj} = (\psi''_j, \phi_k)_w\]

where the test function \(\phi_k \in\) chebyshev.bases.ShenBiharmonic, the trial function \(\psi_j \in\) chebyshev.bases.ShenDirichlet, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

class shenfun.chebyshev.matrices.ASBTmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Stiffness matrix \(A=(a_{kj}) \in \mathbb{R}^{M \times N}\), where

\[a_{kj} = (T''_j, \phi_k)_w\]

where the test function \(\phi_k \in\) chebyshev.bases.ShenBiharmonic, the trial function \(T_j \in\) chebyshev.bases.Orthogonal, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

class shenfun.chebyshev.matrices.ASDHHmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Stiffness matrix \(A=(a_{kj}) \in \mathbb{R}^{M \times N}\), where

\[a_{kj} = (\psi''_j, \phi_k)_w\]

where the test function \(\phi_k \in\) chebyshev.bases.ShenDirichlet, the trial function \(\psi_j \in\) chebyshev.bases.Heinrichs, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

get_solver()

Return appropriate solver for self

Note

Fall back on generic Solve, which is using Scipy sparse matrices with splu/spsolve. This is still pretty fast.

class shenfun.chebyshev.matrices.ASDMNmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Stiffness matrix \(A=(a_{kj}) \in \mathbb{R}^{M \times N}\), where

\[a_{kj} = (\psi''_j, \phi_k)_w\]

where the test function \(\phi_k \in\) chebyshev.bases.ShenDirichlet, the trial function \(\psi_j \in\) chebyshev.bases.MikNeumann, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

get_solver()

Return appropriate solver for self

Note

Fall back on generic Solve, which is using Scipy sparse matrices with splu/spsolve. This is still pretty fast.

class shenfun.chebyshev.matrices.ASDSDmat(test, trial, scale=1, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Stiffness matrix \(A=(a_{kj}) \in \mathbb{R}^{M \times N}\), where

\[a_{kj} = (\phi''_j, \phi_k)_w\]

where \(\phi_k \in\) chebyshev.bases.ShenDirichlet, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

get_solver()

Return appropriate solver for self

Note

Fall back on generic Solve, which is using Scipy sparse matrices with splu/spsolve. This is still pretty fast.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

class shenfun.chebyshev.matrices.ASNCNmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Stiffness matrix \(A=(a_{kj}) \in \mathbb{R}^{M \times N}\), where

\[a_{kj} = (\psi''_j, \phi_k)_w\]

where the test function \(\phi_k \in\) chebyshev.bases.ShenNeumann, the trial function \(\psi_j \in\) chebyshev.bases.CombinedShenNeumann, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

get_solver()

Return appropriate solver for self

Note

Fall back on generic Solve, which is using Scipy sparse matrices with splu/spsolve. This is still pretty fast.

class shenfun.chebyshev.matrices.ASNSNmat(test, trial, scale=1, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Stiffness matrix \(A=(a_{kj}) \in \mathbb{R}^{M \times N}\), where

\[a_{kj} = (\phi''_j, \phi_k)_w\]

where \(\phi_k \in\) chebyshev.bases.ShenNeumann, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

get_solver()

Return appropriate solver for self

Note

Fall back on generic Solve, which is using Scipy sparse matrices with splu/spsolve. This is still pretty fast.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

class shenfun.chebyshev.matrices.ATSDmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Stiffness matrix \(A=(a_{kj}) \in \mathbb{R}^{M \times N}\), where

\[a_{kj} = (\psi''_j, T_k)_w\]

where the test function \(T_k \in\) chebyshev.bases.Orthogonal, the trial function \(\psi_j \in\) chebyshev.bases.ShenDirichlet, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

class shenfun.chebyshev.matrices.ATSNmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Stiffness matrix \(A=(a_{kj}) \in \mathbb{R}^{M \times N}\), where

\[a_{kj} = (\psi''_j, T_k)_w\]

where the test function \(T_k \in\) chebyshev.bases.Orthogonal, the trial function \(\psi_j \in\) chebyshev.bases.ShenNeumann, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

