Approximate ∫f by weighted sum Σwⱼf(xⱼ). Degree of precision. Trapezoidal O(h²), Simpson O(h⁴), Gaussian quadrature exact for polynomials of degree 2n−1.
graph TD
D1["Def: Quadrature rule\n∫f ≈ Σ wⱼf(xⱼ)"]
D2["Def: Degree of precision\nexact for poly deg ≤ d"]
D3["Def: Composite rule\nsubdivide [a,b]"]
T1["Thm: Trapezoidal\nE = −(b−a)³f''(ξ)/12"]
T2["Thm: Simpson\nE = −(b−a)⁵f⁽⁴⁾(ξ)/2880"]
T3["Thm: Gaussian\nn nodes ⇒ degree 2n−1"]
T4["Thm: Nodes = roots of\nLegendre Pₙ"]
T5["Thm: Error composite\nO(hᵖ) with p=2,4,2n"]
L1["Lemma: Peano kernel"]
D1 --> D2
D1 --> D3
D2 --> T1
D2 --> T2
D2 --> T3
T3 --> T4
T1 --> T5
T2 --> T5
T3 --> T5
L1 --> T1
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: 15
- Edges: 12
- Definitions: 3
- Theorems: 5
- Lemmas: 1
Frontier: Adaptive quadrature, quasi-Monte Carlo, cubature. math.NA