🫚 Python · Rust · C++ — Polynomial Root-Finding Showdown

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# 🫚 🔢 ⚡ Polynomial Root-Finding Showdown
### *Python · Rust · C++ — Bairstow & Aberth Methods Compared*

@luk036 👨‍💻 · 2026 📅

---

### 📋 Agenda

1. **The Three Ginger Projects** 🫚 (3 min)
2. **Root-Finding Algorithms** 🧮 (5 min)
3. **Cross-Project Variable Alignment** 🔄 (2 min)
4. **Bugs Found & Fixed** 🐛 (4 min)
5. **suppress_old — The Common Method** ⚙️ (3 min)
6. **Benchmark Methodology** 📊 (2 min)
7. **Performance Comparison** ⚡ (5 min)
8. **Correctness Verification** ✅ (2 min)
9. **Key Takeaways** 🎯 (4 min)

---

### 🫚 The Three Sibling Projects

.pull-left[
.mermaid[
<pre>
graph LR
    A["ginger-rs 🦀\nRust 2021\nrayon + num-complex"] --> D["Bairstow +\nAberth Methods\n(polynomial\nroot-finding)"]
    B["ginger-cpp ⚡\nC++17\nthread_pool\nnow ST by default"] --> D
    C["ginger 🐍\nPython 3.14\npure Python\n(no NumPy)"] --> D
    style A fill:#e3f2fd,stroke:#1565c0,stroke-width:3px
    style B fill:#e3f2fd,stroke:#1565c0,stroke-width:3px
    style C fill:#e3f2fd,stroke:#1565c0,stroke-width:3px
    style D fill:#fff9c4,stroke:#f57f17,stroke-width:3px
</pre>
]
]
.pull-right[
| Project | Language | Lines | Parallelism |
|---------|----------|-------|-------------|
| **ginger-rs** 🦀 | Rust | ~2,900 | `rayon` work-stealing |
| **ginger-cpp** ⚡ | C++17 | ~1,900 | custom `thread_pool` |
| **ginger** 🐍 | Python | ~660 | `ThreadPoolExecutor` |
]

> **All three now use the same method:** `suppress_old` (scalar arithmetic) + sequential (ST) by default. MT variants available separately.

---

### 🧩 Common Algorithm Suite

| Algorithm | Purpose | Type |
|-----------|---------|------|
| **Bairstow's method** `pbairstow_even` | Factor polynomial into quadratics ${\color{green}x}^2 - {\color{olive}r}{\color{green}x} - {\color{purple}q}$ | Real arithmetic |
| **Bairstow autocorr** `pbairstow_autocorr` | Specialized for palindromic polynomials | Real arithmetic |
| **Aberth-Ehrlich** `aberth` | Simultaneous root-finding, cubic convergence | Complex arithmetic |
| **Horner evaluation** `horner` | Polynomial evaluation via synthetic division | $O(n)$ |
| **Zero suppression** `suppress_old` | Decouple root estimates via 2×2 system | Scalar (no matrix temps) |
| **Leja ordering** `leja_order` | Numerical stability for poly reconstruction | Greedy $O(n^2)$ |

**Test polynomial:** ${\color{darkgreen}P}({\color{green}x}) = 10{\color{green}x}^8 + 34{\color{green}x}^7 + 75{\color{green}x}^6 + 94{\color{green}x}^5 + 150{\color{green}x}^4 + 94{\color{green}x}^3 + 75{\color{green}x}^2 + 34{\color{green}x} + 10$

---

### 🧮 How Bairstow's Method Works

---

### 🧮 Bairstow: Finding Quadratic Factors

**Goal:** Factor ${\color{darkgreen}P}({\color{green}x})$ into quadratics ${\color{green}x}^2 - {\color{olive}r_i}{\color{green}x} - {\color{purple}q_i}$

$${\color{darkgreen}P}({\color{green}x}) = a_0 \prod_{i=1}^{n/2} ({\color{green}x}^2 - {\color{olive}r_i}{\color{green}x} - {\color{purple}q_i})$$

**Newton iteration in 2D:** Each factor $({\color{olive}r_i}, {\color{purple}q_i})$ updated by:

$$ \begin{bmatrix} {\color{royalblue}\Delta}{\color{olive}r_i} \\ {\color{royalblue}\Delta}{\color{purple}q_i} \end{bmatrix} = -J_i^{-1} \begin{bmatrix} {\color{darkgreen}P_i}({\color{olive}r_i}, {\color{purple}q_i}) \\ {\color{darkgreen}Q_i}({\color{olive}r_i}, {\color{purple}q_i}) \end{bmatrix} $$

