ShaneLogic/complex-root-finding

Complex Root Finding

The Problem: Spurious Roots

A polynomial approximating a fun on an interval (e.g., [-1, 1]) will have complex roots that are spurious - unrelated to the actual function due to approximation properties.

Finding Genuine Complex Roots

Basic Command

% Find genuine complex roots near the interval
f = chebfun(@(x) exp(x) - 2, [-1, 1]);
roots_complex = roots(f, 'complex');

How Filtering Works

1. First computes all roots: roots(f, 'all')
2. Filters results using "Chebfun ellipses"
3. Returns only roots inside the specific ellipse

Chebfun Ellipse Definition

For interval [-1, 1] and fun length L:

• Map: (z + 1/z) / 2
• Circle: |z| = r where r^(-L) = delta
• Parameter: delta = sqrt(eps) (Chebfun tolerance)

The ellipse is the image of this circle under the map.

Accuracy Control

Understanding Accuracy Degradation

Complex roots lose accuracy as they move away from the definition interval:

• Near interval: ~11 digits accuracy
• Far from interval: ~5 digits accuracy
• Very far: roots may disappear entirely

Using the 'norecursion' Flag

For complicated chebfuns with many complex roots:

% Standard computation (may miss some roots)
r1 = roots(f, 'complex');

% With recursion control (more accurate, slower)
r2 = roots(f, 'complex', 'norecursion');

Trade-offs

| Setting | Speed | Accuracy | |---------|-------|----------| | Default | Fast | May miss roots near interfaces | | 'norecursion' | Slower | Better accuracy, fewer misses |

Verification

% Check accuracy by evaluating at roots
residuals = abs(f(r2));
max_residual = max(residuals);

% Lower residuals indicate better accuracy

When to Use

• Finding complex roots of functions defined on real intervals
• Avoiding spurious polynomial roots
• Maximizing accuracy for complex root finding
• Analyzing root distribution in complex plane
• Working with functions having many complex roots