refactor(topics): 멀티 에이전트용 지식 재편 — _Common(공통 기본기) + Domain_* 구조
에이전트 8종(대화형/프로그래머 C·S/디자이너/설계자/기획자/QA/PD/PM)에게 [공통 기본 능력 + 롤별 Specialty] 2층으로 지식을 주입하기 위한 재분류. 문서 내용·포맷은 무수정, 폴더 이동만 (6,372개 문서 수 보존 확인). - Topic_Programming → Domain_Programming (내부 구조 보존) - Topic_Graphic → Domain_Design - Topic_Business → Domain_Product - Topic_General → Domain_General - _Common 신설: Math(구 Topic_Math_Specialty), Reasoning(구 General/From_Thinking & Reasoning), Reasoning_Creativity(구 General/From_창의성), Communication(Poetic_Blog_Writing + From_writing) - 타 도메인의 From_* 폴더는 유지 (출처 표기일 뿐, 이미 도메인에 맞게 분류된 문서) - 빈 폴더 정리 (memory/procedures) - 에이전트→폴더 매핑은 workspace의 .astra/agent-knowledge-map.json (9개 에이전트) Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
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Antigravity Agent
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---
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id: wiki-2026-0508-precision-recursion
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title: Precision Recursion
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category: 10_Wiki/Topics
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status: verified
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canonical_id: self
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aliases: [Mixed-Precision Recursive Refinement, Iterative Refinement]
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duplicate_of: none
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source_trust_level: A
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confidence_score: 0.85
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verification_status: applied
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tags: [numerical-methods, mixed-precision, iterative-refinement, ML]
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raw_sources: []
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last_reinforced: 2026-05-10
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github_commit: pending
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tech_stack:
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language: python
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framework: pytorch-mlx-cuda
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---
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# Precision Recursion
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## 매 한 줄
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> **"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.
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## 매 핵심
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### 매 기본 mechanism
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```
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1. Solve A x_lo = b in low precision (FP16/FP8) — fast
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2. Compute residual r = b - A x_lo in high precision (FP32/FP64)
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3. Solve A d = r in low precision — fast
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4. x ← x_lo + d
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5. Repeat until ||r|| < tol
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```
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### 매 핵심 invariant
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- **Residual computation**: 매 high precision 필수 (X cancellation error).
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- **Solve**: 매 low precision OK (errors absorbed by refinement).
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- **Convergence**: 매 condition number κ(A) 적절시 매 quadratic.
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### 매 응용
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1. **Linear solve**: GMRES-IR (Carson & Higham 2018).
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2. **LLM inference**: FP8 forward + FP32 residual streams.
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3. **Optimization**: Adam in FP16 + FP32 master weights.
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4. **Eigensolve**: 매 inverse iteration 매 mixed precision.
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## 💻 패턴
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### Iterative refinement (linear solve)
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```python
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import numpy as np
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def iterative_refinement(A, b, tol=1e-12, max_iter=10):
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"""매 mixed-precision linear solve."""
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A_lo = A.astype(np.float16)
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x = np.zeros_like(b)
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for k in range(max_iter):
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r = b - A @ x # 매 high-precision residual
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if np.linalg.norm(r) < tol:
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break
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d = np.linalg.solve(A_lo.astype(np.float32), r.astype(np.float32))
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x = x + d.astype(b.dtype)
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return x, k + 1
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```
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### PyTorch AMP (Automatic Mixed Precision)
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```python
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import torch
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from torch.cuda.amp import autocast, GradScaler
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scaler = GradScaler()
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for batch in loader:
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optim.zero_grad()
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with autocast(dtype=torch.float16):
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loss = model(batch).loss # 매 FP16 forward
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scaler.scale(loss).backward() # 매 FP32 grad scale
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scaler.step(optim) # 매 FP32 master weight update
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scaler.update()
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```
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### FP8 inference + FP32 accumulation (H100)
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```python
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# Transformer Engine — Hopper FP8
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import transformer_engine.pytorch as te
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from transformer_engine.common.recipe import Format, DelayedScaling
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fp8_recipe = DelayedScaling(
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margin=0, interval=1,
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fp8_format=Format.HYBRID, # 매 E4M3 fwd, E5M2 bwd
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)
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with te.fp8_autocast(enabled=True, fp8_recipe=fp8_recipe):
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out = model(x) # FP8 GEMMs, FP32 reductions
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```
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### GMRES with iterative refinement
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```python
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from scipy.sparse.linalg import gmres
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def gmres_ir(A, b, tol=1e-12, outer=5):
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"""매 outer IR loop, 매 inner GMRES low-prec."""
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x = np.zeros_like(b)
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A_lo = A.astype(np.float32)
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for _ in range(outer):
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r = b - A @ x
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if np.linalg.norm(r) < tol:
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return x
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d, _ = gmres(A_lo, r.astype(np.float32), atol=1e-6)
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x = x + d.astype(b.dtype)
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return x
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```
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### Adam with FP32 master weights
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```python
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class MixedPrecisionAdam:
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def __init__(self, params, lr=1e-3):
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self.params_fp16 = params # 매 storage
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self.params_fp32 = [p.detach().clone().float() for p in params]
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self.m = [torch.zeros_like(p) for p in self.params_fp32]
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self.v = [torch.zeros_like(p) for p in self.params_fp32]
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self.lr = lr; self.t = 0
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def step(self):
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self.t += 1
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for p16, p32, m, v in zip(self.params_fp16, self.params_fp32, self.m, self.v):
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g = p16.grad.float()
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m.mul_(0.9).add_(g, alpha=0.1)
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v.mul_(0.999).addcmul_(g, g, value=0.001)
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p32.addcdiv_(m, v.sqrt().add_(1e-8), value=-self.lr)
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p16.data.copy_(p32.half()) # 매 sync back
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```
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## 매 결정 기준
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| 상황 | Strategy |
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|---|---|
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| 매 ill-conditioned linear system | GMRES-IR mixed precision |
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| 매 LLM training | AMP (FP16/BF16 + FP32 master) |
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| 매 Hopper / Blackwell inference | FP8 + FP32 accumulate |
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| 매 well-conditioned + FP64 needed | 매 single-precision solve OK |
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**기본값**: 매 BF16 forward + FP32 master weights (training), FP8 inference (Hopper+).
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## 🔗 Graph
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- 변형: [[Iterative-Refinement]]
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## 🤖 LLM 활용
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**언제**: 매 numerical stability debugging, 매 mixed-precision recipe selection, 매 condition number analysis.
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**언제 X**: 매 integer / discrete optimization — 매 precision concept 무관.
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## ❌ 안티패턴
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- **Low-precision residual**: 매 cancellation error 폭발 → 매 refinement 무용.
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- **Ill-conditioned + low-prec**: 매 κ(A) > 10⁶ + FP16 → 매 발산.
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- **No master weights**: 매 FP16 weight update 매 underflow.
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- **Skip warmup**: 매 FP8 매 calibration 없이 → 매 NaN.
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## 🧪 검증 / 중복
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- Verified (Higham 1997 *Accuracy and Stability*; Carson & Higham 2018 GMRES-IR; NVIDIA Transformer Engine docs 2024).
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- 신뢰도 A.
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## 🕓 Changelog
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| 날짜 | 변경 |
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|---|---|
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| 2026-05-08 | Phase 1 |
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| 2026-05-10 | Manual cleanup — iterative refinement + modern AMP/FP8 stack |
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