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E. T. S. I. Caminos, Canales y Puertos1 Engineering Computation Part 2

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E. T. S. I. Caminos, Canales y Puertos2 Objective: Solve for x, given that f(x) = 0 -or- Equivalently, solve for x such that g(x) = h(x) ==> f(x) = g(x) – h(x) = 0 Roots of Equations

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E. T. S. I. Caminos, Canales y Puertos3 p = pressure, T = temperature, R = universal gas constant, a & b = empirical constants Chemical Engineering (C&C 8.1, p. 187): van der Waals equation; v = V/n (= volume/# moles) Find the molal volume v such that Roots of Equations

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E. T. S. I. Caminos, Canales y Puertos4 Civil Engineering (C&C Prob. 8.17, p. 205): Find horizontal component of tension, H, in a cable that passes through (0,y 0 ) and (x,y) w = weight per unit length of cable Roots of Equations

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E. T. S. I. Caminos, Canales y Puertos5 L = inductance, C = capacitance, q 0 = initial charge Electrical Engineering (C&C 8.3, p. 194): Find the resistance, R, of a circuit such that the charge reaches q at specified time t Roots of Equations

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E. T. S. I. Caminos, Canales y Puertos6 Mechanical Engineering (C&C 8.4, p. 196): Find the value of stiffness k of a vibrating mechanical system such that the displacement x(t) becomes zero at t= 0.5sec. The initial displacement is x 0 and the initial velocity is zero. The mass m and damping c are known, and λ = c/(2m). in which Roots of Equations

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E. T. S. I. Caminos, Canales y Puertos7 Determine real roots of : Algebraic equations (including polynomials) Transcendental equations such as f(x) = sin(x) + e -x Combinations thereof Roots of Equations

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E. T. S. I. Caminos, Canales y Puertos8 in which: PV= present value or purchase price = $7,500 A= annual payment = $1,000/yr n= number of years = 20 yrs i= interest rate = ? (as a fraction, e.g., 0.05 = 5%) Engineering Economics Example: A municipal bond has an annual payout of $1,000 for 20 years. It costs $7,500 to purchase now. What is the implicit interest rate, i ? Solution: Present-value, PV, is: Roots of Equations

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E. T. S. I. Caminos, Canales y Puertos9 Engineering Economics Example (cont.): We need to solve the equation for i: Equivalently, find the root of: Roots of Equations

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E. T. S. I. Caminos, Canales y Puertos10 Excel Roots of Equations

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E. T. S. I. Caminos, Canales y Puertos11 Roots of Equations Graphical methods: –Determine the friction coefficient c necessary for a parachutist of mass 68.1 kg to have a speed of 40 m/seg at 10 seconds. –Reorganizing.

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E. T. S. I. Caminos, Canales y Puertos12 Two Fundamental Approaches 1. Bracketing or Closed Methods - Bisection Method - False-position Method 2. Open Methods - One Point Iteration - Newton-Raphson Iteration - Secant Method Roots of Equations

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E. T. S. I. Caminos, Canales y Puertos13 Bracketing Methods f(x) xlxl xuxu a) b) c) d) x x x x In Figure a) we have the case of f(x l ) and f(x u ) with the same sign, and there is no root in the interval (x l,x u ). In Figure b) we have the case of f(x l ) and f(x u ) With different sign, and there is a root in the interval (x l,x u ). In Figure c) we have the case of f(x l ) and f(x u ) with the same sign, and there are two roots. In Figure d) we have the case of f(x l ) and f(x u ) with different sign, and there is an odd number of roots.

