LECTURE 06 - Lecture 6 Lecture 6 MEEN 357 Engineering...

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Lecture 6 MEEN 357 Engineering Analysis for Mechanical Engineers 1 Lecture-06-MEEN357-2009.doc RMB Lecture 6 Roots to Nonlinear Equations Chapter 5 and 6 of the textbook. We are looking at various ways to find the roots of the equation ( ) 0 fx = (6.1) in some given interval of the real axis. As explained during Lecture 5, there are two methods that are used to find the roots of the function ( ) yfx = . Bracketing method: This method involves first establishing that the root lies within some interval and the use of an iteration scheme to find the root. Open method: An open method utilizes formulas that only require a single starting value or two starting values that do not necessarily bracket the root. During Lecture 5, we discussed the first of two bracketing methods, namely the Bisection Method. The other method, which we shall discuss today, is the False Position Method. During this lecture, we shall also discuss the open methods Newton-Raphson method Secant method Bisection Method Summary (Repeated from Lecture 5 notes.) Section 5.4 of the textbook Bisection Method : A bracketing method that can be structured as a systematic method of finding the distinct roots of continuous functions Basic Idea : Near a root, one of the following graphs is correct: a b a b
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Lecture 6 MEEN 357 Engineering Analysis for Mechanical Engineers 2 Lecture-06-MEEN357-2009.doc RMB a b r x ( ) ( ) 0 r fafx > a b r x ( ) ( ) 0 r < The analytical condition which reflects both of these graphs is that ( ) ( ) 0 fafb < (6.2) By dividing the interval into small and smaller segments, one gets closer to the actual root. The bisection method is one where the interval [ ] , ab is always divided into half . At each iteration, the test (6.2) is them applied to determine which half of the divided interval contains the root. When this interval is identified, the new interval is divided in half and so forth. The following figure suggests the geometric arrangement. The solution algorithm consists of the following sequence of steps: Step 1: Choose lower a and upper b points in the neighborhood of the root such that () () 0 < . Step 2: Estimate that the root occurs at 2 r x + = Step 3: You next need to determine which subinterval contains the actual root. Make the following evaluations to determine which subinterval contains the root: a. If () ( ) 0 r < , the root is contained in the left or lower subinterval. The next step is to set the new b to be the current r x and return to Step 2. b. If 0 r > , the root is not contained in the left or lower subinterval. It lies in the right or upper subinterval. The next step is to set the new a to the current r x and to return to Step 2.
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Lecture 6 MEEN 357 Engineering Analysis for Mechanical Engineers 3 Lecture-06-MEEN357-2009.doc RMB c. If () ( ) 0 r fafx = (or is sufficiently close to zero by some prescribed rule), the root equals r x .
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LECTURE 06 - Lecture 6 Lecture 6 MEEN 357 Engineering...

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