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9/20/2015 http://numericalmethods.eng.usf.edu 1 Secant Method Civil Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates

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Secant Method http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu

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3 Secant Method – Derivation Newton’s Method Approximate the derivative Substituting Equation (2) into Equation (1) gives the Secant method (1) (2) Figure 1 Geometrical illustration of the Newton-Raphson method.

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http://numericalmethods.eng.usf.edu4 Secant Method – Derivation The Geometric Similar Triangles Figure 2 Geometrical representation of the Secant method. The secant method can also be derived from geometry: can be written as On rearranging, the secant method is given as

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http://numericalmethods.eng.usf.edu5 Algorithm for Secant Method

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http://numericalmethods.eng.usf.edu6 Step 1 Calculate the next estimate of the root from two initial guesses Find the absolute relative approximate error

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http://numericalmethods.eng.usf.edu7 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.

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http://numericalmethods.eng.usf.edu8 Example 1 You are making a bookshelf to carry books that range from 8 ½ ” to 11” in height and would take 29”of space along length. The material is wood having Young’s Modulus 3.667 Msi, thickness 3/8 ” and width 12”. You want to find the maximum vertical deflection of the bookshelf. The vertical deflection of the shelf is given by where x is the position where the deflection is maximum. Hence to find the maximum deflection we need to find where and conduct the second derivative test.

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http://numericalmethods.eng.usf.edu9 Example 1 Cont. The equation that gives the position x where the deflection is maximum is given by Use the secant method of finding roots of equations to find the position where the deflection is maximum. Conduct three iterations to estimate the root of the above equation. Find the absolute relative approximate error at the end of each iteration and the number of significant digits at least correct at the end of each iteration. Figure 2 A loaded bookshelf.

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http://numericalmethods.eng.usf.edu10 Example 1 Cont. Figure 3 Graph of the function f(x).

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http://numericalmethods.eng.usf.edu11 Example 1 Cont. Let us take the initial guesses of the root of as and. Iteration 1 The estimate of the root is Solution

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http://numericalmethods.eng.usf.edu12 Example 1 Cont. Figure 4 Graph of the estimated root after Iteration 1.

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http://numericalmethods.eng.usf.edu13 Example 1 Cont. The absolute relative approximate error at the end of Iteration 1 is The number of significant digits at least correct is 1, because the absolute relative approximate error is less than 5%.

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http://numericalmethods.eng.usf.edu14 Example 1 Cont. Iteration 2 The estimate of the root is

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http://numericalmethods.eng.usf.edu15 Example 1 Cont. Figure 5 Graph of the estimate root after Iteration 2.

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http://numericalmethods.eng.usf.edu16 Example 1 Cont. The absolute relative approximate error at the end of Iteration 2 is The number of significant digits at least correct is 2, because the absolute relative approximate error is less than 0.5%.

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http://numericalmethods.eng.usf.edu17 Example 1 Cont. Iteration 3 The estimate of the root is

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http://numericalmethods.eng.usf.edu18 Example 1 Cont. Figure 6 Graph of the estimate root after Iteration 3.

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http://numericalmethods.eng.usf.edu19 Example 1 Cont. The absolute relative approximate error at the end of Iteration 3 is The number of significant digits at least correct is 6, because the absolute relative approximate error is less than 0.00005%.

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http://numericalmethods.eng.usf.edu20 Advantages Converges fast, if it converges Requires two guesses that do not need to bracket the root

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http://numericalmethods.eng.usf.edu21 Drawbacks Division by zero

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http://numericalmethods.eng.usf.edu22 Drawbacks (continued) Root Jumping

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Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit http://numericalmethods.eng.usf.edu/topics/secant_me thod.html

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THE END http://numericalmethods.eng.usf.edu

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