Bisection Method
A bracketing method that repeatedly bisects an interval and selects the subinterval containing the root. Guarantees convergence for continuous functions.
Use:
x^2 sin(x) cos(x) exp(x) log(x) sqrt(x)
Function Visualization
f(x)
Root
Iteration History
| Iter | a | b | c (midpoint) | f(c) | |b - a| |
|---|
Newton-Raphson Method
An open method using derivatives to find successively better approximations. Offers quadratic convergence near the root when conditions are met.
Convergence Path
f(x)
Tangent Lines
Root
Iteration History
| Iter | xᵢ | f(xᵢ) | f'(xᵢ) | xᵢ₊₁ | |xᵢ₊₁ - xᵢ| |
|---|
Secant Method
A derivative-free method that approximates the derivative using two previous points. Faster than bisection, no derivative needed.
Convergence Path
f(x)
Secant Lines
Root
Iteration History
| Iter | xᵢ₋₁ | xᵢ | f(xᵢ) | xᵢ₊₁ | |xᵢ₊₁ - xᵢ| |
|---|
Fixed Point Iteration
Transforms f(x) = 0 into x = g(x) and iterates until convergence. Convergence depends on |g'(x)| < 1 near the root.
Rewrite f(x)=0 as x=g(x). For x³-x-2=0, one form is x=(x+2)^(1/3)
Fixed Point Visualization
g(x)
y = x
Cobweb
Iteration History
| Iter | xᵢ | g(xᵢ) = xᵢ₊₁ | |xᵢ₊₁ - xᵢ| |
|---|