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Construct a quadratically convergent method for calculating the n-th root of a positive number A, where n is a positive integer.

Construct a quadratically convergent method for calculating the n-th root of a positive number A, where n is a positive integer.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Construct a quadratically convergent method for calculating the n-th root of a positive number A, where n is a positive integer.
 
Please show all work and keep in mind Newton's Method 
 
Expert Solution
Step 1: Newton-Raphson method formula

               Let  the n-th root of a positive number A is x. That means

                        n-th root of A space equals space x     rightwards double arrow       A to the power of 1 over n end exponent space equals space x

                                           rightwards double arrow       x to the power of n space equals space A

                                          rightwards double arrow       x to the power of n space minus space A space equals space 0

                         Let  f open parentheses x close parentheses space equals space x to the power of n space minus space A

                          rightwards double arrow  f apostrophe open parentheses x close parentheses space equals space n x to the power of n minus 1 end exponent

             for Newton-Raphson method, the k+1 iteration formula is

                           x subscript k plus 1 end subscript space equals space x subscript n space minus space fraction numerator f open parentheses x subscript k close parentheses over denominator f apostrophe open parentheses x subscript k close parentheses end fraction


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