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# Import required packages
import numpy as np
import matplotlib.pyplot as plt
# ---------------------------------------------------------------------------
# PART 1: Visualizing Truncation Error and Approximate Integral
# ---------------------------------------------------------------------------
# Generate true curve: y = sin(x)
x_fine = np.linspace(0, np.pi/2, 400)
y_fine = np.sin(x_fine)
# Linear approximation (straight line between 0 and pi/2)
x_line = np.array([0, np.pi/2])
y_line = np.sin(x_line)
# Plot true curve and linear approximation
plt.figure(figsize=(8, 6))
plt.plot(x_fine, y_fine, label='True function: $y = \sin(x)$', color='black', linewidth=3)
plt.plot(x_line, y_line, label='Linear approximation', color='red', linestyle='--', linewidth=2)
# Shade area between true curve and linear approximation (truncation error)
plt.fill_between(x_fine, y_fine, np.interp(x_fine, x_line, y_line),
where=(y_fine > np.interp(x_fine, x_line, y_line)),
interpolate=True, color='gray', alpha=0.5, label='Truncation Error')
# Shade approximate integral (area under linear approximation)
plt.fill_between(x_fine, 0, np.interp(x_fine, x_line, y_line),
color='lightblue', alpha=0.5, label='Approximate Integral')
# Remove ticks
plt.xticks([])
plt.yticks([])
plt.grid(False)
# Labels and title
plt.xlabel('x', fontsize=20)
plt.ylabel('y', fontsize=20)
plt.title('Truncation Error', fontsize=22)
plt.legend(fontsize=16, loc='lower right')
plt.tight_layout()
plt.show()
# ---------------------------------------------------------------------------
# PART 2: Error Plot for Truncation, Round-off, and Total Errors
# ---------------------------------------------------------------------------
# Set up for log-log error plot
h = np.logspace(-16, -1, 400)
truncation_error = 1e-1 * h
roundoff_error = 1e-16 / h
total_error = truncation_error + roundoff_error
# Find minimum total error point
min_idx = np.argmin(total_error)
min_h = h[min_idx]
arrow_target_h = 2.5 * min_h
arrow_target_err = total_error[np.argmin(np.abs(h - arrow_target_h))]
# Plot errors
plt.figure(figsize=(10, 7))
plt.loglog(h, truncation_error, label='Truncation error', linewidth=3, color='blue')
plt.loglog(h, roundoff_error, label='Round-off error', linewidth=3, color='orange')
plt.loglog(h, total_error, label='Total error', linewidth=4, color='black')
# Annotation
plt.annotate('Point of\n diminishing\n returns',
xy=(arrow_target_h, arrow_target_err), xycoords='data',
xytext=(arrow_target_h, 15 * arrow_target_err), textcoords='data',
arrowprops=dict(arrowstyle='->', lw=2),
fontsize=20, ha='center')
# Remove ticks and grid
plt.xticks([])
plt.yticks([])
plt.grid(False)
# Labels and legend
plt.xlabel('Log step size', fontsize=22)
plt.ylabel('Log error', fontsize=22)
plt.legend(fontsize=20, loc='upper right')
plt.tight_layout()
plt.show()
# ---------------------------------------------------------------------------
# PART 3: Error Plot for Truncation with more steps
# ---------------------------------------------------------------------------
# Fix legend duplication and improve clarity
import numpy as np
import matplotlib.pyplot as plt
# Generate true curve: y = sin(x) over 0 to π
x_fine = np.linspace(0, np.pi, 800)
y_fine = np.sin(x_fine)
# Choose multiple linear segments
n_steps = 3 # number of straight line segments
x_steps = np.linspace(0, np.pi, n_steps + 1)
y_steps = np.sin(x_steps)
# Plot true curve
plt.figure(figsize=(8, 6))
plt.plot(x_fine, y_fine, label='True function: $y = \\sin(x)$', color='black', linewidth=3)
# Plot piecewise linear approximation
plt.plot(x_steps, y_steps, label='Linear approximation', color='red', linestyle='--', linewidth=2)
# Track whether we've added the truncation label already
truncation_label_added = False
# Shade truncation error (avoid duplicate legend entry)
for i in range(n_steps):
x_segment = np.linspace(x_steps[i], x_steps[i+1], 100)
y_segment_line = np.interp(x_segment, [x_steps[i], x_steps[i+1]], [y_steps[i], y_steps[i+1]])
y_segment_true = np.sin(x_segment)
label = 'Truncation error' if not truncation_label_added else None
plt.fill_between(x_segment, y_segment_true, y_segment_line,
where=(y_segment_true > y_segment_line),
interpolate=True, color='gray', alpha=0.5, label=label)
truncation_label_added = True
# Shade approximate integral under the piecewise linear approximation
plt.fill_between(x_fine, 0, np.interp(x_fine, x_steps, y_steps),
color='lightblue', alpha=0.5, label='Approximate integral')
# Remove ticks
plt.xticks([])
plt.yticks([])
plt.grid(False)
# Labels and title
plt.xlabel('x', fontsize=20)
plt.ylabel('y', fontsize=20)
plt.title('Truncation Error', fontsize=24)
plt.legend(fontsize=16, loc='lower right')
plt.tight_layout()
plt.show()
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