def test_quad_vs_cq():
"""
Проверяем, как работают ИКФ и СКФ при одинакомом количестве обращений к функции
"""
x0, x1 = 0, np.pi/2
p = Harmonic(0, 1)
y0 = p[x0, x1]
n_nodes = 2
ys_nt_ct = []
ys_gauss = []
ys_cquad = []
n_intervals = 2 ** np.arange(8)
for n in n_intervals:
n_evals = n * n_nodes
ys_nt_ct.append(quad(p, x0, x1, np.linspace(0, 1, n_evals) * (x1-x0) + x0))
ys_gauss.append(quad_gauss(p, x0, x1, n_evals))
ys_cquad.append(composite_quad(p, x0, x1, n, n_nodes))
for ys, label in zip((ys_nt_ct, ys_gauss, ys_cquad),
(f'newton-cotes', f'gauss', f'{n_nodes}-node CQ')):
acc = get_accuracy(ys, y0 * np.ones_like(ys))
acc[acc > 17] = 17
plt.plot(np.log10(n_intervals), acc, '.:', label=label)
plt.legend()
plt.ylabel('accuracy')
plt.xlabel('log10(n_evals)')
plt.suptitle(f'test quad vs composite quad')
plt.show()
def test_composite_quad(n_nodes):
"""
Проверяем 2-, 3-, 5-узловые СКФ
Q: объясните скорость сходимости для каждого случая
"""
fig, ax = plt.subplots(1, 2)
x0, x1 = 0, 1
L = 2
n_intervals = [L ** q for q in range(0, 8)]
for i, degree in enumerate((5, 6)):
p = Monome(degree)
Y = [composite_quad(p, x0, x1, n_intervals=n, n_nodes=n_nodes) for n in n_intervals]
accuracy = get_accuracy(Y, p[x0, x1] * np.ones_like(Y))
x = np.log10(n_intervals)
# оценка сходимости
ind = np.isfinite(x) & np.isfinite(accuracy)
k, b = np.polyfit(x[ind], accuracy[ind], 1)
aitken_degree = aitken(*Y[0:6:2], L ** 2)
ax[i].plot(x, k*x+b, 'b:', label=f'{k:.2f}*x+{b:.2f}')
ax[i].plot(x, aitken_degree*x+b, 'm:', label=f'aitken ({aitken_degree:.2f})')
ax[i].plot(x, accuracy, 'kh', label=f'accuracy for x^{degree}')
ax[i].set_title(f'{n_nodes}-node CQ for x^{degree}')
ax[i].set_xlabel('log10(n_intervals)')
ax[i].set_ylabel('accuracy')
ax[i].legend()
if n_nodes < degree:
assert np.abs(aitken_degree - k) < 0.5, \
f'Aitken estimation {aitken_degree:.2f} is too far from actual {k:.2f}'
plt.show()
def test_composite_quad_degree(v):
"""
Проверяем сходимость СКФ при наличии неравных весов
Q: скорость сходимости оказывается дробной, почему?
"""
from .variants import params
a, b, alpha, beta, f = params(v)
x0, x1 = a, b
# a, b = -10, 10
L = 2
n_intervals = [L**q for q in range(2, 11)]
n_nodes = 3
exact = sp_quad(lambda x: f(x) / (x - a)**alpha / (b - x)**beta, x0, x1)[0]
Y = [
composite_quad(f,
x0,
x1,
n_intervals=n,
n_nodes=n_nodes,
a=a,
b=b,
alpha=alpha,
beta=beta) for n in n_intervals
]
accuracy = get_accuracy(Y, exact * np.ones_like(Y))
x = np.log10(n_intervals)
aitken_degree = aitken(*Y[5:8], L)
a1, a0 = np.polyfit(x, accuracy, 1)
assert a1 > 1, 'composite quad did not converge!'
fig, (ax1, ax2) = plt.subplots(1, 2)
# график весовой функции
xs = np.linspace(x0, x1, n_intervals[-1] + 1)
ys = 1 / ((xs - a)**alpha * (b - xs)**beta)
ax1.plot(xs, ys, label='weights')
ax = list(ax1.axis())
ax[2] = 0.
