To determine To find: Aquadratic convergence method to calculate the nth root of a positive number A by Newton’s method where n is a positive integer and prove quadratic convergence. Explanation Given information : n is the positive integer. Calculation: Let us consider the following function. f ( x ) = x n − A ......... ( 1 ) Now differentiate equation (1) with respect tox. f ′ ( x ) = d d x ( x n − A ) = n x n − 1 Now, the Newton’s method will give the following x i + 1 = x i − f ( x i ) f ′ ( x i ) = x i − x i n − A n x i n − 1 Let us simplify it further as shown below. x i + 1 = x i − f ( x i ) f ′ ( x i ) = n x i n − 1 x i − x i n + A n x i n − 1 = n x i n − x i n + A n x i n − 1 = x i n ( n − 1 ) + A n x i n − 1 = 1 n ( ( n − 1 ) x i + A x i n − 1 ) If now, there is a fixed pointrto x i + 1 = 1 n ( ( n − 1 ) x i + A x i n − 1 )

3rd Edition

ISBN: 9780134696454

Author: Sauer, Tim

Publisher: Pearson,

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Chapter 1.4, Problem 11E

To determine

To find: Aquadratic convergence method to calculate the nth root of a positive number A by Newton’s method where n is a positive integer and prove quadratic convergence.

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Chapter 1.4, Problem 10E

Chapter 1.4, Problem 12E

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