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Numerical Methods for Eng [ENGR 391] [Lyes KADEM 2008] CHAPTER III Roots of Equations The objective of this chapter is to introduce methods to solve linear and non-linear equations. We will see how do these methods work and compute their errors. I. GRAPHICAL METHODS This is the simplest method to determine the root of an equation The procedure is quite straightforward: - Plot the function f ( x) = 0 . f ( x) - Observe when it crosses the x-axis, this point represents the value for which N
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Numerical Methods for Eng [ENGR 391] [Lyes KADEM 2008] CHAPTER IIIRoots of Equations The objective of this chapter is to introduce methods to solve linear and non-linear equations. We will see how do these methods work and compute their errors. I. GRAPHICAL METHODS This is the simplest method to determine the root of an equation ( ) 0 =  x f  .The procedure is quite straightforward:- Plot the function ( )  x f  - Observe when it crosses the x-axis, this point represents the value for which ( ) 0 =  x f  .Note 1: This will provide only a rough approximation of the root.Note 2: you can remark that the function has changed sign after the root. 0 2 4 6 8 10 12 14 16 18 20-100102030405060 X       Y    =      f      (    x      ) Figure.3.1. graphical method. However, this rough approximation of the root can be used as a first guess for other more accuratetechniques. Roots of Equations 15 root  Numerical Methods for Eng [ENGR 391] [Lyes KADEM 2008] BRACKETING METHODS II. Bisection method One of the first numerical methods developed to find the root of a nonlinear equation 0)( =  x f   was the bisection method (also called Binary-Search method). The method is based on thefollowing theorem:Since the method is based on finding the root between two points, the method falls under thecategory of bracketing methods.    x l   f(x)  x u    x Figure.3.2.  At least one root exists between two points if the function is real, continuous, and changes sign.    x       f(x)  x u    x Figure.3.3. If function )(  x f  does not change sign between two points, roots of  0)( =  x f  may still exist between thetwo points. Roots of Equations 16 Theorem:  An equation 0)( =  x f  , where )(  x f  is a real continuous function, has at least oneroot between   x and u  x if  0)()( < u  x f  x f   ( Figure 3.2).Note that if  0)()( > u  x f  x f   , there may or may not be any root between   x and u  x (Figures3.3 and 3.4). If  0)()( < u  x f  x f   , then there may be more than one root between   x and u  x  (Figure 3.5). So the theorem only guarantees one root between   x and u  x .  Numerical Methods for Eng [ENGR 391] [Lyes KADEM 2008]    x       f(x)  x u    x  x       f(x)  x u    x Figure.3.4. If the function )(  x f  does not change sign between two points, there may not be any roots 0)( =  x f   between the two points.    x      f(x) x u   x Figure.3.5. If the function )(  x f  changes sign between two points, more than one root for  0)( =  x f  may exist betweenthe two points.  A general rule is:- If the f(x l ) and f(x u ) have the same sign ( 0)()( > u  x f  x f   ):- There is no root between x l and x u .- There is an even number of roots between x l and x u .- If the f(x l ) and f(x u ) have different signs ( 0)()( < u  x f  x f   ):- The is an odd number of roots. Roots of Equations 17  Numerical Methods for Eng [ENGR 391] [Lyes KADEM 2008] Exceptions: - Multiple roots 0 1 2 3 4 5 6-20-15-10-50510 X       Y    =      f      (    x      ) Figure.3.6. Multiple roots. Example: 223 )1)(3()(35)( −+=⇔+−+= x x x f  x x x x f  - Discontinuous functions 0 1 2 3 4 5 6 7 8 9 10-2-101234 X       Y    =      f      (      X      ) Figure.3.7. Discontinuous function. Since the root is bracketed between two points,   x and u  x , one can find the mid-point, m  x between   x and u  x . This gives us two new intervals 1.   x and m  x , and 2. m  x and u  x .Is the root now between   x and m  x , or between m  x and u  x ? One can find the sign of  )()( m  x f  x f   , and if  0)()( < m  x f  x f   then the new bracket is between   x and m  x , otherwise, it is between m  x and u  x . So, you can see that you are literally halving the interval. As one repeats this process, the Roots of Equations 18
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