MATH342: Numerical Analysis Sunjae Kim

Root-finding Method Bisection Method Newton Method Secant Method

Bisection Method f(x) = 0 for the real variable x, where f is a continuous function defined on an interval [a, b] and f(a) and f(b) have opposite signs By the intermediate value theorem, the continuous function f must have at least one root in the interval (a, b) At each step, for the midpoint c = (a+b)/2, either f(a) and f(c) have opposite signs (root in the interval (a, c)) or f(c) and f(b) have opposite signs (root in the interval (c, b)) The process is continued until the interval is sufficiently small

Bisection Method

Relatively slow (interval is reduced in width by 50% at each step) Often used to obtain a rough approximation This method is also called the interval halving method, the binary search method, or the dichotomy method

Newton’s Method Given a function f defined over the reals x, its derivative f’, and a first guess x0 for a root of the function f, a better approximation x1 is x1 = x0 – f(x0)/f’(x0) One starts with an initial guess which is reasonably close to the root then the function is approximated by its tangent line

Newton’s Method

Secant Method Using succession of roots of secant lines to better approximate a root of a function f Finite difference approximation of Newton’s method

Secant Method

Newton’s Method for Nonlinear Equation

Steepest Descent Method Steepest descent method finds a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient

The SIR Model

Black-Scholes Equation