The Problem create a Java program that repeatedly asks the user whether they wish to calculate another square root If the response is y , then the program should proceed if it is anything else, then the program should quit Whenever it proceeds, the program should prompt the user for a number (a positive double , and your program may simply assume the input is consistent with this requirement) and then report the square root of that number to within a relative error of no more than 0 01 The computation must be done using Newton iteration The intuitive idea of Newton iteration for computing square roots is fairly straightforward Suppose you have a guess r for x 1 2 that is too large the argument is similar if it is too small If r is too large to be the square root of x , then x r must be too small, so the average of r and x r should be a better guess than either r or x r This suggests that if you repeatedly replace your current guess r by (r x r) 2 , then your sequence of guesses should converge to x 1 2 And indeed it can be proved that it does A good initial guess for x 1 2 is simply r x If you continue updating r until r 2 x x 2 , then the relative error of the guess r will be less than After your initial program works, there are a number of other requirements to change it slightly, one step at a time, as explained below Method Note this is the last project description that will include instructions about basic Eclipse operations you have practiced already seeEnvironment SetupandGetting to Know Eclipse and Java As always, we recommend that you start early, and ask (and answer) questions onPiazzaso everyone learns from everyone Create a new Eclipse project by copyingProjectTemplate(if needed, seeCreating a New Project from a Project Templatefor details) Name the new projectNewton Open thesrcfolder of this project and then open(default package) As a starting point you should useProgramWithIOAndStaticMethod java Rename itNewton1and delete the other files from the project (if needed, seeCreating a Program from a Skeleton (also Renaming a Java Program)for details) EditNewton1 javato satisfy the problem requirements stated above, including updating comments appropriately Estimating the square root should be done in a static method declared as follows Computes estimate of square root of x to within relative error 0 01 param x positive number to compute square root of return estimate of square root private static double sqrt( double x) CopyNewton1 javato createNewton2 java Changesqrt(including its Javadoc comments) so it also works when x 0 Note if your code fromNewton1appears to work without any changes, but it is such that it might execute a division by 0, then it is not correct Division by 0, in general, is undefined and you should not write code that attempts to compute it CopyNewton2 javato createNewton3 java Change it so the main program prompts the user to input the value of (rather than assuming it is 0 0001), just once as the program begins, and so this value is also passed tosqrt CopyNewton3 javato createNewton4 java Change it so the main program does not ask the user whether they wish to calculate another square root, but instead simply asks for a new value of x and interprets a negative value as an indication that it's time to quit

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The Problem create a Java program that repeatedly asks the user whether they wish to calculate another square root. If the response is y, then

The Problem

create a Java program that repeatedly asks the user whether they wish to calculate another square root. If the response is "y", then the program should proceed; if it is anything else, then the program should quit. Whenever it proceeds, the program should prompt the user for a number (a positivedouble, and your program may simply assume the input is consistent with this requirement) and then report the square root of that number to within a relative error of no more than 0.01%. The computation must be done usingNewton iteration.

The intuitive idea of Newton iteration for computing square roots is fairly straightforward. Suppose you have a guessrforx^1/2that is too large; the argument is similar if it is too small. Ifris too large to be the square root ofx, thenx/rmust be too small, so the average ofrandx/rshould be a better guess than eitherrorx/r. This suggests that if you repeatedly replace your current guessrby(r + x/r)/2, then your sequence of guesses should converge tox^1/2. And indeed it can be proved that it does. A good initial guess forx^1/2is simplyr = x. If you continue updatingruntil |r²- x|/x < ², then the relative error of the guessrwill be less than.

After your initial program works, there are a number of other requirements to change it slightly, one step at a time, as explained below.

Method

Note: this is the last project description that will include instructions about basic Eclipse operations you have practiced already; seeEnvironment SetupandGetting to Know Eclipse and Java. As always, we recommend that you start early, and ask (and answer) questions onPiazzaso everyone learns from everyone.

Create a new Eclipse project by copyingProjectTemplate(if needed, seeCreating a New Project from a Project Templatefor details). Name the new projectNewton.
Open thesrcfolder of this project and then open(default package). As a starting point you should useProgramWithIOAndStaticMethod.java. Rename itNewton1and delete the other files from the project (if needed, seeCreating a Program from a Skeleton (also Renaming a Java Program)for details).
EditNewton1.javato satisfy the problem requirements stated above, including updating comments appropriately. Estimating the square root should be done in a static method declared as follows:

/**

* Computes estimate of square root of x to within relative error 0.01%.

* @param x

* positive number to compute square root of

* @return estimate of square root

private static double sqrt(double x) {

...

}

CopyNewton1.javato createNewton2.java. Changesqrt(including its Javadoc comments) so it also works whenx = 0. Note: if your code fromNewton1appears to work without any changes, but it is such that it might execute a division by 0, then it is not correct. Division by 0, in general, is undefined and you should not write code that attempts to compute it.
CopyNewton2.javato createNewton3.java. Change it so the main program prompts the user to input the value of(rather than assuming it is 0.0001), just once as the program begins, and so this value is also passed tosqrt.
CopyNewton3.javato createNewton4.java. Change it so the main program does not ask the user whether they wish to calculate another square root, but instead simply asks for a new value ofxand interprets a negative value as an indication that it's time to quit.