What Is the Numerical Stability Range for Calculating Arctan Using Arcsin?

[happyg1]

Hi,
I asked this question a couple of days ago, and I have been working on it a lot since then. Please see if you know what I can do next


Here is the question:
Suppose a computer program is available which yields values for arcsin y in floating point representation with t decimal mantissa places and for |y|<=1 subject to a relative error eps with |eps|<=10exp-t. In view of the relation

arctan x = arcsin [x/(sqrt(1+x^2)]

this program could also be used to evaluate arctan x. Determine for which values x this procedure is numerically stable by estimating the relative error.


I have calculated the relative error by taking the derivative of the RHS and multiplying that by x/arctan x, which is the realtive error equation, but it makes no sense to me. What do I do with it now? I got

relative error = (x^3/arctanx)[(1+x^2)^3/2 + (1+x^2)^1/2]*(epsilon of x)

Thanks for any input,
CC

 

[anathema]

Dear CC,

Thank you for your question. It seems like you have made some progress in your calculations. To determine the numerical stability of this procedure, we need to consider the behavior of the relative error as the values of x and epsilon change.

First, let's review the definition of relative error. It is given by the formula:

relative error = (approximate value - exact value) / exact value

In this case, the approximate value is arctan x calculated using the arcsin program, and the exact value is the true value of arctan x. As you have correctly noted, the relative error can also be written as:

relative error = (x^3/arctanx)[(1+x^2)^3/2 + (1+x^2)^1/2]*(epsilon of x)

This formula shows how the relative error depends on the values of x and epsilon. We can see that the relative error is directly proportional to x^3, which means that as x increases, the relative error also increases. This indicates that the procedure becomes less numerically stable as x increases.

Additionally, the relative error is also affected by the value of epsilon. We can see that as epsilon increases, the relative error also increases. This means that as the precision of the program decreases (as epsilon gets larger), the numerical stability of the procedure decreases as well.

To determine the values of x for which the procedure is numerically stable, we need to find the range of x values for which the relative error is small. This would mean that the approximate value is close to the exact value, indicating a stable procedure.

From the formula for relative error, we can see that as x gets closer to 0, the relative error gets smaller. This means that the procedure is more numerically stable for values of x close to 0. Additionally, as epsilon gets smaller, the relative error also decreases, indicating a more stable procedure.

In conclusion, the procedure for using the arcsin program to evaluate arctan x is numerically stable for values of x close to 0 and for smaller values of epsilon. I hope this helps you in your calculations and understanding of numerical stability. If you have any further questions, please don't hesitate to ask.

 

[quicknote]

As a scientist, my suggestion would be to first confirm the accuracy of your calculation for the relative error. You can do this by comparing it to the relative error obtained from other methods or by running tests with different values of x. Once you have confirmed the accuracy of your calculation, you can then use it to determine the stability of the program for different values of x.

To do this, you can plot the relative error as a function of x and see at which points the error is within the acceptable range of |eps|<=10exp-t. This will give you a range of values for x that the program is numerically stable for.

Additionally, you can also consider the precision of the floating point representation used in the program. If the precision is not high enough, the error may increase for larger values of x. In this case, you may need to adjust the program or use a higher precision representation.

Overall, it is important to thoroughly test and validate the results of the program to ensure its accuracy and stability for different inputs.