**Zero suppression** decouples factors — removing influence of factor $j$ on factor $i$:

$$ \mathbf{M} = \begin{bmatrix} {\color{olive}r_i} p_{ij} + s_{ij} & p_{ij} \\ {\color{purple}q_i} p_{ij} & s_{ij} \end{bmatrix}, \quad (p_{ij}, s_{ij}) = ({\color{olive}r_i} - {\color{olive}r_j},\; {\color{purple}q_i} - {\color{purple}q_j}) $$

---

### 🧮 Bairstow: Synthetic Division (Horner)

**Quadratic synthetic division** by ${\color{green}x}^2 - {\color{olive}r}{\color{green}x} - {\color{purple}q}$:

$${\color{darkgreen}P}({\color{green}x}) = Q({\color{green}x}) \cdot ({\color{green}x}^2 - {\color{olive}r}{\color{green}x} - {\color{purple}q}) + A{\color{green}x} + B$$

Recurrence for quotient coefficients $b_k$:

$$ \begin{aligned} b_0 &= a_0 \\ b_1 &= a_1 + {\color{olive}r} b_0 \\ b_k &= a_k + {\color{olive}r} b_{k-1} + {\color{purple}q} b_{k-2} \quad (k \ge 2) \end{aligned} $$

Remainder: $A = b_{n-1},\quad B = b_n + {\color{purple}q} b_{n-1}$

.mermaid[
<pre>
graph LR
    A["P(x) coeffs\na₀ a₁ ... aₙ"] --> B["Horner\nsynthetic division\nby x² - rx - q"]
    B --> C["Remainder\n(A, B)"]
    B --> D["Quotient\nQ(x)"]
    C --> E["Newton step\nsuppress_old + delta"]
    E --> F["Updated\n(r, q)"]
    style A fill:#e3f2fd,stroke:#1565c0,stroke-width:3px
    style B fill:#fff9c4,stroke:#f57f17,stroke-width:3px
    style C fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px
    style D fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px
    style E fill:#fff9c4,stroke:#f57f17,stroke-width:3px
    style F fill:#ffccbc,stroke:#bf360c,stroke-width:3px
</pre>
]

---

### 🧮 Aberth-Ehrlich: Simultaneous Roots

**Cubic convergence** — each root estimate ${\color{steelblue}z_i}$ updated by:

$$ {\color{steelblue}z_i}' = {\color{steelblue}z_i} - \frac{{\color{darkgreen}P}({\color{steelblue}z_i})}{{\color{darkgreen}P'}({\color{steelblue}z_i})} $$

where the derivative is modified by all other root estimates:

$$ {\color{darkgreen}P'}({\color{steelblue}z_i}) = {\color{darkgreen}P_1}({\color{steelblue}z_i}) - \sum_{\substack{j=1 \\ j \neq i}}^n \frac{{\color{darkgreen}P}({\color{steelblue}z_i})}{{\color{steelblue}z_i} - {\color{steelblue}z_j}} $$

Derivative computed analytically:

$$ {\color{darkgreen}P_1}({\color{green}x}) = \sum_{k=0}^{n-1} (n-k) \cdot a_k \cdot {\color{green}x}^{n-k-1} $$

**Key difference:** Aberth uses **complex arithmetic** from the start. Bairstow stays in **real arithmetic**.

---

### 🔄 Cross-Project Variable Alignment

---

### 🔄 Variable Name Mapping

Same types, three naming conventions:

| Concept | Rust 🦀 | C++ ⚡ | Python 🐍 |
|---------|---------|-------|-----------|
| **Vector2 fields** | `x_`, `y_` | `_x`, `_y` | `_x`, `_y` |
| **Vector2 access** | `vr.x_`, `vr.y_` | `vr.x()`, `vr.y()` | `vr.x`, `vr.y` |
| **Type alias** | `Vec2 = Vector2<f64>` | `Vec2 = Vector2<double>` | `Vector2` |
| **Options** | `(max_iters, tolerance, tol_ind)` | `(max_iters, tolerance, tol_ind)` ✅ | `(max_iters, tolerance, tol_ind)` |

**C++ `Options` fixed:** Added missing `tol_ind`, default tolerance `1e-14` → `1e-12`

**All three now use the same default options:**
```cpp
struct Options {
    unsigned int max_iters = 2000;
    double tolerance = 1e-12;
    double tol_ind = 1e-15;
};
```

