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Published byMarilyn Black Modified over 4 years ago

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Suppose f(x) is a continuous function of x within interval [a, b]. f(a) = - ive and f(b) = + ive There exist at least a number p in [a, b] with f(p) = 0. Meaning, p is a root of the equation f(x) = 0

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The Bisection Method calls for a repeated halving of subintervals of [a, b] each time locating the half containing p. Bisection Method (Binary Search)

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Bisection Algorithm

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Set a 1 = a and b 1 = b. Find the midpoint between a 1 and b 1. Midpoint,

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If f(p 1 ) = 0, then p 1 is the root of the equation within [a, b]. If f(p 1 ) 0, then what? Then find if f(p 1 ) has the same sign as either f(a 1 ) or f(b 1 ).

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Bisection Algorithm

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IF f(p 1 ) has the same sign as f(a 1 ), then the root is in [p 1, b 1 ]. Set a 2 = p 1 and b 2 = b 1. IF f(p 1 ) has the same sign as f(b 1 ), then the root is in [a 1, p 1 ]. Set a 2 = a 1 and b 2 = p 1.

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The root is in the interval [a 2, b 2 ]. Divide the interval in two halves and repeat the process.

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When do we stop?

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has a root in [1, 2]. nanan pnpn bnbn 11.0 (-)1.5 (+)2.0 (+) 21.0 (-)1.25 (-)1.5 (+) 31.25 (-)1.375 (+)1.5 (+) 41.25 (-)1.3125 (-)1.375 (+) 51.3125 (-)1.34375 (-)1.375 (+)

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The Method of False Position The method is based on bracketing the root between two points. At the beginning choose two points, so that Now draw a line joining The x-intercept of the line is

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Now bracket the root between either Which pair to choose? On the other hand Let us assume that This means that the root is between

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Now draw a line joining The x-intercept of the line is and the process continues …

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has a root in [1, 2]. n 11.26316 21.33883 31.35855 41.36355 51.36481

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Fixed-Point Iteration Rewrite f(x) = 0 in the form of x = g(x) and iterate. has a root in [1, 2].

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We can rewrite f(x) in the form of x = g(x) in the following ways.

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Start with x = 1.5

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Results of the Fixed-point Iteration n(a)(b)(c)(d) 11.5 2-0.8750.81651.28695371.3483997 36.7322.99691.40254081.3673763 4-469.71.34545831.3649570 51.37517021.3652647 61.36009411.3652255 71.36784691.3652305 81.36388701.3652299 91.36591671.3652300

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Why some expressions failed to deliver the root? To deliver the root, g(x) for all x in [a, b] must stay within [a, b].

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Newton’s Method

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Consider the triangle (p 2, 0), (p 1, 0) and (p 1, f(p 1 )).

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Newton’s Method

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Consider the triangle (p 2, 0), (p 1, 0) and (p 1, f(p 1 )). (p 2, 0) (p 1, 0) (p 1, f(p 1 ))

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(p 2, 0) (p 1, 0) (p 1, f(p 1 )) The slope,

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A sequence can be generated as:

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Example:

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