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newton - Second part of Excercise 12.21(Newton-Rhapsons...

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Second part of Excercise 12.21 (Newton-Rhapson’s method) Patrick De Leenheer April 5, 2005 We’ve already proved in class that if I is an open interval, and if f : I R is convex and differentiable in I , then for ξ I , f ( x ) - f ( ξ ) f 0 ( ξ )( x - ξ ) , x I. (1) Now suppose that φ : R R is strictly increasing, convex and differentiable and φ ( ξ ) = 0. Let x 1 > ξ , and consider the iteration x n +1 = x n - φ ( x n ) φ 0 ( x n ) , n = 1 , 2 , . . . (2) Then prove that the sequence x n ξ as n → ∞ . Remark 1 . Observe that ξ is the unique root of φ , since φ is strictly increasing. Also notice that this result yields an algorithm to find an approximation of the root ξ of the function φ (only an approximation, because we assume that an algorithm terminates after a finite number of steps). Proof. We will first show that the sequence x n is decreasing and bounded below (by ξ ) and therefore converges to some l (by Theorem 4 . 17). Then we will show that l is in fact ξ , at which point we’ll be done.
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