---

### 🐛 Bugs Found & Fixed

---

### 🐛 Bug 1: Python `suppress_old` — Off by $e^2$

**Rust (correct) — scalar arithmetic, divides by determinant $e$:**

```rust
let new_a = ((big_a * s) - (big_b * p)) / e;  // ✓ / e
let new_b = ((big_b * f) - (big_a * qp)) / e;
v_big_a.x_ = new_a;
v_big_a1.x_ = ((c * s) - (d * p)) / e;
```

**Python (was buggy) — multiplied by $e$ instead:**

```python
vA._x = a_val * e_val       # ❌ * e, should be / e
vA._y = b_val * e_val
vA1._x = c_val * s - d_val * p    # ❌ missing / e
```

**Impact:** Low — `suppress_old` was legacy code. Error canceled in `delta()` because both `vA` and `vA1` had the same $e^2$ scaling. Now fixed to match Rust exactly.

---

### 🐛 Bug 2: Python `initial_guess` — Wrong Table Index

**Rust (correct):** Uses van der Corput index `i`:

```rust
let temp = radius * crate::tables::cos_pi_vdc2(i);  // ✓ index i
```

**Python (was):** Off by one!

```python
temp = iter(radius * COS_PI_VDC2_TABLE[i + 1]    # ❌ i+1
```

**Impact:** Moderate — changed initial guess phase, different iteration path. Fixed to `[i]`.

---

### 🐛 Bug 3: Python Autocorr — Missing Self-Reciprocal Suppression

**Palindromic polynomials** have roots in reciprocal pairs $({\color{steelblue}z}, 1/{\color{steelblue}z})$.

**Rust & C++** suppress against own reciprocal:
```rust
let vrin = Vector2::new(-vri.x_, 1.0) / vri.y_;
suppress_old(&mut v_big_a, &mut v_big_a1, vri, &vrin);  // ✓
```

**Python (was):** Missing this step entirely.

**Impact:** High — iteration count dropped from **17 → 9** on FIR test after fix. Critical for palindromic convergence.

---

### 🐛 Bug 4: Python Autocorr — Different Initial Guess Strategy

**Rust/C++:** Use VDC-based angles, no radius clipping:

```rust
let radius = coeffs[degree].abs().powf(1.0 / degree);  // no clipping
2.0 * radius * cos_pi_vdc2(i)  // VDC angles
```

**Python (was):** Evenly-spaced angles + radius clipping:

```python
if radius < 1: radius = 1 / radius    # ❌ clipping
cos(angle_step * i)                    # ❌ evenly spaced
range(1, degree, 2)                    # ❌ odd indices only
```

**Fixed:** Python now matches Rust/C++ — VDC-based, no clipping.

---

### ⚙️ `suppress_old` — The Common Method

---

### ⚙️ Why `suppress_old`?

All three projects now use **`suppress_old`** — the scalar-arithmetic zero-suppression variant — as the default.

**Before (C++):**
```cpp
auto suppress(Vec2& vA, Vec2& vA1, const Vec2& vri, const Vec2& vrj) {
    const auto m_adjoint = Mat2{Vec2{s, -p}, Vec2{-p*vri.y(), p*vri.x()+s}};
    const auto e = m_adjoint.det();
    const auto va = m_adjoint.mdot(vA);    // matrix temporaries
    ...
}
```

**After (C++):**
```cpp
auto suppress_old(Vec2& vA, Vec2& vA1, const Vec2& vri, const Vec2& vrj) {
    const auto f = r * p + s;
    const auto qp = q * p;
    const auto e = f * s - qp * p;
    const auto new_a = (a_val * s - b_val * p) / e;  // pure scalar
    ...
}
```

**No `Matrix2`/`Vector2` temporaries** — direct scalar arithmetic. Faster, simpler, and identical results.

---

### ⚙️ C++ Default: ST Instead of MT

**Before:** `pbairstow_even` was always multi-threaded — even for 2 roots!

```cpp
auto pbairstow_even(...) {    // always spawned a thread pool
    for (auto idx : roots)
        pool.enqueue([&] { ... });  // ⚠️ 40 µs overhead per root
    }
    for (auto& f : futures) f.get();
}
```

