def error6(N):
cN = F(1, 6) + F(10, 36 * N)
coeffs = (1, -5, 25, -125, 625, -3125)
p_exact = de_casteljau.basic(cN, coeffs)
p_computed = de_casteljau.basic(float(cN), coeffs)
err_exact = abs((F(p_computed) - p_exact) / p_exact)
return float(err_exact)
def error5(N):
bN = F(1, 5) + F(8, 25 * N)
coeffs = (1, -4, 16, -64, 256, -1024)
p_exact = de_casteljau.basic(bN, coeffs)
p_computed = de_casteljau.basic(float(bN), coeffs)
err_exact = abs((F(p_computed) - p_exact) / p_exact)
return float(err_exact)
def error4(N):
aN = F(1, 4) + F(6, 16 * N)
coeffs = (1, -3, 9, -27, 81, -243)
p_exact = de_casteljau.basic(aN, coeffs)
p_computed = de_casteljau.basic(float(aN), coeffs)
err_exact = abs((F(p_computed) - p_exact) / p_exact)
return float(err_exact)
def count_de_casteljau_basic():
print("de_casteljau.basic() ((3n^2 + 3n + 2) / 2 = 3 T_n + 1):")
for degree in range(1, 5 + 1):
parent = operation_count.Computation()
x = operation_count.Float(0.25, parent)
coeffs = tuple(
operation_count.Float((-1.0)**k, parent)
for k in range(degree + 1))
p = de_casteljau.basic(x, coeffs)
assert p.value == 0.5**degree
assert parent.count == 3 * (degree * (degree + 1) // 2) + 1
print(" degree {}: {}".format(degree, parent.display))
def _main():
s_vals = np.linspace(ROOT - DELTA_S, ROOT + DELTA_S, NUM_POINTS)
horner1 = []
de_casteljau1 = []
for s in s_vals:
horner1.append(horner.basic(s, POLY_COEFFS))
de_casteljau1.append(de_casteljau.basic(s, BEZIER_COEFFS))
figure, (ax1, ax2) = plt.subplots(1, 2, sharex=True, sharey=True)
ax1.plot(s_vals, horner1)
ax2.plot(s_vals, de_casteljau1)
# Since ``sharex=True``, ticks only need to be set once.
ax1.set_xticks([
ROOT - DELTA_S,
ROOT - 0.5 * DELTA_S,
ROOT,
ROOT + 0.5 * DELTA_S,
ROOT + DELTA_S,
])
ax1.set_title(r"$\mathtt{Horner}$", fontsize=plot_utils.TEXT_SIZE)
ax2.set_title(r"$\mathtt{DeCasteljau}$", fontsize=plot_utils.TEXT_SIZE)
ax1.tick_params(labelsize=plot_utils.TICK_SIZE, which="both")
ax2.tick_params(labelsize=plot_utils.TICK_SIZE, which="both")
filename = "horner_inferior.pdf"
# NOTE: These are (intended to be) the same settings used in
# ``compensated_insufficient.py``, so they should probably be
# kept in sync.
figure.set_size_inches(6.0, 2.9)
figure.subplots_adjust(left=0.07,
bottom=0.13,
right=0.97,
top=0.92,
wspace=0.13,
hspace=0.20)
path = plot_utils.get_path("k-compensated", filename)
figure.savefig(path)
print("Saved {}".format(filename))
plt.close(figure)
def main(filename=None):
s_vals = np.linspace(ROOT - DELTA_S, ROOT + DELTA_S, NUM_POINTS)
horner1 = []
de_casteljau1 = []
for s in s_vals:
horner1.append(horner.basic(s, POLY_COEFFS))
de_casteljau1.append(de_casteljau.basic(s, BEZIER_COEFFS))
figure, (ax1, ax2) = plt.subplots(1, 2, sharex=True, sharey=True)
ax1.plot(s_vals, horner1)
ax2.plot(s_vals, de_casteljau1)
# Since ``sharex=True``, ticks only need to be set once.
ax1.set_xticks([
ROOT - DELTA_S,
ROOT - 0.5 * DELTA_S,
ROOT,
ROOT + 0.5 * DELTA_S,
ROOT + DELTA_S,
])
ax1.set_title(r"$\mathtt{Horner}$")
ax2.set_title(r"$\mathtt{DeCasteljau}$")
if filename is None:
plt.show()
else:
figure.set_size_inches(9.87, 4.8)
figure.subplots_adjust(
left=0.06,
bottom=0.12,
right=0.97,
top=0.92,
wspace=0.13,
hspace=0.20,
)
path = plot_utils.get_path(filename)
figure.savefig(path, bbox_inches="tight")
print("Saved {}".format(filename))
plt.close(figure)
def standard_residual(s, coeffs1, t, coeffs2):
x1 = de_casteljau.basic(s, coeffs1[0, :])
y1 = de_casteljau.basic(s, coeffs1[1, :])
x2 = de_casteljau.basic(t, coeffs2[0, :])
y2 = de_casteljau.basic(t, coeffs2[1, :])
return np.array([[x1 - x2], [y1 - y2]])
def do_plot(ax, s_vals, coeffs, title, add_legend=False, add_ylabel=False):
bounds = []
rel_errors1 = []
rel_errors2 = []
n = len(coeffs) - 1
gamma3n = utils.gamma(3 * n)
exact_coeffs = tuple(map(F, coeffs))
abs_coeffs = tuple(map(abs, exact_coeffs))
for s in s_vals:
exact_s = F(s)
exact_p = de_casteljau.basic(exact_s, exact_coeffs)
if not isinstance(exact_p, F):
raise TypeError(exact_p)
exact_p_tilde = de_casteljau.basic(exact_s, abs_coeffs)
a_priori_bound = gamma3n * exact_p_tilde / abs(exact_p)
bounds.append(float(a_priori_bound))
error1 = F(de_casteljau.basic(s, coeffs)) - exact_p
rel_errors1.append(float(abs(error1 / exact_p)))
error2 = F(vs_method.basic(s, coeffs)) - exact_p
rel_errors2.append(float(abs(error2 / exact_p)))
size = 5
ax.semilogy(
s_vals,
bounds,
marker="o",
markersize=size,
linestyle=":",
alpha=0.5,
color="black",
label="Bound",
)
ax.semilogy(
s_vals,
rel_errors1,
marker="o",
markersize=size,
label=r"$\mathtt{DeCasteljau}$",
)
ax.semilogy(
s_vals,
rel_errors2,
marker="s",
markersize=size,
label=r"$\mathtt{VS}$",
)
if add_legend:
ax.legend(
loc="lower center",
framealpha=1.0,
frameon=True,
markerscale=1.25,
fontsize=16,
)
# Label the axes.
ax.set_xlabel("$s$", fontsize=20)
if add_ylabel:
ax.set_ylabel("Relative Forward Error", fontsize=20)
# Set the axis title.
ax.set_title(title, fontsize=20)