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 ```matlab % 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: ```matlab % 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 ```matlab % 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