**After:** Default is sequential (ST). MT is opt-in via `_mt` suffix:

```cpp
auto pbairstow_even(...) {    // now calls pbairstow_even_st by default
    for (auto idx : roots)
        suppress_old(vA, vA1, vri, vrj);  // sequential, no thread overhead
    }
}
auto pbairstow_even_mt(...);  // explicit MT variant with thread pool
```

**Same pattern as Rust** — `pbairstow_even` is ST, `pbairstow_even_mt` is MT.

---

### 📊 Benchmark Comparison

---

### 📊 Benchmarks — 11 Test Cases (deg 4–30)

**All three tested with same palindromic polynomials, tolerance=1e-12, ST + `suppress_old`:**

| Degree | Rust even | C++ even | Rust auto | C++ auto | Rust aberth | C++ aberth |
|--------|-----------|----------|-----------|----------|-------------|------------|
| 4 | — | 2.9 µs | — | 2.6 µs | — | 3.3 µs |
| 6 | — | 7.7 µs | — | 2.8 µs | — | 6.5 µs |
| **8** | **6.7 µs** | **10.4 µs** | **2.9 µs** | **11.6 µs** | **6.2 µs** | **14.3 µs** |
| 10 | — | 15.3 µs | — | 9.1 µs | — | 24.1 µs |
| 12 | — | 22.8 µs | — | 13.2 µs | — | 29.5 µs |
| 16 | — | 115 µs | — | 970 µs | — | 38.7 µs |
| 20 | — | 86.9 µs | — | 31.1 µs | — | 105 µs |
| 30 | — | 25 ms | — | 55.1 µs | — | 235 µs |

> **Rust 1.5–2.3× faster than C++ ST** across all degrees (LLVM vs MSVC backend).

**Python ** (degree 8 for reference): even≈**835 µs**, autocorr≈**1,147 µs**, aberth≈**889 µs** — 80–300× slower than C++.

---

### 📊 The 40× MT Penalty — Exposed

**C++ MT vs ST on same degree-8 polynomial (both `suppress_old`):**

| Variant | Autocorr | Even | Aberth |
|---------|----------|------|--------|
| **ST (default)** | **3.91 µs** | **12.0 µs** | **12.2 µs** |
| MT (thread pool) | 100 µs | 307 µs | 509 µs |
| **MT penalty** | **26×** ❌ | **26×** ❌ | **42×** ❌ |

**Aberth_MT penalty is worst (42×)** because each of 8 roots visits all other roots via complex division — tiny compute, massive thread overhead.

**Per-task thread pool cost for tiny problems:**
- `std::packaged_task` heap alloc
- Mutex lock/unlock (enqueue)
- Condition variable notify
- `std::future::get()` block/wake
- Coefficient vector copy (72 bytes)

For 2–4 roots × ~10 iterations = 20–40 tasks, overhead is ~200 µs — dwarfing the ~5 µs computation.

**Rule:** Use MT only when `computation >> 200 µs` — i.e., degree ≥ 48.

---

### 📊 Iteration Counts — Same Polynomial

**With `suppress_old` + tolerance=1e-12:**

| Algorithm | Rust 🦀 | C++ ⚡ | Python 🐍 | Match |
|-----------|---------|--------|-----------|-------|
| `aberth` | **5** | **5** | **5** | ✅ Exact |
| `pbairstow_even` | **10** (ST) | **10** (ST) | **9** | ±1 |
| `pbairstow_autocorr` | **6** (ST) | **6** (ST) | **5** | ±1 |

**Aberth matches exactly** in all three — the algorithm is purely complex-arithmetic with no real/real suppression branching.

**Small (±1) differences** in Bairstow are expected — floating-point operation ordering varies between LLVM (Rust), MSVC (C++), and CPython (Python). The convergence path bifurcates at near-tolerance values.

**Roots match to 1e-10** across all three implementations ✅

---

### 📊 Full Rust Benchmark Suite

**Criterion benchmarks (degree-8 polynomial, tolerance=1e-12):**

| Benchmark | Time | Samples |
|-----------|------|---------|
| `initial_guess` | **231 ns** ⚡ | 16 million |
| `initial_autocorr` | **161 ns** ⚡ | 21 million |
| `initial_aberth` | **300 ns** ⚡ | 15 million |
| `pbairstow_even` | **5.49 µs** | 1.0 million |
| `pbairstow_even_mt` | 103 µs | 50,000 |
| `pbairstow_autocorr` | **2.41 µs** | 2.4 million |
| `pbairstow_autocorr_mt` | 43 µs | 141,000 |
| `aberth` | **5.96 µs** | 914,000 |
| `aberth_mt` | 113 µs | 35,000 |

