def endpoints(f):
f0 = sp.mirror0(sp.matC(f), -1)
aa = f - sp.unmirror0(
sp.I_space(0, h / 2)(sp.I_space_inv(-h / 2, h / 2)(f0)))
bb = f - sp.unmirror0(
sp.I_space(-h / 2, 0)(sp.I_space_inv(-h / 2, h / 2)(f0)))
return sp.matB(aa) + sp.matB1(bb)
def midpoints(f):
f = sp.mirror0(sp.matC(f), -1)
# Shift function with S, Sinv to avoid division by zero at x=0, x=pi
f = sp.I_space_inv(-h/2, h/2)(f)
f = sp.T_space(+h/2)(f)
f = f/sp.Omega_d(f.shape[0])
f = sp.T_space(-h/2)(f)
f = sp.unmirror0(f)
return f
def midpoints(f):
f = sp.mirror0(sp.matC(f), -1)
# Shift function with S, Sinv to avoid division by zero at x=0, x=pi
f = sp.I_space_inv(-h / 2, h / 2)(f)
f = sp.T_space(+h / 2)(f)
f = f / sp.Omega_d(f.shape[0])
f = sp.T_space(-h / 2)(f)
f = sp.unmirror0(f)
return f
def H0_cheb(f):
'''
>>> sp.to_matrix(H0_cheb, 2)
array([[ 0.75, 0.25],
[ 0.25, 0.75]])
'''
f = sp.mirror0(f, +1)
N = f.shape[0]; h = 2*pi/N
f = sp.F(f)
f = sp.fourier_K(f, 0, h/2)
f = sp.Finv(f)
f = sp.fold0(f, -1)
return real(f)
def H0_cheb(f):
'''
>>> sp.to_matrix(H0_cheb, 2)
array([[ 0.75, 0.25],
[ 0.25, 0.75]])
'''
f = sp.mirror0(f, +1)
N = f.shape[0]
h = 2 * pi / N
f = sp.F(f)
f = sp.fourier_K(f, 0, h / 2)
f = sp.Finv(f)
f = sp.fold0(f, -1)
return real(f)
def S_cheb(f):
'''
Interpolate from primal to dual vertices.
>>> S_cheb(array([-1, 0, 1]))
array([-0.70710678, 0.70710678])
>>> S_cheb(array([1, 0, -1]))
array([ 0.70710678, -0.70710678])
>>> sp.to_matrix(S_cheb, 3).round(3)
array([[ 0.604, 0.5 , -0.104],
[-0.104, 0.5 , 0.604]])
'''
f = sp.mirror0(f, +1)
N = f.shape[0]; h = 2*pi/N
f = sp.Finv(sp.F(f)*sp.S_diag(N, h/2))
f = sp.unmirror1(f)
return real(f)
def H1d_regular(f):
r'''
.. math::
\widetilde{\mathbf{H}}^{1}=
\mathbf{M}_{1}^{\dagger}
{\mathbf{I}^{-\frac{h}{2},\frac{h}{2}}}^{-1}
\mathbf{M}_{1}^{+}
\mathbf{A}^{-1}
'''
f = f/sp.A_diag(f.shape[0])
f = sp.mirror0(f, +1)
N = f.shape[0]; h = 2*pi/N
f = sp.I_space_inv(-h/2, +h/2)(f)
f = sp.unmirror0(f)
return f
def H0_regular(f):
r'''
.. math::
\mathbf{H}^{0}=
\mathbf{A}
\mathbf{M}_{0}^{\dagger}
\mathbf{I}^{-\frac{h}{2},\frac{h}{2}}
\mathbf{M}_{0}^{+}
'''
f = sp.mirror0(f, +1)
N = f.shape[0]; h = 2*pi/N
f = sp.I_space(-h/2, h/2)(f)
f = sp.unmirror0(f)
f = f*sp.A_diag(f.shape[0])
return f
def S_cheb(f):
'''
Interpolate from primal to dual vertices.
>>> S_cheb(array([-1, 0, 1]))
array([-0.70710678, 0.70710678])
>>> S_cheb(array([1, 0, -1]))
array([ 0.70710678, -0.70710678])
>>> sp.to_matrix(S_cheb, 3).round(3)
array([[ 0.604, 0.5 , -0.104],
[-0.104, 0.5 , 0.604]])
'''
f = sp.mirror0(f, +1)
N = f.shape[0]
h = 2 * pi / N
f = sp.Finv(sp.F(f) * sp.S_diag(N, h / 2))
f = sp.unmirror1(f)
return real(f)
def H0_regular(f):
r'''
.. math::
\mathbf{H}^{0}=
\mathbf{A}
\mathbf{M}_{0}^{\dagger}
\mathbf{I}^{-\frac{h}{2},\frac{h}{2}}
\mathbf{M}_{0}^{+}
'''
f = sp.mirror0(f, +1)
N = f.shape[0]
h = 2 * pi / N
f = sp.I_space(-h / 2, h / 2)(f)
f = sp.unmirror0(f)
f = f * sp.A_diag(f.shape[0])
return f
def H1d_regular(f):
r'''
.. math::
\widetilde{\mathbf{H}}^{1}=
\mathbf{M}_{1}^{\dagger}
{\mathbf{I}^{-\frac{h}{2},\frac{h}{2}}}^{-1}
\mathbf{M}_{1}^{+}
\mathbf{A}^{-1}
'''
f = f / sp.A_diag(f.shape[0])
f = sp.mirror0(f, +1)
N = f.shape[0]
h = 2 * pi / N
f = sp.I_space_inv(-h / 2, +h / 2)(f)
f = sp.unmirror0(f)
return f
def endpoints(f):
f0 = sp.mirror0(sp.matC(f), -1)
aa = f - sp.unmirror0(sp.I_space(0, h/2)(sp.I_space_inv(-h/2, h/2)(f0)))
bb = f - sp.unmirror0(sp.I_space(-h/2, 0)(sp.I_space_inv(-h/2, h/2)(f0)))
return sp.matB(aa) + sp.matB1(bb)
