---
id: wiki-2026-0508-precision-recursion
title: Precision Recursion
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [Mixed-Precision Recursive Refinement, Iterative Refinement]
duplicate_of: none
source_trust_level: A
confidence_score: 0.85
verification_status: applied
tags: [numerical-methods, mixed-precision, iterative-refinement, ML]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: pytorch-mlx-cuda
---

# Precision Recursion

## 매 한 줄
> **"매 lower precision 으로 fast 계산 → 매 higher precision 으로 residual 매 correct → 매 recurse"**. 매 numerical iterative refinement 의 modern variant — 매 H100/H200/MI300X 의 FP8/FP16 throughput 을 활용하면서 매 FP64-equivalent accuracy 를 달성. 매 Higham (1997) 의 classical refinement 매 GPU mixed-precision 시대에서 매 부활.

## 매 핵심

### 매 기본 mechanism
```
1. Solve A x_lo = b in low precision (FP16/FP8) — fast
2. Compute residual r = b - A x_lo in high precision (FP32/FP64)
3. Solve A d = r in low precision — fast
4. x ← x_lo + d
5. Repeat until ||r|| < tol
```

### 매 핵심 invariant
- **Residual computation**: 매 high precision 필수 (X cancellation error).
- **Solve**: 매 low precision OK (errors absorbed by refinement).
- **Convergence**: 매 condition number κ(A) 적절시 매 quadratic.

### 매 응용
1. **Linear solve**: GMRES-IR (Carson & Higham 2018).
2. **LLM inference**: FP8 forward + FP32 residual streams.
3. **Optimization**: Adam in FP16 + FP32 master weights.
4. **Eigensolve**: 매 inverse iteration 매 mixed precision.

## 💻 패턴

### Iterative refinement (linear solve)
```python
import numpy as np

def iterative_refinement(A, b, tol=1e-12, max_iter=10):
    """매 mixed-precision linear solve."""
    A_lo = A.astype(np.float16)
    x = np.zeros_like(b)
    for k in range(max_iter):
        r = b - A @ x  # 매 high-precision residual
        if np.linalg.norm(r) < tol:
            break
        d = np.linalg.solve(A_lo.astype(np.float32), r.astype(np.float32))
        x = x + d.astype(b.dtype)
    return x, k + 1
```

### PyTorch AMP (Automatic Mixed Precision)
```python
import torch
from torch.cuda.amp import autocast, GradScaler

scaler = GradScaler()
for batch in loader:
    optim.zero_grad()
    with autocast(dtype=torch.float16):
        loss = model(batch).loss  # 매 FP16 forward
    scaler.scale(loss).backward()  # 매 FP32 grad scale
    scaler.step(optim)              # 매 FP32 master weight update
    scaler.update()
```

### FP8 inference + FP32 accumulation (H100)
```python
# Transformer Engine — Hopper FP8
import transformer_engine.pytorch as te
from transformer_engine.common.recipe import Format, DelayedScaling

fp8_recipe = DelayedScaling(
    margin=0, interval=1,
    fp8_format=Format.HYBRID,  # 매 E4M3 fwd, E5M2 bwd
)

with te.fp8_autocast(enabled=True, fp8_recipe=fp8_recipe):
    out = model(x)  # FP8 GEMMs, FP32 reductions
```

### GMRES with iterative refinement
```python
from scipy.sparse.linalg import gmres

def gmres_ir(A, b, tol=1e-12, outer=5):
    """매 outer IR loop, 매 inner GMRES low-prec."""
    x = np.zeros_like(b)
    A_lo = A.astype(np.float32)
    for _ in range(outer):
        r = b - A @ x
        if np.linalg.norm(r) < tol:
            return x
        d, _ = gmres(A_lo, r.astype(np.float32), atol=1e-6)
        x = x + d.astype(b.dtype)
    return x
```

### Adam with FP32 master weights
```python
class MixedPrecisionAdam:
    def __init__(self, params, lr=1e-3):
        self.params_fp16 = params  # 매 storage
        self.params_fp32 = [p.detach().clone().float() for p in params]
        self.m = [torch.zeros_like(p) for p in self.params_fp32]
        self.v = [torch.zeros_like(p) for p in self.params_fp32]
        self.lr = lr; self.t = 0
    def step(self):
        self.t += 1
        for p16, p32, m, v in zip(self.params_fp16, self.params_fp32, self.m, self.v):
            g = p16.grad.float()
            m.mul_(0.9).add_(g, alpha=0.1)
            v.mul_(0.999).addcmul_(g, g, value=0.001)
            p32.addcdiv_(m, v.sqrt().add_(1e-8), value=-self.lr)
            p16.data.copy_(p32.half())  # 매 sync back
```

## 매 결정 기준
| 상황 | Strategy |
|---|---|
| 매 ill-conditioned linear system | GMRES-IR mixed precision |
| 매 LLM training | AMP (FP16/BF16 + FP32 master) |
| 매 Hopper / Blackwell inference | FP8 + FP32 accumulate |
| 매 well-conditioned + FP64 needed | 매 single-precision solve OK |

**기본값**: 매 BF16 forward + FP32 master weights (training), FP8 inference (Hopper+).

## 🔗 Graph
- 변형: [[Iterative-Refinement]]

## 🤖 LLM 활용
**언제**: 매 numerical stability debugging, 매 mixed-precision recipe selection, 매 condition number analysis.
**언제 X**: 매 integer / discrete optimization — 매 precision concept 무관.

## ❌ 안티패턴
- **Low-precision residual**: 매 cancellation error 폭발 → 매 refinement 무용.
- **Ill-conditioned + low-prec**: 매 κ(A) > 10⁶ + FP16 → 매 발산.
- **No master weights**: 매 FP16 weight update 매 underflow.
- **Skip warmup**: 매 FP8 매 calibration 없이 → 매 NaN.

## 🧪 검증 / 중복
- Verified (Higham 1997 *Accuracy and Stability*; Carson & Higham 2018 GMRES-IR; NVIDIA Transformer Engine docs 2024).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — iterative refinement + modern AMP/FP8 stack |