class shenfun.chebyshev.matrices.ATTmat(test, trial, scale=1, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Stiffness matrix \(A=(a_{kj}) \in \mathbb{R}^{M \times N}\), where

\[a_{kj} = (T''_j, T_k)_w\]

where \(T_k \in\) chebyshev.bases.Orthogonal, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

class shenfun.chebyshev.matrices.BLDLDmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Mass matrix \(B=(b_{kj}) \in \mathbb{R}^{M \times N}\), where

\[b_{kj}=(\phi_j, \phi_k)_w,\]

where \(\phi_k \in\) chebyshev.bases.LowerDirichlet, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

class shenfun.chebyshev.matrices.BSBSBmat(test, trial, scale=1, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Mass matrix \(B=(b_{kj}) \in \mathbb{R}^{M \times N}\), where

\[b_{kj}=(\phi_j, \phi_k)_w,\]

where \(\phi_k \in\) chebyshev.bases.ShenBiharmonic, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

get_solver()

Return appropriate solver for self

Note

Fall back on generic Solve, which is using Scipy sparse matrices with splu/spsolve. This is still pretty fast.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

class shenfun.chebyshev.matrices.BSBSDmat(test, trial, scale=1, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Mass matrix \(B=(b_{kj}) \in \mathbb{R}^{M \times N}\), where

\[b_{kj}=(\psi_j, \phi_k)_w,\]

where the test function \(\phi_k \in\) chebyshev.bases.ShenBiharmonic, the trial \(\psi_j \in\) chebyshev.bases.ShenDirichlet, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

class shenfun.chebyshev.matrices.BSBTmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Mass matrix \(B=(b_{kj}) \in \mathbb{R}^{M \times N}\), where

\[b_{kj}=(T_j, \phi_k)_w,\]

where the test function \(\phi_k \in\) chebyshev.bases.ShenBiharmonic, the trial \(T_j \in\) chebyshev.bases.Orthogonal, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

class shenfun.chebyshev.matrices.BSDSDmat(test, trial, scale=1, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Mass matrix \(B=(b_{kj}) \in \mathbb{R}^{M \times N}\), where

\[b_{kj}=(\phi_j, \phi_k)_w,\]

where \(\phi_j \in\) chebyshev.bases.ShenDirichlet, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

get_solver()

Return appropriate solver for self

Note

Fall back on generic Solve, which is using Scipy sparse matrices with splu/spsolve. This is still pretty fast.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

class shenfun.chebyshev.matrices.BSDSNmat(test, trial, scale=1, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Mass matrix \(B=(b_{kj}) \in \mathbb{R}^{M \times N}\), where

\[b_{kj}=(\psi_j, \phi_k)_w,\]

where the test function \(\phi_k \in\) chebyshev.bases.ShenDirichlet, the trial \(\psi_j \in\) chebyshev.bases.ShenNeumann, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

class shenfun.chebyshev.matrices.BSDTmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Mass matrix \(B=(b_{kj}) \in \mathbb{R}^{M \times N}\), where

\[b_{kj}=(T_j, \phi_k)_w,\]

where the test function \(\phi_k \in\) chebyshev.bases.ShenDirichlet, the trial \(\psi_j \in\) chebyshev.bases.Orthogonal, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

class shenfun.chebyshev.matrices.BSNSDmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Mass matrix \(B=(b_{kj}) \in \mathbb{R}^{M \times N}\), where

\[b_{kj}=(\psi_j, \phi_k)_w,\]

where the test function \(\phi_k \in\) chebyshev.bases.ShenNeumann, the trial \(\psi_j \in\) chebyshev.bases.ShenDirichlet, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

class shenfun.chebyshev.matrices.BSNSNmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Mass matrix \(B=(b_{kj}) \in \mathbb{R}^{M \times N}\), where

\[b_{kj}=(\phi_j, \phi_k)_w,\]

where \(\phi_k \in\) chebyshev.bases.ShenNeumann, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

get_solver()

