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E. T. S. I. Caminos, Canales y Puertos1 Engineering Computation Lecture 4

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E. T. S. I. Caminos, Canales y Puertos2 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 Puertos3 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 Puertos4 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 Puertos5 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 Puertos6 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 Puertos7 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 Puertos8 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 Puertos9 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 Puertos10 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 Puertos11 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 Puertos12 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 Puertos13 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 Puertos14 Open Methods (Fixed point method)

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E. T. S. I. Caminos, Canales y Puertos15 Open Methods

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E. T. S. I. Caminos, Canales y Puertos16 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 Puertos17 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 Puertos18 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 Puertos19 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 Puertos20 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 Puertos21 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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