Root Finding Methods
Find roots of nonlinear equations \( f(x) = 0 \) using various numerical algorithms. Each method has different convergence properties and stability characteristics.
Bisection MethodLinear
A reliable bracketing method that guarantees convergence
Complexity:O(log n)
Stability:Very Stable
Key Formula: f(a) × f(b) < 0
Regula FalsiSuperlinear
False position method with faster convergence than bisection
Complexity:O(log n)
Stability:Stable
Key Formula: Linear interpolation
Fixed Point IterationLinear
Iterative method for solving g(x) = x equations
Complexity:O(n)
Stability:Conditional
Key Formula: x_{n+1} = g(x_n)
Newton RaphsonQuadratic
Fast convergence method using derivatives
Complexity:O(n)
Stability:Good
Key Formula: x_{n+1} = x_n - f(x_n)/f'(x_n)
Secant MethodSuperlinear
Newton's method without derivative calculation
Complexity:O(n)
Stability:Good
Key Formula: Uses finite differences
About Root Finding
Root finding is the process of finding values of \( x \) such that \( f(x) = 0 \). These methods are fundamental in numerical analysis and have applications in:
- Engineering design and optimization
- Scientific simulations and modeling
- Economics and financial modeling
- Computer graphics and game development