Return appropriate solver for self

Note

Fall back on generic Solve, which is using Scipy sparse matrices with splu/spsolve. This is still pretty fast.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

class shenfun.chebyshev.matrices.BTLmat(test, trial, scale=1, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Mass matrix \(B=(b_{kj}) \in \mathbb{R}^{M \times N}\), where

\[b_{kj}=(L_j, T_k)_w,\]

where \(T_k \in\) chebyshev.bases.Orthogonal, \(L_j \in\) shenfun.legendre.bases.Orthogonal, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

class shenfun.chebyshev.matrices.BTSDmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Mass matrix \(B=(b_{kj}) \in \mathbb{R}^{M \times N}\), where

\[b_{kj}=(\psi_j, T_k)_w,\]

where the test function \(T_k \in\) chebyshev.bases.Orthogonal, the trial \(\psi_j \in\) chebyshev.bases.ShenDirichlet, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

class shenfun.chebyshev.matrices.BTSNmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Mass matrix \(B=(b_{kj}) \in \mathbb{R}^{M \times N}\), where

\[b_{kj}=(\psi_j, T_k)_w,\]

where the test function \(T_k \in\) chebyshev.bases.Orthogonal, the trial \(\psi_j \in\) chebyshev.bases.ShenNeumann, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

class shenfun.chebyshev.matrices.BTTmat(test, trial, scale=1, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Mass matrix \(B=(b_{kj}) \in \mathbb{R}^{M \times N}\), where

\[b_{kj}=(T_j, T_k)_w,\]

where \(T_k \in\) chebyshev.bases.Orthogonal, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

matvec(v, c, format='csr', axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

solve(b, u=None, axis=0, constraints=())

Solve matrix system Au = b

where A is the current matrix (self)

Parameters:

• b (array) – Array of right hand side on entry and solution on exit is u is None

• u (array, optional) – Solution array if provided

• axis (int, optional) – The axis over which to solve for if u is multi- dimensional

• constraints (tuple of 2-tuples) – The 2-tuples represent (row, val) The constraint indents the matrix row and sets b[row] = val

class shenfun.chebyshev.matrices.CSBSDmat(test, trial, scale=1, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Derivative matrix \(C=(c_{kj}) \in \mathbb{R}^{M \times N}\), where

\[c_{kj}=(\psi'_j, \phi_k)_w,\]

where the test function \(\phi_k \in\) chebyshev.bases.ShenBiharmonic, the trial \(\psi_j \in\) chebyshev.bases.ShenDirichlet, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

class shenfun.chebyshev.matrices.CSDSBmat(test, trial, scale=1, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Derivative matrix \(C=(c_{kj}) \in \mathbb{R}^{M \times N}\), where

\[c_{kj}=(\psi'_j, \phi_k)_w,\]

where the test function \(\phi_k \in\) chebyshev.bases.ShenDirichlet, the trial \(\psi_j \in\) chebyshev.bases.ShenBiharmonic, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

class shenfun.chebyshev.matrices.CSDSDmat(test, trial, scale=1, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Derivative matrix \(C=(c_{kj}) \in \mathbb{R}^{M \times N}\), where

\[c_{kj}=(\phi'_j, \phi_k)_w,\]

where \(\phi_k \in\) chebyshev.bases.ShenDirichlet, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

class shenfun.chebyshev.matrices.CSDSNmat(test, trial, scale=1, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Derivative matrix \(C=(c_{kj}) \in \mathbb{R}^{M \times N}\), where

\[c_{kj}=(\psi'_j, \phi_k)_w,\]

where the test function \(\phi_k \in\) chebyshev.bases.ShenDirichlet, the trial \(\psi_j \in\) chebyshev.bases.ShenNeumann, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

class shenfun.chebyshev.matrices.CSDTmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Derivative matrix \(C=(c_{kj}) \in \mathbb{R}^{M \times N}\), where

\[c_{kj}=(T'_j, \phi_k)_w,\]

