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Click to edit Master title style Click to edit Master subtitle style * * * Roots of a Nonlinear Equation Topic: Secant Method Major: General Engineering http://numericalmethods.eng.usf.edu * * * Secant Method Newton’s Method Approximate the derivative http://numericalmethods.eng.usf.edu f(x) f(xi) f(xi-1) � EMBED Equation.3 ��� xi+2 xi+1 xi X _1075143848.unknown * * * Secant Method Geometric Similar Triangles http://numericalmethods.eng.usf.edu f(x) f(xi) f(xi-1) A D E C B xi+1 xi-1 xi X * * * Algorithm for Secant Method http://numericalmethods.eng.usf.edu * * * Step 1 Calculate the next estimate of the root from two initial guesses Find the absolute relative approximate error http://numericalmethods.eng.usf.edu * * * Step 2 Find if the absolute relative approximate error is greater than the prespecified relative error tolerance. If so, go back to step 1, else stop the algorithm. Also check if the number of iterations has exceeded the maximum number of iterations. http://numericalmethods.eng.usf.edu * * * Example You are working for ‘DOWN THE TOILET COMPANY’ that makes floats for ABC commodes. The ball has a specific gravity of 0.6 and has a radius of 5.5 cm. You are asked to find the distance to which the ball will get submerged when floating in water. http://numericalmethods.eng.usf.edu * * * Solution The equation that gives the depth ‘x’ to which the ball is submerged under water is given by Use the Secant method of finding roots of equations to find the depth ‘x’ to which the ball is submerged under water. Conduct three iterations to estimate the root of the above equation. http://numericalmethods.eng.usf.edu * * * Graph of function f(x) http://numericalmethods.eng.usf.edu * * * Iteration #1 http://numericalmethods.eng.usf.edu * * * Iteration #2 http://numericalmethods.eng.usf.edu * * * Iteration #3 http://numericalmethods.eng.usf.edu * * * Advantages Converges fast, if it converges Requires two guesses that do not need to bracket the root http://numericalmethods.eng.usf.edu * * * Drawbacks Division by zero http://numericalmethods.eng.usf.edu * * * Drawbacks (continued) Root Jumping http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu
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