def cond_number(x):
# 2: [-1, y0, -1]
# (3, 4): [-1, y0, -y0, -1], [-1, -y0, 0, y0, 1],
# (5, 6): [-1, y0, y1, -y1, -y0, -1], [-1, y0, y1, 0, -y1, -y0, -1]
# One-d optimzation
if vary:
x_ = chebyshev_points(deg)
x_[vary] = x
x_[deg-vary] = -x
dof_set = x_
# Multi-d optimization
else:
if deg == 2:
dof_set = np.r_[-1, x, 1]
# Combine no zero
elif deg % 2 == 1:
dof_set = np.r_[-1, x, -x[::-1], 1]
# Combine w/ zero
else:
dof_set = np.r_[-1, x, 0, -x[::-1], 1]
element = LagrangeElement(poly_set, dof_set)
alpha = element.alpha
M_ = alpha.dot(M.dot(alpha.T))
return np.linalg.cond(M_)
def cg_optimal_dofs_restricted(deg, vary=0):
'''
What are optimal dofs that give smallest condition number in L^2 norm.
By CG I mean that I constraint two dofs to be at (-1, 1). The remaining
deg-1 points are to be determined. For even degrees symmetry is dof at 0 is
also forced
For vary None: the symmetry is used and `all` the points are used for
search, i.e. this is multi-d problem.
For vary=int: that guy and its mirror are changed, i.e. this is
1-d optimization problem.
'''
poly_set = leg.basis_functions(deg)
M = leg.mass_matrix(deg)
# Only allow negative (make life easier for mirroring) and not midpoint or -1
if vary: assert 0 < vary < (deg/2 + 1 if deg % 2 else deg/2)
# Want to minimize this
def cond_number(x):
# 2: [-1, y0, -1]
# (3, 4): [-1, y0, -y0, -1], [-1, -y0, 0, y0, 1],
# (5, 6): [-1, y0, y1, -y1, -y0, -1], [-1, y0, y1, 0, -y1, -y0, -1]
# One-d optimzation
if vary:
x_ = chebyshev_points(deg)
x_[vary] = x
x_[deg-vary] = -x
dof_set = x_
# Multi-d optimization
else:
if deg == 2:
dof_set = np.r_[-1, x, 1]
# Combine no zero
elif deg % 2 == 1:
dof_set = np.r_[-1, x, -x[::-1], 1]
# Combine w/ zero
else:
dof_set = np.r_[-1, x, 0, -x[::-1], 1]
element = LagrangeElement(poly_set, dof_set)
alpha = element.alpha
M_ = alpha.dot(M.dot(alpha.T))
return np.linalg.cond(M_)
# Initial guess
# 1d optim
x0 = chebyshev_points(deg)
if vary:
x0 = x0[vary]
else:
# Multi-d optim
if deg == 2:
x0 = x0[1]
# Combine no zero
elif deg % 2 == 1:
x0 = x0[1:deg/2+1]
else:
x0 = x0[1:deg/2]
# Optimize
res = minimize(cond_number, x0)
return res, x0, cond_number(x0)