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E. T. S. I. Caminos, Canales y Puertos14 Though the cases above are generally valid, there are cases in which they do not hold. In Figure a) we have the case of f(x l ) and f(x u ) with different sign, but there is a double root. f(x) xlxl xuxu a) b) c) x x x In Figure b) We have the case of f(x l ) and f(x u ) With different sign, but there are two discontinuities. In Figure c) we have the case of f(x l ) and f(x u ) with the same sign, but there is a multiple root. Bracketing Methods

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E. T. S. I. Caminos, Canales y Puertos15 f(x 1 ) f(x r ) > 0 x f(x) f(x u ) (x u ) (x 1 ) f(x 1 ) f(x r ) f(x u ) f(x 1 ) (x 1 ) (x u ) (x r ) f(x r ) x f(x) x r => x 1 Bracketing Methods (Bisection method) Bisection Method

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E. T. S. I. Caminos, Canales y Puertos16 Bisection Method: Given lower and upper bounds, x l and x u, which bracket the root: f(x l ) f(x u ) < 0 1) Estimate the Root by midpoint: 2) Revise the bracket: f(x l ) f(x r ) x u, f(x l ) f(x r ) > 0,x r –> x l 3) Repeat steps 1-2 until: a) |f(x r )| < k,b) a < s, with a = c) | x l – x u | < d) maximum # of iterations is reached. (Always do this in iteration algorithms.) Bracketing Methods (Bisection method)

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E. T. S. I. Caminos, Canales y Puertos17 Make conservative guesses at the upper and lower bounds: 100% interest rate,f(1.0) = 6,500 0% interest rate, f(0.0) = -12,500 Engineering Economics Example: We need to solve the equation for i: Equivalently, find the root of: Bracketing Methods (Bisection method)

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E. T. S. I. Caminos, Canales y Puertos18 Excel Bracketing Methods (Bisection method)

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E. T. S. I. Caminos, Canales y Puertos19 Engineering Economics Example: Score Sheet for Rootfinding Example: MethodInitial Est(s). s = 2 E-2 s = 2 E-7 Bisection(0.00, 1.00)926 (0.05, 0.15)622 Convergence guaranteed: Bracketing Methods (Bisection method)

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E. T. S. I. Caminos, Canales y Puertos20 Bracketing Methods (Bisection method) One important advantage of this method is that one can calculate the number of required iterations for a given error.

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E. T. S. I. Caminos, Canales y Puertos21 Parachutist Example: Bracketing Methods (Bisection method)

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E. T. S. I. Caminos, Canales y Puertos22 Parachutist Example: Bracketing Methods (Bisection method)

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E. T. S. I. Caminos, Canales y Puertos23 Bisection Method Advantages: 1. Simple 2. Good estimate of maximum error: 3. Convergence guaranteed Disadvantages: 1. Slow 2. Requires two good initial estimates which define an interval around root: use graph of function, incremental search, or trial & error Bracketing Methods (Bisection method)

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E. T. S. I. Caminos, Canales y Puertos24 x False-position Method f(x) f(x u ) (x u ) (x 1 ) f(x 1 ) f(x r ) f(x 1 ) f(x r ) > 0 x 1 = x r f(x) f(x u ) f(x 1 ) (x 1 ) (x u ) (x r ) f(x r ) Bracketing Methods (False-position Method)

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E. T. S. I. Caminos, Canales y Puertos25 Similar to bisection. Uses linear interpolation to approximate the root x r : 1) 2) Revise the bracket:f(x 1 ) f(x r ) x u, f(x 1 ) f(x r ) > 0,x r –> x 1 3) Repeat steps 1-2 until: a) |f(x r )| < k, b) a < s, with a = c) |x u – x 1 | = d d) maximum # of iterations is reached. (Always do this in iteration algorithms.) Bracketing Methods (False-position Method)

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E. T. S. I. Caminos, Canales y Puertos26 Excel Bracketing Methods (False-position Method)

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E. T. S. I. Caminos, Canales y Puertos27 Bracketing Methods (False-position Method)

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E. T. S. I. Caminos, Canales y Puertos28 Score Sheet for False-Position Example: MethodInitial Est(s). s = 2 E-2 s = 2 E-7 Bisection(0.00, 1.00)926 (0.05, 0.15)622 False-pos.(0.00, 1.00)1128 (0.05, 0.15)314 Bracketing Methods (False-position Method)

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E. T. S. I. Caminos, Canales y Puertos29 Parachutist Example: Bracketing Methods (False-position Method)