where the test function \(\phi_k \in\) chebyshev.bases.ShenDirichlet, the trial \(T_j \in\) chebyshev.bases.Orthogonal, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

class shenfun.chebyshev.matrices.CSNSDmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Derivative matrix \(C=(c_{kj}) \in \mathbb{R}^{M \times N}\), where

\[c_{kj}=(\psi'_j, \phi_k)_w,\]

where the test function \(\phi_k \in\) chebyshev.bases.ShenNeumann, the trial \(\psi_j \in\) chebyshev.bases.ShenDirichlet, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

class shenfun.chebyshev.matrices.CTSDmat(test, trial, scale=1.0, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Derivative matrix \(C=(c_{kj}) \in \mathbb{R}^{M \times N}\), where

\[c_{kj}=(\psi'_j, T_k)_w,\]

where the test function \(T_k \in\) chebyshev.bases.Orthogonal, the trial \(\psi_j \in\) chebyshev.bases.ShenDirichlet, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

class shenfun.chebyshev.matrices.CTTmat(test, trial, scale=1, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Derivative matrix \(C=(c_{kj}) \in \mathbb{R}^{M \times N}\), where

\[c_{kj}=(T'_j, T_k)_w,\]

where \(T_j \in\) chebyshev.bases.Orthogonal, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

class shenfun.chebyshev.matrices.SSBSBmat(test, trial, scale=1, measure=1, assemble=None, kind=None, fixed_resolution=None)

Bases: SpectralMatrix

Biharmonic matrix \(S=(s_{kj}) \in \mathbb{R}^{M \times N}\), where

\[s_{kj}=(\phi''''_j, \phi_k)_w,\]

where \(\phi_k \in\) chebyshev.bases.ShenBiharmonic, and test and trial spaces have dimensions of M and N, respectively.

assemble(method)

Return diagonals of SpectralMatrix

Parameters:

method (str) – Type of integration

• ‘exact’

• ‘quadrature’

Note

Subclass SpectralMatrix and overload this method in order to provide a fast and accurate implementation of the matrix representing an inner product. See the matrix modules in either one of

• legendre.matrix

• chebyshev.matrix

• chebyshevu.matrix

• ultraspherical.matrix

• fourier.matrix

• laguerre.matrix

• hermite.matrix

• jacobi.matrix

Example

The mass matrix for Chebyshev polynomials is

\[(T_j, T_i)_{\omega} = \frac{c_i \pi}{2}\delta_{ij},\]

where \(c_0=2\) and \(c_i=1\) for integer \(i>0\). We can implement this as

>>> from shenfun import SpectralMatrix
>>> class Bmat(SpectralMatrix):
...     def assemble(self, method):
...         test, trial = self.testfunction, self.trialfunction
...         ci = np.ones(test[0].N)
...         ci[0] = 2
...         if test[0].quad == 'GL' and method != 'exact':
...             # Gauss-Lobatto quadrature inexact at highest polynomial order
...             ci[-1] = 2
...         return {0: ci*np.pi/2}

Here {0: ci*np.pi/2} is the 0’th diagonal of the matrix. Note that test and trial are two-tuples of (instance of :class:.SpectralBase`, number)`, where the number represents the number of derivatives. For the mass matrix the number will be 0. Also note that the length of the diagonal must be correct.

matvec(v, c, format=None, axis=0)

Matrix vector product

Returns c = dot(self, v)

Parameters:

• v (array) – Numpy input array of ndim>=1

• c (array) – Numpy output array of same shape as v

• format (str, optional) – Choice for computation

  • csr - Compressed sparse row format

  • dia - Sparse matrix with DIAgonal storage

  • python - Use numpy and vectorization

  • self - To be implemented in subclass

  • cython - Cython implementation that may be implemented in subclass

  • numba - Numba implementation that may be implemented in subclass

  Using config['matrix']['sparse']['matvec'] setting if format is None

• axis (int, optional) – The axis over which to take the matrix vector product

shenfun.chebyshev.matrices.dmax(N, M, d)

Module contents

Functionality for working with Chebyshev bases