Ax=b. LU factorization, pivoting. Condition number κ(A)=‖A‖‖A⁻¹‖. Backward stability.
graph TD
D1["Def: LU factorization\nA = LU lower/upper tri"]
D2["Def: Condition number\nκA = ‖A‖‖A⁻¹‖"]
D3["Def: Backward stable\n(A+δA)x̂ = b+δb"]
T1["Thm: LU exists\nif all leading minors ≠0"]
T2["Thm: PA = LU\npartial pivoting"]
T3["Thm: Error bound\n‖x−x̂‖/‖x‖ ≤ κ·ε"]
T4["Thm: Cholesky\nA pos def ⇒ A=LL*"]
T5["Thm: QR for least squares\nmin ‖Ax−b‖"]
L1["Lemma: Growth factor"]
D1 --> T1
D1 --> T2
D2 --> T3
D3 --> T3
T1 --> T4
T2 --> L1
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
classDef lemma fill:#74c0fc,color:#fff
class D1,D2,D3 definition
class T1,T2,T3,T4,T5 theorem
class L1 lemma
Process Statistics
- Nodes: 14
- Edges: 11
- Definitions: 3
- Theorems: 5
- Lemmas: 1
Frontier: Randomized linear algebra, fast solvers for structured matrices. math.NA