It is desired to determine a recursive recursive expression (difference equation

Question

It is desired to determine a recursive recursive expression (difference equation) for finding the rth root of a real number N. We start by writing the Taylor series expansion of function f(x) about a point xn: where if the series is truncated after two terms, we have Let x represent the next iterate xn+1, and let x also represent a solution of the equation f(x) = 0. Then eq. (p2-4-2) becomes Equation (p2-4-4) is a difference equation useful for solving f(x)=0. choose f(x) appropriately and determine the corresponding difference equation for finding the rth root of a number N. Do five iterations to calculate the cube root of six using x0 = 1 as the initial guess. Repeat part a to find the cube root of seven.

Explanation / Answer

choose f(x)=xr-N

f'(x)=rxr-1

thus iteration rule becomes

xn+1=xn-(xnr-N)/rxnr-1.

now putting r=3 and N=6 and x0=1

we get x1= x0-(x03-6)/3x02=8/3

similarly putting x2 in place of x1 and x1 in place of x0

we get x2= 251/96= 2.61

after that x3= 2.03

x4=1.83

x5=1.8171.

thus after 5 iterationscube root of 6 is 1.817.

similarly you can find cube root of 7 by putting r=3 and N=7.

NOTE: this method is konown as newton raphson iteration . you can find it on wikipedia.

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