Lecture 2 Typeset Notes

Lecture 2 Typeset Notes - ) <0, THEN x R = x App...

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2. False Position (Regula – Falsi) Method Here obtain [ x L , x R ], such that f(x L ) . f(x R ) <0. Instead of successively bisecting the interval, as in the bisection method, linear interpolation is used to obtain x App such that ( x App , 0) lies on the line joining ( x L , f(x L ) ) and ( x R , f(x R ) ). Since, the points ( x L , f(x L ) ), ( x App , 0) and ( x R , f(x R ) ) all lie on the same line, it is clear that: L R L App L R L x x x x x f x f x f ! ! " ! ! ) ( ) ( ) ( 0 , and hence, ) ( ) ( ) ( R L R L L L App x f x f x x x f x x ! ! ! " . Next we determine which of the 2 sub-intervals [ x L , x App ] or [ x App , x R ] contains the root of f(x) . Algorithm: Given the above, REPEAT ) ( ) ( ) ( R L R L L L App x f x f x x x f x x ! ! ! " IF f(x L ) . f(x App
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Unformatted text preview: ) &lt;0, THEN x R = x App ELSE x L = x App UNTIL Stopping criteria are met END Possible stopping criteria # | x L x R | &lt; e for some small e (means we have narrowed down the interval so much that for all intents and purposes we do not need to continue. # n&gt;N limit (means we have exceeded the maximal allowed number of iterations N limit ) # | f(x App ) | &lt; e 2 for some small e 2 (this means that my middle point is almost the root of this function, so no need to continue...
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Lecture 2 Typeset Notes - ) &amp;amp;lt;0, THEN x R = x App...

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