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E. T. S. I. Caminos, Canales y Puertos30 Parachutist Example: Bracketing Methods (False-position Method)

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E. T. S. I. Caminos, Canales y Puertos31 There are some cases in which the false position method is very slow, and the bisection method gives a faster solution. Bracketing Methods (False-position Method)

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E. T. S. I. Caminos, Canales y Puertos32 Summary of False-Position Method: Advantages: 1. Simple 2. Brackets the Root Disadvantages: 1. Can be VERY slow 2. Like Bisection, need an initial interval around the root. Bracketing Methods (False-position Method)

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E. T. S. I. Caminos, Canales y Puertos33 Roots of Equations - Open Methods Characteristics: 1. Initial estimates need not bracket the root 2. Generally converge faster 3. NOT guaranteed to converge Open Methods Considered: - One Point Iteration - Newton-Raphson Iteration - Secant Method Open Methods

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E. T. S. I. Caminos, Canales y Puertos34 Open Methods An alternative method consists of separating the function into two parts.

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E. T. S. I. Caminos, Canales y Puertos35 Fixed point Method predict a value of x i+1 as a function of x i. Convert f(x) = 0 to x = g(x) iteration steps:x i+1 = g(x i ) x(new) = g(x(old) ) Open Methods (Fixed point method)

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E. T. S. I. Caminos, Canales y Puertos36 Example II: x = sin(x) –> x i+1 = sin(x i ) OR x = arcsin(x) –> x i+1 = arcsin(x i ) Example I: Open Methods (Fixed point method)

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E. T. S. I. Caminos, Canales y Puertos37 Convergence: Does x move closer to real root (?) Depends on: 1. nature of the function 2. accuracy of the initial estimate Interested in: 1. Will it converge or will it diverge? 2. How fast will it converge ? (rate of convergence) Open Methods (Fixed point method)

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E. T. S. I. Caminos, Canales y Puertos38 Convergence of the Fixed point Method: Root satisfies: x r = g(x r ) The Taylor series for function g is: x i+1 = g(x r ) + g'(x)(x i - x r )x r < x < x i Subtracting the second equation from the first yields (x r – x i+1 ) = g'(x) (x r – x i ) or 1. True error for next iteration is smaller than the true error in the previous iteration if |g'(x)| < 1.0 (it will converge). 2. Because g'(x) is almost constant, the new error is directly proportional to the old error (linear rate of convergence). Open Methods (Fixed point method)

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E. T. S. I. Caminos, Canales y Puertos39 Further Considerations: Convergence depends on how f(x) = 0 is converted into x = g(x) So... Convergence may be improved by recasting the problem. Open Methods (Fixed point method)

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E. T. S. I. Caminos, Canales y Puertos40 can be small, even though x new is not close to root. Remedy: Do not completely rely on a to ensure that the problem is solved. Check to make sure |f(x new ) | < . Convergence Problem: For slowly converging functions Open Methods (Fixed point method)

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E. T. S. I. Caminos, Canales y Puertos41 Open Methods (Fixed point method)

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E. T. S. I. Caminos, Canales y Puertos42 Open Methods

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E. T. S. I. Caminos, Canales y Puertos43 Two Fundamental Approaches 1. Bracketing or Closed Methods - Bisection Method - False-position Method 2. Open Methods - One Point Iteration - Newton-Raphson Iteration - Secant Method Roots of Equations

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E. T. S. I. Caminos, Canales y Puertos44 Newton-Raphson Method: Geometrical Derivation: Slope of tangent at x i is Solve for x i+1 : [Note that this is the same form as the generalized one- point iteration, x i+1 = g(x i )] Open Methods (Newton-Raphson Method)

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E. T. S. I. Caminos, Canales y Puertos45 Newton-Raphson Method x i = x i+1 Tangent w/slope=f '(x i ) x f(x) f(x i ) xixi f(x i+1 ) x f(x) f(x i ) (x i ) f(x i+1 ) x i+ 1 Open Methods (Newton-Raphson Method)

