Newton-Raphson Method

graph TD A1["f x, f' x\nInitial guess x0, tolerance eps"] B1["x = x0"] C1["Compute x_new = x - f x / f' x"] D1{"Converged?\nabs x_new - x less than eps"} D2{"Max iterations\nreached?"} E1["Update x = x_new"] F1["Return x\nRoot found"] F2["Return failure\nNo convergence"] A1 --> B1 B1 --> C1 C1 --> D1 D1 -->|Yes| F1 D1 -->|No| D2 D2 -->|Yes: sequential AND| F2 D2 -->|No: sequential AND| E1 E1 --> C1 classDef red fill:#ff6b6b,color:#fff,stroke:#c0392b classDef yellow fill:#ffd43b,color:#000,stroke:#f59f00 classDef green fill:#51cf66,color:#fff,stroke:#40c057 classDef lightblue fill:#74c0fc,color:#fff,stroke:#4dabf7 classDef violet fill:#b197fc,color:#fff,stroke:#9775fa classDef lavender fill:#e6e6fa,color:#333,stroke:#b19cd9 class A1 red class B1 yellow class C1 green class E1 lightblue class F1,F2 violet class D1,D2 lavender