91Ó°ÊÓ

Calculate the error, relative error, and number of significant digits in the following approximations \(x_{A} \approx x_{T}:\) (a) \(x_{T}=28.254, x_{A}=28.271\) (b) \(\quad x_{T}=0.028254, x_{A}=0.028271\) (c) \(\quad x_{T}=e, x_{A}=19 / 7\) (d) \(x_{T}=\sqrt{2}, x_{A}=1.414\) (e) \(x_{T}=\log (2), x_{A}=0.7\)

Discuss the possible loss-of-significance error that may be encountered in solving the quadratic equation \(a x^{2}+b x+c=0\). How might that loss-of- significance error be avoided?

For what values of \(x\) does \(x^{10}\) underflow using IEEE double precision normalized floating-point arithmetic. When does it overflow?