Practice

Euler's Method

Design and code a main program that prompts for and accepts a floating-point number and an error margin and calculates the approximate square root of the number using Euclid's method.  Do not use the sqrt function from the math library.

Euclid's method keeps guessing the square root of a number, say x, using the formula

 ` quotient = x / estimate`
where estimate is the most recent estimate of the square root of x.  The method improves each estimate by taking the average of the estimate and the quotient as defined above.  The improvement process terminates once the difference between the estimate and the quotient is within a prescribed error margin, say epsilon:
 ` | quotient - estimate | <= epsilon`
Euclid's method starts with an initial estimate of
 ` estimate = x / 2`
If the number input by the user is positive or zero, your program displays successive estimates in tabular form, something like:
 ``` Find the Square Root of : 34567.0 Acceptable Error : 0.000000001 Iteration Estimate 1 17283.500000000000 2 8642.750000000000 3 4323.374768592173 4 2165.685071502284 5 1090.823151320520 6 561.256032187495 7 311.422339911770 8 211.209757516741 9 187.435851925555 10 185.928139870331 11 185.922026767594 12 185.922026667095 The approximate square root after 12 iterations is 185.922027```
If the number input is negative, your program does not calculate a square root and simply displays a message to that effect.

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