The document discusses the bisection method, a numerical method for finding the roots of a polynomial function. The bisection method works by repeatedly bisecting an interval where the function changes sign and narrowing in on a root. It involves finding an interval [a,b] where f(a) and f(b) have opposite signs, taking the midpoint x1, and discarding half of the interval based on the sign of f(x1). This process is repeated, halving the interval width each time, until a root is isolated within the desired accuracy. An example applying the bisection method to find the roots of f(x) = x^3 + 3x - 5 on the interval [1,2

1. BRACKETING or CLOSED METHODS BISECTION METHOD & FALSE POSITION METHOD!!!!!

2. BISECTION METHOD! The Bisection Method is a numerical method for estimating the roots of a polynomialf(x). It is one of the simplest and most reliable but it is not the fastest method. Assume that f(x) is continuous.

3. LET’S DO IT STEP BY STEP! Given a continuous function f(x) 1- Find points a and b such that a < b and f(a) * f(b) < 0. 2- Take the interval [a, b] and find its midpoint x1. 3- If f(x1) = 0 then x1 is an exact root, else if f(x1) * f(b) < 0 then let a = x1, else if f(a) * f(x1) < 0 then let b = x1. 4- Repeat steps 2 & 3 until f(xi) = 0 or |f(xi)| <= DOA, where DOA stands for degree of accuracy.

4. LET’S DO IT STEP BY STEP!

5. LET’S DO IT STEP BY STEP! For the ith iteration, where i = 1, 2, . . . , n, the interval width is xi = 0.5 xi–1 = ( 0.5 )i ( b – a ) and the new midpoint is xi = ai–1 + xi

6. EXAMPLE! Consider f(x) = x3 + 3x – 5, where [a = 1, b = 2] and DOA = 0.001.

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