lec20 - 20. Newtons Method P. K. Lamm Lecture Notes:...

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20. Newton’s Method P. K. Lamm 11/01/11 (14:36) p. 1 / 6 Lecture Notes: 20. Newton’s Method These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1. Newton’s Method Newton’s Method uses the idea of linear approximation in a very important way. Suppose we are given y = f ( x ) where we know there is some c for which f ( c ) = 0; that is, the point x = c solves the equation f ( x ) = 0 , or c is a root of f ( x ). The goal is to actually estimate c . We’ll start with a guess x 0 for c ; hopefully the initial guess is reasonably close to c . If we then construct the tangent line to the curve at ( x 0 ,f ( x 0 )) we have the line y - f ( x 0 ) = f 0 ( x 0 )( x - x 0 ) , or y = f ( x 0 ) + f 0 ( x 0 )( x - x 0 ) . Now for x near x 0 , we know the y -value of the curve is well-approximated by the y -value of the tangent line. That is, f ( x ) f ( x 0 ) + f 0 ( x 0 )( x - x 0 ) , for x x 0 . Since our goal is to find c such that y = f ( x ) crosses the x -axis at c , we could try to approximate c by finding where this tangent line to the curve at x 0 crosses the x -axis. That is, we set 0 = f ( x 0 ) + f 0 ( x 0 )( x - x 0 ) and solve for x . Then f 0 ( x 0 )( x - x 0 ) = - f ( x 0 ) x - x 0 = - f ( x 0 ) f 0 ( x 0 ) x = x 0 - f ( x 0 ) f 0 ( x 0 ) , provided f 0 ( x 0 ) 6 = 0. 1
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20. Newton’s Method P. K. Lamm 11/01/11 (14:36) p. 2 / 6 Hopefully this point where the tangent line crosses the x -axis is closer to c than our initial guess x 0 was. We will call x 1 this new x -value, x 1 = x 0 - f ( x 0 ) f 0 ( x 0 ) . But now we can repeat the process with our new initial guess
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lec20 - 20. Newtons Method P. K. Lamm Lecture Notes:...

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