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Unformatted text preview: HOMEWORK 4: NEWTON ITERATIONS FOR THE SQUARE ROOT JAN DE LEEUW Newton iterations to find the root of a real valued function f , i.e. a number x for which f ( x ) = 0, are of the form x ( k + 1) = x ( k )- f ( x ( k ) ) f ( x ( k ) ) . Example. To find the square root of a positive number y we can use New- ton’s method to solve the equation f ( x ) = x 2- y = 0. Since f ( x ) = 2 x we see that x ( k + 1) = x ( k )- ( x ( k ) ) 2- y 2 x ( k ) = ( x ( k ) ) 2 + y 2 x ( k ) . Global Convergence. Does Zangwill’s Theorem apply in this example ? What follows from it ? We can also use simple calculations here. Show that, if we start with x (0) > 0, x ( k + 1) > √ y , for all k > 0 and x ( k + 1) < x ( k ) . for all k > 0. This implies the sequence decreases monotonically and con- verges to √ y from any (positive) starting point. Speed of Convergence. Does Ostrowski’s Theorem apply in this example ?...
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This note was uploaded on 01/14/2011 for the course STATS 102A 102A taught by Professor Jandeleeuw during the Fall '10 term at UCLA.
- Fall '10