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E. T. S. I. Caminos, Canales y Puertos46 First order Taylor Series Derivation: 0 = f( x r ) f(x i ) + f '(x i ) (x r – x i ) solve for x r to yield next guess x i+1 : This has the form x i+1 = g(x i ) with: Open Methods (Newton-Raphson Method)

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E. T. S. I. Caminos, Canales y Puertos47 Newton-Raphson iteration: This iteration process is repeated until: 1.f(x i+1 ) 0, i.e., | f(x i+1 ) | < , with = small number Maximum number of iterations is reached. Open Methods (Newton-Raphson Method)

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E. T. S. I. Caminos, Canales y Puertos48 Open Methods a) Inflection point in the neighboor of a root. b) Oscilation in the neighboor of a maximum or minimum. c) Jumps in functions with several roots. d) Existence of a null derivative.

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E. T. S. I. Caminos, Canales y Puertos49 Bond Example: To apply Newton-Raphson method to: We need the derivative of the function: Open Methods (Newton-Raphson Method)

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E. T. S. I. Caminos, Canales y Puertos50 Score Sheet for Newton-Raphson Example: MethodInitial Est(s). s = 2 E-2 s = 2 E-7 Bisection(0.00, 1.00)926 (0.05, 0.15)622 False-pos.(0.00, 1.00)1128 (0.05, 0.15)314 N-R1.0divergesdiverges 0. 52, but wrong EXCEL Open Methods (Newton-Raphson Method)

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E. T. S. I. Caminos, Canales y Puertos51 Error Analysis for N-R : Recall that Taylor Series gives: where x r x x i and f(x r ) = 0 Open Methods (Newton-Raphson Method)

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E. T. S. I. Caminos, Canales y Puertos52 Dividing through by f '(x i ) yields E i+1 is proportional to E i 2 ==> quadratic rate of convergence. OR Open Methods (Newton-Raphson Method)

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E. T. S. I. Caminos, Canales y Puertos53 Summary of Newton-Raphson Method: Advantages: 1.Can be fast Disadvantages: 1.May not converge 2. Requires a derivative Open Methods (Newton-Raphson Method)

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E. T. S. I. Caminos, Canales y Puertos54 Secant Method Approx. f '(x) with backward FDD: Substitute this into the N-R equation: to obtain the iterative expression: Open Methods (Secant Method)

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E. T. S. I. Caminos, Canales y Puertos55 Secant Method x i = x i+1 x f(x) f(x i ) xixi f(x i-1 ) f(x) x i-1 x i+ 1 x f(x i ) xixi f(x i-1 ) x i-1 x i+ 1 Open Methods (Secant Method)

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E. T. S. I. Caminos, Canales y Puertos56 1) Requires two initial estimates: x i-1 and x i These do NOT have to bracket root ! 2) Maintains a strict sequence: Repeated until: a. | f(x i+1 ) | < k with k = small number b. c. Max. number of iterations is reached. 3. If x i and x i+1 were to bracket the root, this would be the same as the False-Position Method. BUT WE DON'T! Open Methods (Secant Method)

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E. T. S. I. Caminos, Canales y Puertos57 Open Methods (Secant Method) –In the secant method, the values are replaced in a strict sequence, x i+1 to x i, and this to x i-1. Thus, the new values can be on the de same sode of the root, and sometimes diverge.

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E. T. S. I. Caminos, Canales y Puertos58 Score Sheet for Secant Example: MethodInitial Est(s). s = 2 E-2 s = 2 E-7 Bisection(0.00, 1.00)926 (0.05, 0.15)622 False-pos. (0.00, 1.00)1128 (0.05, 0.15)314 N-R1.0divergesdiverges 0.52, but wrong Secant(0, 1)divergesdiverges (0.00, 0.50)4, but wrong (chaotic)27 (0.05, 0.15)36 Open Methods (Secant Method)