ax1.axis(ax)
ax1.set_xlabel('x')
ax1.set_ylabel('p(x)')
ax1.legend()
# график точности
ax2.plot(x, accuracy, 'kh')
ax2.plot(x, a1 * x + a0, 'b:', label=f'{a1:.2f}*x+{a0:.2f}')
ax2.set_xlabel('log10(n_intervals)')
ax2.set_ylabel('accuracy')
ax2.legend()
fig.suptitle(f'variant #{v} (alpha={alpha:4.2f}, beta={beta:4.2f})\n'
f'aitken estimation: {aitken_degree:.2f}')
fig.tight_layout()
plt.show()
def test_composite_quad():
plt.figure()
x0, x1 = 0, 1
L = 2
n_intervals = [L**q for q in range(0, 8)]
n_nodes = 3
for i, degree in enumerate((5, 6)):
p = Monome(degree)
Y = [
composite_quad(p, x0, x1, n_intervals=n, n_nodes=n_nodes)
for n in n_intervals
]
accuracy = get_log_error(Y, p[x0, x1] * np.ones_like(Y))
x = np.log10(n_intervals)
# check convergence
ind = np.isfinite(x) & np.isfinite(accuracy)
k, b = np.polyfit(x[ind], accuracy[ind], 1)
aitken_degree = aitken(*Y[0:6:2], L**2)
plt.subplot(1, 2, i + 1)
plt.plot(x, k * x + b, 'b:', label=f'{k:.2f}*x+{b:.2f}')
plt.plot(x, aitken_degree * x + b, 'm:', label=f'aitken estimation')
plt.plot(x, accuracy, 'kh', label=f'accuracy for x^{degree}')
plt.xlabel('log10(node count)')
plt.ylabel('accuracy')
plt.legend()
assert np.abs(aitken_degree - k) < 0.5, \
f'Aitken estimation {aitken_degree:.2f} is too far from actual {k:.2f}'
plt.show()
def test_gauss_vs_cq():
"""
check quad_gauss() versus composite_quad() on the same number of function evaluations
"""
x0, x1 = 0, np.pi/2
p = Harmonic(0, 1)
y0 = p[x0, x1]
n_nodes = 2
Y_gauss = []
Y_cquad = []
n_intervals = np.arange(1, 256, 5)
for n in n_intervals:
n_evals = n * n_nodes
Y_gauss.append(quad_gauss(p, x0, x1, n_evals))
Y_cquad.append(composite_quad(p, x0, x1, n, n_nodes))
accuracy_gauss = get_log_error(Y_gauss, y0 * np.ones_like(Y_gauss))
accuracy_gauss[accuracy_gauss > 17] = 17
accuracy_cquad = get_log_error(Y_cquad, y0 * np.ones_like(Y_cquad))
accuracy_cquad[accuracy_cquad > 17] = 17
plt.plot(np.log10(n_intervals), accuracy_gauss, '.:', label=f'gauss')
plt.plot(np.log10(n_intervals), accuracy_cquad, '.:', label=f'2-node CQ')
plt.legend()
plt.ylabel('accuracy')
plt.xlabel('log10(n_evals)')
plt.suptitle(f'test gauss vs CQ')
plt.show()
def test_composite_quad(n_nodes):
"""
test composite 2-, 3-, 5-node quads
Q: explain converge speed for each case
"""
fig, ax = plt.subplots(1, 2)
x0, x1 = 0, 1
L = 2
n_intervals = [L ** q for q in range(0, 8)]
for i, degree in enumerate((5, 6)):
p = Monome(degree)
Y = [composite_quad(p, x0, x1, n_intervals=n, n_nodes=n_nodes) for n in n_intervals]
accuracy = get_log_error(Y, p[x0, x1] * np.ones_like(Y))
x = np.log10(n_intervals)
# check convergence
ind = np.isfinite(x) & np.isfinite(accuracy)
k, b = np.polyfit(x[ind], accuracy[ind], 1)
aitken_degree = aitken(*Y[0:6:2], L ** 2)
ax[i].plot(x, k*x+b, 'b:', label=f'{k:.2f}*x+{b:.2f}')
ax[i].plot(x, aitken_degree*x+b, 'm:', label=f'aitken ({aitken_degree:.2f})')
ax[i].plot(x, accuracy, 'kh', label=f'accuracy for x^{degree}')
ax[i].set_title(f'{n_nodes}-node CQ for x^{degree}')
ax[i].set_xlabel('log10(n_intervals)')
ax[i].set_ylabel('accuracy')
ax[i].legend()
if n_nodes < degree:
assert np.abs(aitken_degree - k) < 0.5, \
f'Aitken estimation {aitken_degree:.2f} is too far from actual {k:.2f}'
plt.show()
def test_gauss_vs_cq():
"""
Проверяем, как работают quad_gauss() и composite_quad() при одинакомом количестве обращений к функции
"""
x0, x1 = 0, np.pi/2
p = Harmonic(0, 1)
y0 = p[x0, x1]
n_nodes = 2
Y_gauss = []
Y_cquad = []
n_intervals = np.arange(1, 256, 5)
for n in n_intervals:
n_evals = n * n_nodes
Y_gauss.append(quad_gauss(p, x0, x1, n_evals))
Y_cquad.append(composite_quad(p, x0, x1, n, n_nodes))
accuracy_gauss = get_accuracy(Y_gauss, y0 * np.ones_like(Y_gauss))
accuracy_gauss[accuracy_gauss > 17] = 17
accuracy_cquad = get_accuracy(Y_cquad, y0 * np.ones_like(Y_cquad))
accuracy_cquad[accuracy_cquad > 17] = 17
plt.plot(np.log10(n_intervals), accuracy_gauss, '.:', label=f'gauss')
plt.plot(np.log10(n_intervals), accuracy_cquad, '.:', label=f'2-node CQ')
plt.legend()
plt.ylabel('accuracy')
plt.xlabel('log10(n_evals)')
plt.suptitle(f'test gauss vs CQ')
plt.show()
def test_composite_quad_degree(v):
"""
Q: convergence maybe somewhat between 3 and 4, why?
