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Unformatted text preview: Lecture 6 Secant Methods* In this lecture we introduce two additional methods to find numerical solutions of the equation f ( x ) = 0. Both of these methods are based on approximating the function by secant lines just as Newtons method was based on approximating the function by tangent lines. The Secant Method The secant method requires two initial points x and x 1 which are both reasonably close to the solution x . Preferably the signs of y = f ( x ) and y 1 = f ( x 1 ) should be different. Once x and x 1 are determined the method proceeds by the following formula: x i +1 = x i- x i- x i 1 y i- y i 1 y i (6.1) Example: Suppose f ( x ) = x 4- 5 for which the true solution is x = 4 5. Plotting this function reveals that the solution is at about 1 . 25. If we let x = 1 and x 1 = 2 then we know that the root is in between x and x 1 . Next we have that y = f (1) =- 4 and y 1 = f (2) = 11. We may then calculate x 2 from the formula (6.1): x 2 = 2- 2- 1 11- (- 4) 11 = 19...
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- Fall '08