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E. T. S. I. Caminos, Canales y Puertos59 Why do open methods fail? Function may not look linear. Remedy: recast into a linear form. For example, Is a poorly constrained problem in that there is a large, nearly flat zone for which the derivative is near zero. Recast as: i f(i) = 0 = 7,500 i [ 1 - (1+i) -20 ] Open Methods

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E. T. S. I. Caminos, Canales y Puertos60 Recast as:i f(i) = 0 = 7,500 i [ 1 - (1+i) -20 ] –The recast function, "i f(i) will have the same roots as f(i) plus an additional root at i = 0. –It will not have a large, flat zone, thus: h(i) = i f(i) = 7,500 i – 1000 [ 1 – (1+ i) –20 ] – To apply N-R we also need the first derivative: h'(i) = 7, ,000 (1+ i) -21 Open Methods

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E. T. S. I. Caminos, Canales y Puertos61 Score Sheet for Open Methods: MethodInitial Est(s). s = 2 E-2 s = 2 E-7 N-R1.0divergesdiverges 0. 52, but wrong Secant(0.00, 0.50)4, but wrong27 (0.05, 0.15)36 N-R [as i f(i)] crazy results** 0.03converges to i=0** Open Methods

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E. T. S. I. Caminos, Canales y Puertos62 Cases of Multiple Roots Multiple Roots: f(x) = (x – 2) 2 (x – 4) x = 2 represents two of the three roots. Open Methods

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E. T. S. I. Caminos, Canales y Puertos63 Problems and Approaches: Cases of Multiple Roots 1.Bracketing Methods fail locating x = 2. Note that f(x ) f(x r ) > At x = 2, f(x) = f '(x) = 0. Newton-Raphson and Secant methods may experience problems. Rate of convergence drops to linear. Luckily, f(x) 0 faster than f '(x) 0 3. Other remedies, recasting problem: Find x such that u(x) = 0 where : Note that u(x) and f(x) have same roots. Open Methods

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E. T. S. I. Caminos, Canales y Puertos64 m = 1: linear convergence m = 2: quadratic convergence Method m Bisection1 False Position1 Secant, mult. root1 NR, mult. root1 Secant, single root1.618"super linear" NR, single root2 Accel. NR, mult. root (f(x)/f'(x)=0)2 Summary -- Rates of Convergence

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E. T. S. I. Caminos, Canales y Puertos65 Solve f i (x 1,..., x n ) = 0 for i = 1,...,n Let X = (x 1,..., x n ) T Given intial guess X t, try to solve where: Obtain X = (X i+1 – X i ) from linear equations: Multivariate (Multidimensional) Equations

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E. T. S. I. Caminos, Canales y Puertos66 1.Always limit number of iterations using an outer DO loop. The problem may not converge and could try to go on forever. 2.Absolute error criteria for "small" differences: | x t - x t-1 | < 3. Relative error criteria for "relatively small" changes | x t – x t-1 | < | x t | 4. Can combined error criteria 2 & 3 for large and small x-values: | x t – x t-1 | < + | x t | 5. Converge on zero residual | f(x t ) | < Alternative Stopping Criteria

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E. T. S. I. Caminos, Canales y Puertos67 Stopping Criteria: | x i – x i-1 | < + |x i | or| f(x i ) | < orMax. iterations Convergence Criteria: | x i – x i-1 | < + |x i | and| f(x i ) | < N-R and Secant Confirmation of Convergent Behavior: x in feasible region and| f(x i ) | ≤ 0.5 | f(x i-1 ) | and| x i – x i-1 | ≤ 0.6 | x i-1 – x i-2 | otherwise, do Bisection for a while. Three Performance Criteria

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E. T. S. I. Caminos, Canales y Puertos68 A real rootfinding problem can be viewed as having three phases: 1) Opening moves: One needs to find the region of the parameter space in which desired root can be found. Understanding of problem, physical insight, and common sense are valuable. 2) Middle Game: Use robust algorithm to reduce initial region of uncertainty. 3) End game: Generate a highly accurate solution in a few iterations. Three Phase Rootfinding Strategy

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