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 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 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 S_cheb_pinv(f):
'''
Interpolate from dual to primal vertices.
Since there are smaller number of dual vertices,
this is only a pseudo inverse of S_cheb, and not an
exact inverse.
>>> allclose( S_cheb_pinv(array([ -1/sqrt(2), 1/sqrt(2)])),
... array([ -1, 0, 1]))
True
>>> allclose( S_cheb_pinv(array([ 1/sqrt(2), -1/sqrt(2)])),
... array([ 1, 0, -1]))
True
>>> sp.to_matrix(S_cheb_pinv, 2).round(3)
array([[ 1.207, -0.207],
[ 0.5 , 0.5 ],
[-0.207, 1.207]])
'''
f = sp.mirror1(f, +1)
N = f.shape[0]; h = 2*pi/N
f = sp.Finv(sp.F(f)*sp.S_diag(N, -h/2))
f = sp.unmirror0(f)
return real(f)
def S_cheb_pinv(f):
'''
Interpolate from dual to primal vertices.
Since there are smaller number of dual vertices,
this is only a pseudo inverse of S_cheb, and not an
exact inverse.
>>> allclose( S_cheb_pinv(array([ -1/sqrt(2), 1/sqrt(2)])),
... array([ -1, 0, 1]))
True
>>> allclose( S_cheb_pinv(array([ 1/sqrt(2), -1/sqrt(2)])),
... array([ 1, 0, -1]))
True
>>> sp.to_matrix(S_cheb_pinv, 2).round(3)
array([[ 1.207, -0.207],
[ 0.5 , 0.5 ],
[-0.207, 1.207]])
'''
f = sp.mirror1(f, +1)
N = f.shape[0]
h = 2 * pi / N
f = sp.Finv(sp.F(f) * sp.S_diag(N, -h / 2))
f = sp.unmirror0(f)
return real(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)
