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Secant Method. Civil Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates. Secant Method http://numericalmethods.eng.usf.edu. Secant Method – Derivation. Newton’s Method. (1).
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Secant Method Civil Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates http://numericalmethods.eng.usf.edu
Secant Method – Derivation Newton’s Method (1) Approximate the derivative (2) Substituting Equation (2) into Equation (1) gives the Secant method Figure 1 Geometrical illustration of the Newton-Raphson method. http://numericalmethods.eng.usf.edu
Secant Method – Derivation The secant method can also be derived from geometry: The Geometric Similar Triangles can be written as On rearranging, the secant method is given as Figure 2 Geometrical representation of the Secant method. http://numericalmethods.eng.usf.edu
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 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. http://numericalmethods.eng.usf.edu
Example 1 Cont. The equation that gives the position x where the deflection is maximum is given by Figure 2 A loaded bookshelf. 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. http://numericalmethods.eng.usf.edu
Example 1 Cont. Figure 3 Graph of the function f(x). http://numericalmethods.eng.usf.edu
Example 1 Cont. Solution Let us take the initial guesses of the root of as and . Iteration 1 The estimate of the root is http://numericalmethods.eng.usf.edu
Example 1 Cont. Figure 4 Graph of the estimated root after Iteration 1. http://numericalmethods.eng.usf.edu
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%. http://numericalmethods.eng.usf.edu
Example 1 Cont. Iteration 2 The estimate of the root is http://numericalmethods.eng.usf.edu
Example 1 Cont. Figure 5 Graph of the estimate root after Iteration 2. http://numericalmethods.eng.usf.edu
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%. http://numericalmethods.eng.usf.edu
Example 1 Cont. Iteration 3 The estimate of the root is http://numericalmethods.eng.usf.edu
Example 1 Cont. Figure 6 Graph of the estimate root after Iteration 3. http://numericalmethods.eng.usf.edu
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%. 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
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_method.html
THE END http://numericalmethods.eng.usf.edu