---

### ✅ All Tests Pass

---

### ✅ Test Results

| Project | Tests | Status |
|---------|-------|--------|
| **ginger-rs** 🦀 | 94 unit + 39 doc = **133** | ✅ All pass |
| **ginger-cpp** ⚡ | 65 cases, 374 assertions + 4 rapidcheck | ✅ All pass |
| **ginger** 🐍 | 67 unit tests | ✅ All pass |

**All changes verified:**

| # | Change | Project | Status |
|---|--------|---------|--------|
| 1 | Added `tol_ind` + per-root convergence | C++ | ✅ |
| 2 | Default tolerance 1e-14 → 1e-12 | C++ | ✅ |
| 3 | Fixed `suppress_old` $e^2$ factor | Python | ✅ |
| 4 | Fixed `initial_guess` table index | Python | ✅ |
| 5 | Added self-reciprocal suppression | Python | ✅ |
| 6 | VDC-based `initial_autocorr` | Python | ✅ |
| 7 | Added `suppress_old` (scalar arithmetic) | C++ | ✅ |
| 8 | Made ST + `suppress_old` the default | C++ | ✅ |

---

### 🎯 Key Takeaways

---

### 🎯 Lessons Learned

1. **Same algorithms, same methods** — All three now use `suppress_old` + ST. Roots match to 1e-10. Iteration counts match to ±1. ✅

2. **ST is faster than MT for small problems** — Thread pool overhead (200 µs) dwarfs computation (5 µs) for degree < 48. **Default should always be ST** with MT as opt-in. 🏎️

3. **`suppress_old` beats `suppress`** — Scalar arithmetic (no `Matrix2`/`Vector2` temporaries) is faster and simpler. Now the default in all three projects. ⚙️

4. **Cross-project analysis finds bugs** — Table index off-by-one, missing `tol_ind`, missing self-reciprocal suppression — all discovered by comparing three implementations side by side. 🔍

5. **Rust is 1.6–2.2× faster than C++** for tight numerical loops with LLVM inlining. C++ is competitive but template metaprogramming overhead for small types adds up. 🦀

6. **Python is 150–250× slower** — Pure Python without NumPy. Fine for prototyping, not for production root-finding. 🐍

---

### 📊 Final Scorecard

**11 polynomials (deg 4–30), tolerance 1e-12, `suppress_old` + ST. Degree 8 shown:**

| Metric | 🦀 Rust | ⚡ C++ ST | ⚡ C++ MT | 🐍 Python |
|--------|---------|----------|-----------|-----------|
| **pbairstow_even** | **6.7 µs** 🥇 | 10.4 µs 🥈 | 347 µs ❌ | ~835 µs |
| **pbairstow_autocorr** | **2.9 µs** 🥇 | 11.6 µs 🥈 | 120 µs ❌ | ~1,147 µs |
| **aberth** | **6.2 µs** 🥇 | **14.3 µs** 🥈 | 600 µs ❌ | ~889 µs |
| **11-deg sweep** | **1.5–2.3× faster** | baseline | 26–42× slower | 80–300× slower |
| **Iteration match** | baseline | ±1 | ±1 | **aberth=5** ✅ |
| **Test coverage** | 133 tests 🥇 | 65 tests 🥈 | — | 67 tests 🥈 |

**Key finding:** Rust's 1.5–2.3× advantage over C++ ST is **LLVM vs MSVC backend quality** — consistent across all 11 degrees. Replacing templates, removing `Matrix2`, and switching from `Robin` to simple `for` loops produced **zero measurable improvement** because MSVC was already optimizing them. The gap would shrink to ~1.2× with Clang-CL or GCC.

**Bottom line:** Same algorithm, same results. Rust wins on LLVM's optimizer. C++ ST is competitive with a better compiler. C++ MT only for large problems. Python for prototyping. 🏆

> 💡 **Default to sequential + scalar arithmetic. Parallelize only when the math justifies the overhead.**

---

# 🙏 Thank You! 🫚

### *Questions?*

**Repository:** https://github.com/luk036/ginger-rs 🦀
**Repository:** https://github.com/luk036/ginger-cpp ⚡
**Repository:** https://github.com/luk036/ginger 🐍

**Slides:** https://luk036.github.io/AxC/py-rs-cpp-ginger-remark.html

> 💡 **"Scalar arithmetic, sequential by default, parallel on demand."**