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# Lecture7 - 1 LECTURE 2.3 Overview(Covers a part of section...

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1 LECTURE 2.3 Dr. Vyas Overview: (Covers a part of section 2.4) Newton-Raphson Method Secant Method Examples/Demos NEWTON-RAPHSON METHOD: A. Algebraic Description 1. Theorem 2.5 : Assume that 2 [ , ] f C a b ( meaning function is continuous upto its second derivative in the given interval ) and that there exists a number [ , ] p a b such that ( ) 0 f p = . If ( ) 0 f p , then there exists a 0 δ such that the sequence { } 0 k k p = defined by the iteration 1 1 1 1 ( ) ( ) ( ) k k k k k f p p g p p f p - - - - = = - for 1, 2, k = K where, ( ) ( ) ( ) f x g x x f x - will converge to p for any initial approximation 0 [ , ] p p p δ δ - + . 2. The function ( ) g x is called the Newton-Raphson iteration function . 3. As can be observed the recursive relation 1 ( ) k k p g p - = is an indicator of the application of the fixed point iteration method . Therefore, one can also think of this process as that of finding a fixed point of the function ( ) g x (Not ( ) f x !) 4. Recall, the Taylor expansion of a function about a point 0 x x = may be expressed as ( ) ( 1) 1 0 0 0 0 ( ) ( ) ( ) ( ) ( ) ! ( 1)! k n n k n k f x f c f x x x x x k n + = = - + - + . The assumptions that underlies this expression are that, 1 0 0 [ , ], [ , ], ( ) ( , ) n f C a b x a b c c x x x + = 5. Let 0 0 x p = be the starting point of the iterative process. Therefore, the Taylor expansion (approximated to two term expansion plus remainder) can be rewritten as, (2) (1) 2 0 0 0 0 ( ) ( ) ( ) ( ) ( ) ( ) 2! f c f x f p x p f p x p = + - + - . The superscript numbers are the order of the derivatives of the function. The primed notation will be followed henceforth to stay with the textbook nomenclature. 6. If p is the fixed point of the function then, ( ) ( ) 0 g p p f p = = . Therefore, using this information in the Taylor expansion, 2 0 0 0 0 ( ) 0 ( ) ( ) ( ) ( ) ( ) 2! f c f p f p p p f p p p ′′ = = + - + - . 7. Further, if the starting point 0 p is close enough (practically, sufficiently less than unity) to the fixed point p , then the last term in the above expression can be neglected. As a result, the following approximation results, 0 0 0 0 0 0 ( ) ( ) ( ) ( ) 0 ( ) f p f p p p f p p p f p + - - . Geometrically, we are

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2 approximating the curve by its tangent at a point. This is valid in so far one is close to the point of tangency. B. Geometric Description 8. The slope of tangent line between points 0 0 ( , ( )) p f p and 1 ( ,0) p is given by 0 1 0 0 ( ) f p slope p p - = - . However, recall that for a function ( ) y f x = , the slope is also given by its first derivative ( ) y f x = . The first derivative evaluated at the point of tangency gives the slope of the tangent line. Therefore, the slope of the tangent line is 0 0 0 1 0 1 0 0 0 ( ) ( ) ( ) ( ) f p f p f p p p p p f p - = = - - 9. This approximation can be used to refine until fixed point (or root) is located.
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