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Second Term 05/061 Roots of Equations Bracketing Methods

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Second Term 05/062 Root We are given f(x), a function of x, and we want to find α such that f(α) = 0 α is called the root of the equation f(x) = 0, or the zero of the function f(x)

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Second Term 05/063 Example: Interest Rate Suppose you want to buy an electronic appliance from a shop and you can either pay an amount of 12,000 or have a monthly payment of 1,065 for 12 months. What is the corresponding interest rate? A is the monthly payment P is the loan amount x is the interest rate per period of time n is the loan period To find the yearly interest rate, x, you have to find the zero of We know the payment formulae is:

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Second Term 05/064 Finding Roots Graphically Not accurate However, graphical view can provide useful info about a function. –Multiple roots? –Continuous? –etc.

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Second Term 05/065 The following root finding methods will be introduced: A. Bracketing Methods A.1. Bisection Method A.2. Regula Falsi B. Open Methods B.1. Fixed Point Iteration B.2. Newton Raphson's Method B.3. Secant Method

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Second Term 05/066 Bracketing Methods By Mean Value Theorem, we know that if a function f(x) is continuous in the interval [a, b] and f(a)f(b) < 0, then the equation f(x) = 0 has at least one real root in the interval (a, b).

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Second Term 05/067 Usually f(a)f(b) > 0 implies zero or even number of roots –[figure (a) and (c)] f(a)f(b) < 0 implies odd number of roots –[figure (b) and (d)]

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Second Term 05/068 Exceptional Cases Multiple roots –Roots that overlap at one point. –e.g.: f(x) = (x-1)(x-1)(x-2) has a multiple root at x=1. Functions that discontinue within the interval

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Second Term 05/069 Algorithm for bracketing methods Step 1: Choose two points x l and x u such that f(x l )f(x u ) < 0 Step 2: Estimate the root x r (note: x l < x r < x u ) Step 3: Determine which subinterval the root lies: if f(x l )f(x r ) < 0 // the root lies in the lower subinterval set x u to x r and goto step 2 if f(x l )f(x r ) > 0 // the root lies in the upper subinterval set x l to x r and goto step 2 if f(x l )f(x r ) = 0 x r is the root

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Second Term 05/0610 How to select x r in step 2? 1.Bisection Method Guess without considering the characteristics of f(x) in (xl, xu) 2.False Position Method (Regula Falsi) Use "average slope" to predict the root

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Second Term 05/0611 A.1. Bisection Method Each guess reduce the search interval by half

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Second Term 05/0612 Bisection Method – Example Find the root of f(x) = 0 with an approximated error below 0.5%. (True root: α=14.7802 )

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Second Term 05/0613 Example (continue) nxlxl xrxr xuxu f(xl)f(xl)f(xr)f(xr)f(xu)f(xu)f(xl)f(xu)f(xl)f(xu)εaεa 012166.067-2.269 11214166.0671.569-2.269> 0 21415161.569-0.425-2.269< 06.667% 31414.5151.5690.552-0.425> 03.448% 414.514.75150.5520.0590-0.425> 01.695% 514.7514.875150.0590-0.184-0.425< 00.840% 614.7514.812514.8750.0590-0.0629-0.184< 00.422%

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Second Term 05/0614 Error Bounds The true root, α, must lie between x l and x u. xlxl xuxu xrxr x l (1) x u (1) After the 1 st iteration, the solution, x r (1), should be within an accuracy of x r (1) Let x r (n) denotes x r in the n th iteration

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Second Term 05/0615 Error Bounds x l (2) x u (1) x u (2) Suppose the root lies in the lower subinterval. x r (2) After the 2 nd iteration, the solution, x r (2), should be within an accuracy of

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Second Term 05/0616 Error Bounds In general, after the n th iteration, the solution, x r (n), should be within an accuracy of If we want to achieve an absolute error of no more than E α

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Second Term 05/0617 Implementation Issues The condition f(x l )f(x r ) = 0 (in step 3) is difficult to achieve due to errors. We should repeat until x r is close enough to the root, but we don't know what the root is! Therefore, we have to estimate the error as and repeat until e a < e s (acceptable error)

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Second Term 05/0618 Bisection Method (as C function) // xl, xu: Lower and upper bound of the interval // es: Acceptable relative percentage error // xr: Estimated root in zero iteration (can simply // take xl or xu) // iter_max: Maximum # of iterations double Bisect(double xl, double xu, double es, double xr, int iter_max) { double xr_old; // Est. root in the previous step double ea; // Est. error int iter = 0; // Keep track of # of iterations do { iter++; xr_old = xr;

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Second Term 05/0619 xr = (xl + xu) / 2; // Estimate root if (xr != 0) ea = fabs((xr – xr_old) / xr) * 100; test = f(xl) * f(xr); if (test < 0) xu = xr; else if (test > 0) xl = xr; else ea = 0; } while (ea > es && iter < iter_max); return xr; }

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Second Term 05/0620 Additional Implementation Issues Function call is a relatively slow operation. In the previous example, function f() is called twice in each iteration. Is it necessary? –We only need to update one of the bounds (see step 3 in the algorithm for the bracketing method).

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Second Term 05/0621 Revised Bisection Method (as C function) double Bisect(double xl, double xu, double es, double xr, int iter_max) { double xr_old; // Est. root in the previous step double ea; // Est. error int iter = 0; // Keep track of # of iterations double fl, fr; // Save values of f(xl) and f(xr) fl = f(xl); do { iter++; xr_old = xr; xr = (xl + xu) / 2; // Estimate root fr = f(xr);

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Second Term 05/0622 if (xr != 0) ea = fabs((xr – xr_old) / xr) * 100; test = fl * fr; if (test < 0) xu = xr; else if (test > 0) { xl = xr; fl = fr; } else ea = 0; } while (ea > es && iter < iter_max); return xr; }

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Second Term 05/0623 Comments on Bisection Method The method is guaranteed to converge. However, the convergence is slow as we gain only one binary digit in accuracy in each iteration.

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Second Term 05/0624 A.2. Regula Falsi Method Also known as the false-position method, or linear interpolation method. Unlike the bisection method which divides the search interval by half, regula falsi interpolates f(x u ) and f(x l ) by a straight line and the intersection of this line with the x-axis Is used as the new search position. The slope of the line connecting f(x u ) and f(x l ) represents the "average slope" (i.e., the value of f'(x)) of the points in [x l, x u ].

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Second Term 05/0625

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Second Term 05/0626 False-position vs Bisection False position in general performs better than bisection method. Exceptional Cases: –(Usually) When the deviation of f'(x) is high and the end points of the interval are selected poorly. –For example,

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Second Term 05/0627 Iteration xlxl xuxu xrxr ε a (%)ε t (%) 101.30.6535 20.651.30.97533.325 30.9751.31.137514.313.8 40.9751.13751.056257.75.6 50.9751.056251.0156254.01.6 Iteration xlxl xuxu xrxr ε a (%)ε t (%) 101.30.0943090.6 20.094301.30.1817648.181.8 30.181761.30.2628730.973.7 40.262871.30.3381122.366.2 50.338111.30.4078817.159.2 Bisection Method (Converge quicker) False-position Method

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Second Term 05/0628

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Second Term 05/0629 Summary Bracketing Methods –f(x) has the be continuous in the interval [ x l, x u ] and f(x l )f(x u ) < 0 –Always converge –Usually slower than open methods Bisection Method –Slow but guarantee the best worst-case convergent rate. False-position method –In general performs better than bisection method (with some exceptions).

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