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# Lecture8 - 1 LECTURE 2.4 Dr Vyas Overview(Covers the...

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Unformatted text preview: 1 LECTURE 2.4 Dr. Vyas Overview: (Covers the remaining part of section 2.4) • Simple and Multiple Roots • Rate of convergence of root finding methods • Examples/Demos SIMPLE & MULTIPLE ROOTS 1. Before beginning on the concept of convergence, it is instructive to discuss the concept of multiple real roots. We say ( ) f x = has a root of order M at x p = if and only if, ( 1) ( ) ( ) 0, ( ) 0, ( ) 0,....... ( ) 0,& ( ) M M f p f p f p f p f p- ′ ′′ = = = = ≠ . Of course, we are assuming that all the relevant derivatives exist and are continuous on an interval containing x p = . 2. The root of order 1 M = is called a simple root, else it is a multiple root. 3. If the equation ( ) f x = has a root of order M at x p = , then there exists a continuous function ( ) h x such that, ( ) ( ) ( ), ( ) M f x x p h x h p =- ≠ 4. For example, 3 2 3 2 ( 1) ( 2) x x x x-- = +- , 1 x = - is a multiple root of order 2. >> x = -1; (Please note that this is a multiple root) >> f = x^3 - 3*x - 2 f = 0 >> df = 3*x^2 - 3 df = 0 >> ddf = 6*x ddf = -6 >> x = 2; (Please note that this is a simple root) >> f = x^3 - 3*x - 2 f = 0 >> df = 3*x^2 - 3 df = 9 5. For simple root, iteration to the final answer is faster than that for a multiple root. This brings forth the concept of the rate (order) of convergence for the methods. 2 ORDER/RATE/SPEED OF CONVERGENCE 1 : 1. Assume that { } n n p ∞ = converges to p and set n n E p p =- for n ≥ . If two positive constants A ≠ and R exist, and if 1 1 lim lim (asymptotic error constant) n n R R n n n n p p E A p p E + + →∞ →∞- = =- , then the sequence is said to converge to p with the order of convergence R . 2. If 1 R = the convergence is linear, if 2 R = the convergence is quadratic, cubic if 3 R = and so forth. 3. Theorem 2.6: Assume that Newton-Raphson iteration produces a sequence { } n n p ∞ = that converges to the root p of the function ( ) f x . If p is a simple root, the convergence is quadratic and 2 1 ( ) 2 ( ) n n f p E E f p + ′′ ≈ ′ for n sufficiently large. However, if p is a multiple root of order M , the convergence is linear and 1 1 n n M E E M +- ≈ for n sufficiently large. 4. Assuming similar sequence is produced by the Secant method iteration, the error terms satisfy the relationship 2 , 0.618 1.618 1 ( ) 2 ( ) n n f p E E f p + ′′ ≈ ′ 5. Bisection method and Regula-Falsi method always converge linearly for the simple root. Because the multiple root touches the horizontal axes, these methods cannot be used. However, the Bisection method has the asymptotic error constant that is always approximately 1 2 . 6. Next, we consider simple and multiple roots of the equation, 3 ( ) 3 2 f x x x =-- 1 Observe that the iterations are counted from n = , and therefore the subscript 1 n + as next iteration....
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Lecture8 - 1 LECTURE 2.4 Dr Vyas Overview(Covers the...

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