# bisection.pdf - The Bisection Method Introduction Bisection...

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The Bisection Method Introduction Bisection Method: Bisection Method = a numerical method in Mathematics to find a root of a given function Root of a function: Root of a function f(x) = a value a such that: f(a) = 0 Example: Function: f(x) = x 2 - 4 Roots: x = -2, x = 2 Because: f(-2) = (-2) 2 - 4 = 4 - 4 = 0 f(2) = (2) 2 - 4 = 4 - 4 = 0 A Mathematical Property Well-known Mathematical Property: If a function f(x) is continuous on the interval [ a .. b ] and sign of f(a) ≠ sign of f(b) , then: There is a value c [ a .. b ] such that: f(c) = 0 I.e., there is a root c in the interval [ a .. b ] Example:
The Bisection Method The Bisection Method is a successive approximation method that narrows down an interval that contains a root of the function f(x) The Bisection Method is given an initial interval [ a .. b ] that contains a root (We can use the property sign of f(a) ≠ sign of f(b) to find such an initial interval ) The Bisection Method will cut the interval into 2 halves and check which half interval contains a root of the function The Bisection Method will keep cut the interval in halves until the resulting interval is extremely small The root is then approximately equal to any value in the final (very small) interval . Example: Suppose the interval [ a .. b ] is as follows:
We cut the interval [ a .. b ] in the middle : m = (a+b)/2 Because sign of f(m) ≠ sign of f(a) , we proceed with the search in the new interval [ a .. b ] : We can use this statement to change to the new interval : b = m; In the above example, we have changed the end point b to obtain a smaller interval that still contains a root
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