the quantity y=1-\sqrt{1+x^2} for x=10^{-6} is not so accurate due to numerical cancellation. indeed, the relative error associated to it is, approximately: in matlab

To find the numerical cancellation and the associated relative error for the quantity y using MATLAB, let's first calculate the exact value of y and then compare it with the approximate value.

The given formula is y = 1 - sqrt(1 + x^2), and x = 10^(-6).

Using MATLAB, we can calculate the exact value of y as follows:

main.m
x = 1e-6;
y_exact = 1 - sqrt(1 + x^2);
39 chars
3 lines

Next, we will calculate the approximate value of y using the simplified formula:

main.m
y_approx = -x^2 / (2 * (sqrt(1 + x^2) + 1));
45 chars
2 lines

Now, we can calculate the relative error using the formula:

main.m
relative_error = abs((y_approx - y_exact) / y_exact);
54 chars
2 lines

The relative error will give us an idea of how accurate the approximation is compared to the exact value. A larger relative error indicates a less accurate result due to numerical cancellation.

To display the relative error in MATLAB, you can use the following code:

main.m
fprintf('Relative Error: %.16f\n', relative_error);
52 chars
2 lines

Keep in mind that the relative error calculated using MATLAB is an approximation and may vary depending on the precision of your system.

Note: The negative sign in the y_approx formula is due to the difference in the sign convention between MATLAB and the formula given.

I hope this helps!