bisection_method_posts.txt

# Social Media Posts: Bisection Method for Root Finding

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## SHORT-FORM POSTS
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### Twitter/X (280 chars)
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The bisection method: when in doubt, split it in half!

Finding where f(x) = x³ - x - 2 crosses zero:
→ Start with [1, 2]
→ 37 iterations later: x ≈ 1.521379706804568

Simple, robust, guaranteed to work.

#Python #NumericalMethods #Math

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### Bluesky (300 chars)
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Implemented the bisection method for root finding today.

Given f(x) = x³ - x - 2 on [1, 2]:
• Error halves each iteration: εₙ = (b₀ - a₀)/2ⁿ⁺¹
• Found root x ≈ 1.521379706804568 in 37 iterations
• Linear convergence, but guaranteed to work for any continuous function

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### Threads (500 chars)
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Ever need to find where a function equals zero? The bisection method is your reliable friend.

Here's how it works:
1. Pick an interval [a, b] where f(a) and f(b) have opposite signs
2. Check the midpoint c = (a + b)/2
3. Narrow down to whichever half contains the root
4. Repeat until you're close enough

For f(x) = x³ - x - 2, starting from [1, 2]:
→ 37 iterations
→ Root: x ≈ 1.521379706804568
→ Error bound: 10⁻¹⁰

Not the fastest method, but it ALWAYS works. That's the beauty of it.

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### Mastodon (500 chars)
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Exploring the bisection method for root finding.

For a continuous f(x) with f(a)·f(b) < 0, the Intermediate Value Theorem guarantees a root in (a, b).

Algorithm:
• Compute midpoint: c = (a + b)/2
• Update interval based on sign of f(c)
• Error after n iterations: |r - cₙ| ≤ (b₀ - a₀)/2ⁿ⁺¹

Results for f(x) = x³ - x - 2 on [1, 2]:
• Root: 1.521379706804568
• 37 iterations for ε = 10⁻¹⁰
• Linear convergence (rate = 1/2)

Also found the Dottie number solving cos(x) - x = 0!

#NumericalAnalysis #Python #Mathematics

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## LONG-FORM POSTS
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### Reddit (r/learnpython or r/math)
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**Title:** Visualizing the Bisection Method: A Simple but Powerful Root-Finding Algorithm

**Body:**

I created a Jupyter notebook exploring the bisection method, one of the most fundamental algorithms in numerical analysis. Here's what I learned:

**The Basic Idea (ELI5)**

Imagine you're playing a number guessing game. Someone picks a number between 1 and 100, and tells you "higher" or "lower" after each guess. The optimal strategy? Always guess the middle.

That's exactly what the bisection method does for finding roots (where a function equals zero).

**How It Works**

1. Start with an interval [a, b] where f(a) and f(b) have opposite signs
2. The Intermediate Value Theorem guarantees a root exists between them
3. Compute the midpoint c = (a + b)/2
4. Check which half contains the root (based on sign of f(c))
5. Repeat with the new, smaller interval

**Key Results**

For f(x) = x³ - x - 2 on [1, 2]:
- Found root x ≈ 1.521379706804568
- Required 37 iterations for tolerance 10⁻¹⁰
- Error bound after n iterations: (b₀ - a₀)/2ⁿ⁺¹

**Why It Matters**

The bisection method has LINEAR convergence (error halves each step), which is slower than Newton-Raphson's quadratic convergence. But here's the tradeoff:

✓ Guaranteed to converge for continuous functions
✓ No derivatives needed
✓ Numerically stable
✗ Needs initial bracketing with sign change
✗ Can't find tangent roots

**Bonus:** Also found the Dottie number (≈0.739) by solving cos(x) - x = 0. It's the unique fixed point of cosine!

Check out the full notebook with visualizations:
https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/bisection_method.ipynb

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### Facebook (500 chars)
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Ever wondered how computers find exact solutions to equations?

The bisection method is beautifully simple: keep cutting the search space in half until you find the answer.

For the equation x³ - x - 2 = 0:
• Start between 1 and 2
• 37 halvings later → x ≈ 1.5214

It's like a high-low guessing game, but for math! Slow but guaranteed to work.

View the interactive notebook:
https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/bisection_method.ipynb

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### LinkedIn (1000 chars)
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Exploring Numerical Methods: The Bisection Algorithm

Just completed an implementation of the bisection method for root finding—a foundational algorithm in numerical analysis that demonstrates key principles of computational mathematics.

**Technical Implementation:**
• Python with NumPy/Matplotlib
• Full iteration history tracking
• Convergence visualization and error analysis

**Key Findings:**
For f(x) = x³ - x - 2 on interval [1, 2]:
• Converged to x = 1.521379706804568 in 37 iterations
• Achieved error bound of 10⁻¹⁰
• Verified theoretical convergence: εₙ = (b₀ - a₀)/2ⁿ⁺¹

**Why This Matters:**
The bisection method offers guaranteed convergence for continuous functions—critical when reliability matters more than speed. It's often used to:
• Provide initial estimates for faster algorithms (Newton-Raphson)
• Solve problems where derivatives are unavailable
• Handle functions with numerical instabilities

**Skills Demonstrated:**
• Numerical analysis theory and implementation
• Scientific visualization
• Algorithm convergence analysis

Full notebook with interactive code:
https://cocalc.com/github/Ok-landscape/computational-pipeline/blob/main/notebooks/published/bisection_method.ipynb

#NumericalMethods #Python #DataScience #ComputationalMathematics #ScientificComputing

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### Instagram (500 chars)
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Finding where math meets zero

The bisection method is elegant in its simplicity:
→ Pick an interval
→ Split in half
→ Keep the half with the answer
→ Repeat

For x³ - x - 2 = 0:
Starting interval: [1, 2]
After 37 halvings: x ≈ 1.5214

The error halves every single time.
Guaranteed convergence.
No fancy calculus required.

Sometimes the simplest approach
is the most reliable.

Swipe to see the convergence plots →

#NumericalMethods
#Mathematics
#Python
#DataVisualization
#CodingLife
#STEM
#ScienceIsBeautiful

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