Chapter 1: False-Position Method of Solving a Nonlinear Equation http://numericalmethods.eng.usf.edu Numerical Methods for STEM undergraduates 11/30/2018 http://numericalmethods.eng.usf.edu 1

Introduction (1) In the Bisection method (2) (3) Figure 1 False-Position Method 2 http://numericalmethods.eng.usf.edu

False-Position Method Based on two similar triangles, shown in Figure 1, one gets: (4) The signs for both sides of Eq. (4) is consistent, since: http://numericalmethods.eng.usf.edu

The above equation can be solved to obtain the next predicted root From Eq. (4), one obtains The above equation can be solved to obtain the next predicted root , as (5) http://numericalmethods.eng.usf.edu

The above equation, (6) http://numericalmethods.eng.usf.edu

Step-By-Step False-Position Algorithms 1. Choose and as two guesses for the root such that 2. Estimate the root, 3. Now check the following (a) If , then the root lies between and ; then and (b) If , then the root lies between and ; then and http://numericalmethods.eng.usf.edu

Stop the algorithm if this is true. (c) If , then the root is Stop the algorithm if this is true. 4. Find the new estimate of the root Find the absolute relative approximate error as http://numericalmethods.eng.usf.edu

= estimated root from present iteration where = estimated root from present iteration = estimated root from previous iteration 5. If , then go to step 3, else stop the algorithm. Notes: The False-Position and Bisection algorithms are quite similar. The only difference is the formula used to calculate the new estimate of the root shown in steps #2 and 4! http://numericalmethods.eng.usf.edu

Example 1 The non linear equation that gives the depth Use the false-position method of finding roots of equations. Conduct three iterations to estimate the root of the above equation. Find the absolute relative approximate error at the end of each iteration. http://numericalmethods.eng.usf.edu

Iteration 1 http://numericalmethods.eng.usf.edu

Iteration 2 Hence, http://numericalmethods.eng.usf.edu

Iteration 3 http://numericalmethods.eng.usf.edu

Hence, http://numericalmethods.eng.usf.edu

for False-Position Method. Table 1: Root of for False-Position Method. Iteration 1 0.0000 0.1100 0.0660 N/A -3.1944x10-5 2 0.0611 8.00 1.1320x10-5 3 0.0624 2.05 -1.1313x10-7 4 0.0632377619 0.02 -3.3471x10-10 http://numericalmethods.eng.usf.edu

Exercise 1 The non linear equation that gives as Use the false-position method of finding roots of equations. Conduct three iterations to estimate the root of the above equation. Find the absolute relative approximate error at the end of each iteration. http://numericalmethods.eng.usf.edu

Let us assume Hence, http://numericalmethods.eng.usf.edu

Iteration 1 http://numericalmethods.eng.usf.edu

References S.C. Chapra, R.P. Canale, Numerical Methods for Engineers, Fourth Edition, Mc-Graw Hill. http://numericalmethods.eng.usf.edu

The End http://numericalmethods.eng.usf.edu

Acknowledgement This instructional power point brought to you by Numerical Methods for STEM undergraduate http://numericalmethods.eng.usf.edu Committed to bringing numerical methods to the undergraduate