"""
from .variants import params
fig, (ax1, ax2) = plt.subplots(1, 2)
a, b, alpha, beta, f = params(v)
x0, x1 = a, b
# a, b = -10, 10
exact = sp_quad(lambda x: f(x) / (x - a)**alpha / (b - x)**beta, x0, x1)[0]
# plot weights
xs = np.linspace(x0, x1, 101)
ys = 1 / ((xs - a)**alpha * (b - xs)**beta)
ax1.plot(xs, ys, label='weights')
ax = list(ax1.axis())
ax[2] = 0.
ax1.axis(ax)
ax1.set_xlabel('x')
ax1.set_ylabel('p(x)')
ax1.legend()
L = 2
n_intervals = [L**q for q in range(2, 10)]
n_nodes = 4
Y = [
composite_quad(f,
x0,
x1,
n_intervals=n,
n_nodes=n_nodes,
a=a,
b=b,
alpha=alpha,
beta=beta) for n in n_intervals
]
accuracy = get_log_error(Y, exact * np.ones_like(Y))
x = np.log10(n_intervals)
k, b = np.polyfit(x, accuracy, 1)
assert k > 1, 'composite quad did not converge!'
aitken_degree = aitken(*Y[5:8], L)
# plot acc
ax2.plot(x, accuracy, 'kh')
ax2.plot(x, k * x + b, 'b:', label=f'{k:.2f}*x+{b:.2f}')
ax2.set_xlabel('log10(n_intervals)')
ax2.set_ylabel('accuracy')
ax2.legend()
fig.suptitle(f'variant #{v} (alpha={alpha:4.2f}, beta={beta:4.2f})\n'
f'aitken estimation: {aitken_degree:.2f}')
plt.show()
def test_composite_quad_degree(v):
"""
Q: convergence maybe somewhat between 3 and 4, why?
"""
from .variants import params
from .variants import exactint
plt.figure()
a, b, alpha, beta, f = params(v)
x0, x1 = a, b
#a, b = -10, 10
exact = exactint(
v) #sp_quad(lambda x: f(x) / (x-a)**alpha / (b-x)**beta, x0, x1)[0]
# plot weights
xs = np.linspace(x0, x1, 101)
ys = 1 / (xs - a)**alpha / (b - xs)**beta
plt.subplot(1, 2, 1)
plt.plot(xs, ys, label='weights')
ax = list(plt.axis())
ax[2] = 0.
plt.axis(ax)
plt.xlabel('x')
plt.ylabel('p(x)')
plt.legend()
L = 2
n_intervals = [L**q for q in range(2, 10)]
n_nodes = 3
Y = [
composite_quad(f,
x0,
x1,
n_intervals=n,
n_nodes=n_nodes,
a=a,
b=b,
alpha=alpha,
beta=beta) for n in n_intervals
]
accuracy = get_log_error(Y, exact * np.ones_like(Y))
x = np.log10(n_intervals)
aitken_degree = aitken(*Y[5:8], L)
# plot acc
plt.subplot(1, 2, 2)
plt.plot(x, accuracy, 'kh')
plt.xlabel(Y)
plt.ylabel(exact)
plt.suptitle(f'variant #{v} (alpha={alpha:4.2f}, beta={beta:4.2f})\n'
f'aitken estimation: {aitken_degree:.2f}')
plt.show()