Let f(x) = tan(x). What is the formula for the conditioning number of f?
Evaluate this formula for x =(π/4), 1.01, 1.26, 1.51 (in radians, working to 3 signifi�cant �figures, the precision that is implicit in this data). Replace each of the expressions for x by its approximation ^x to 2 signifi�cant figures. Compute the relative error of the approximation ^x and the relative error of the approximation f(^x) to f(x).
Note; ^x is x-hat.
To determine the formula for the conditioning number of a function f(x), we need to consider the concept of numerical stability. The conditioning number measures how sensitive the output of a function is to small changes in its input. In other words, it quantifies the amplification of errors in the input to errors in the output.
Let's derive the formula for the conditioning number of the function f(x) = tan(x):
1. Start with the definition of the derivative of f(x):
f'(x) = sec^2(x)
2. The absolute condition number (K_abs) of f(x) is defined as:
K_abs = |x| * |f'(x)| / |f(x)|
Where |x| denotes the norm or absolute value of x.
3. Substitute the values specific to f(x) = tan(x):
f'(x) = sec^2(x)
f(x) = tan(x)
4. Rewrite K_abs in terms of tan(x) and sec^2(x):
K_abs = |x| * |sec^2(x)| / |tan(x)|
5. Simplify the expression:
K_abs = |x| / |sin(x)|
Now, let's evaluate the formula using the given values of x and their approximations, and compute the relative errors.
Given values of x: π/4, 1.01, 1.26, 1.51
1. Substitute the values into the formula:
K_abs(π/4) = |π/4| / |sin(π/4)|
K_abs(1.01) = |1.01| / |sin(1.01)|
K_abs(1.26) = |1.26| / |sin(1.26)|
K_abs(1.51) = |1.51| / |sin(1.51)|
2. Replace each of the expressions for x by its approximation ^x to 2 significant figures:
K_abs(π/4) ≈ |0.79| / |0.71|
K_abs(1.01) ≈ |1.0| / |0.85|
K_abs(1.26) ≈ |1.3| / |0.95|
K_abs(1.51) ≈ |1.5| / |0.99|
3. Compute the relative error of the approximation ^x:
Relative Error of ^x = |^x - x| / |x|
4. Compute the relative error of the approximation f(^x) to f(x):
Relative Error of f(^x) = |f(^x) - f(x)| / |f(x)|
By following these steps, we have derived the formula for the conditioning number of f(x) = tan(x), evaluated it for the given values, and calculated the relative errors.