def integrate_clenshaw(f, a, b, maxn = 2**16, tol = 10e-16,
debug_info = True, debug_plot = True):
n = 3
prevI = None
nodes = None
cm = convergence_monitor()
while n <= maxn:
coeffs = clenshaw_coefficients(n)
if nodes is None:
nodes = cheb_nodes(n, a, b)
fs = f(nodes)
else:
new_nodes = incremental_cheb_nodes(n, a, b)
new_fs = f(new_nodes)
nodes, fs = combine_interpolation_nodes_fast(nodes, fs,
new_nodes, new_fs)
I = dot(fs, coeffs) * 0.5 * (b - a)
if prevI is not None:
err = abs(I - prevI)
cm.add(err, I)
if cm.test_convergence()[0]:
break
prevI = I
n = 2 * n - 1
if debug_info:
print "===="
if debug_plot and n >= maxn: # problems with integration
debug_plot(a, b, nodes, fs, coeffs)
return I, err
def integrate_fejer2(f,
a,
b,
par=None,
maxn=2**10,
tol=finfo(double).eps,
debug_info=False,
debug_plot=False):
if par is not None:
maxn = par.maxn
debug_plot = par.debug_plot
debug_info = par.debug_info
cm = convergence_monitor(par=par.convergence)
else:
cm = convergence_monitor()
n = 65
prevI = None
nodes = None
while n <= maxn:
coeffs = fejer2_coefficients(n)
if nodes is None:
nodes = cheb_nodes(n, a, b)[1:-1]
fs = f(nodes)
else:
new_nodes = incremental_cheb_nodes(n, a, b)
new_fs = f(new_nodes)
# roles of new and old nodes are reversed in the call below
nodes, fs = combine_interpolation_nodes_fast(
new_nodes, new_fs, nodes, fs)
I = dot(fs, coeffs) * (b - a) / 2
if prevI is not None:
err = abs(I - prevI)
if debug_info:
print repr(I), err, n, I + err == I, err <= abs(I) * tol, min(
nodes), max(nodes), min(fs), max(fs)
cm.add(err, I)
if cm.test_convergence()[0]:
break
prevI = I
n = 2 * n - 1
if debug_info:
print "===="
if debug_plot and n >= maxn: # problems with integration
debug_plot(a, b, nodes, fs, coeffs)
# return currently best result
I, err, _extra = cm.get_best_result()
return I, err
def integrate_fejer2(f, a, b, par = None, maxn = 2**10, tol = finfo(double).eps,
debug_info = False, debug_plot = False):
if par is not None:
maxn = par.maxn
debug_plot = par.debug_plot
debug_info = par.debug_info
cm = convergence_monitor(par = par.convergence)
else:
cm = convergence_monitor()
n = 65
prevI = None
nodes = None
while n <= maxn:
coeffs = fejer2_coefficients(n)
if nodes is None:
nodes = cheb_nodes(n, a, b)[1:-1]
fs = f(nodes)
else:
new_nodes = incremental_cheb_nodes(n, a, b)
new_fs = f(new_nodes)
# roles of new and old nodes are reversed in the call below
nodes, fs = combine_interpolation_nodes_fast(new_nodes, new_fs,
nodes, fs)
I = dot(fs, coeffs) * (b - a) / 2
if prevI is not None:
err = abs(I - prevI)
if debug_info:
print repr(I), err, n, I + err == I, err <= abs(I) * tol, min(nodes), max(nodes), min(fs), max(fs)
cm.add(err, I)
if cm.test_convergence()[0]:
break
prevI = I
n = 2 * n - 1
if debug_info:
print "===="
if debug_plot and n >= maxn: # problems with integration
debug_plot(a, b, nodes, fs, coeffs)
# return currently best result
I, err, _extra = cm.get_best_result()
return I, err
def add_nodes(self, new_Xs, new_Ys):
"""Add new interpolation nodes. A faster version assuming nesting is possible here."""
self.Xs, self.Ys = combine_interpolation_nodes_fast(self.Xs, self.Ys, new_Xs, new_Ys)
self.init_weights(self.Xs)