Question

a) Evaluate the polynomial \(\displaystyle{y}={x}^{{3}}-{7}{x}^{{2}}+{8}{x}-{0.35}\)
at \(\displaystyle{x}={1.37}\) . Use 3-digit arithmetic with chopping. Evaluate the percent relative error. b) Repeat (a) but express y as \(\displaystyle{y}={\left({\left({x}-{7}\right)}{x}+{8}\right)}{x}-{0.35}\) Evaluate the error and compare with part (a).

Answer 1

(a)First determine the powers of x needed for evaluating the polynomial. \(\displaystyle{x}={1.37}{7}{x}^{{2}}={13.1383}\) Chopping to 3 digit arithmetic yields \(\displaystyle{x}^{{2}}={13.1}{8}{x}={10.96}\) Chopping to 3 digit arithmetic yields \(\displaystyle{8}{x}={10.9}{x}^{{3}}={2.571353}\) Chopping to 3 digit arithmetic yields \(\displaystyle{x}^{{3}}={2.57}\) Insert these values in the given polynomial to find \(\displaystyle{y}{\left({1.37}\right)}.{y}{\left({1.37}\right)}={2.57}−{13.1}+{10.9}−{0.35}={0.02}\) Whereas the exact result is, \(\displaystyle{y}{\left({1.37}\right)}={2.571353}−{13.1383}+{10.96}−{0.35}={0.043053}\) Hence, the percent relative error is, \(\displaystyle\xi={\frac{{{0.043053}-{0.02}}}{{{0.043053}}}}\times{100}\%={53.54}\%\) (b) \(\displaystyle{y}={\left({\left({x}-{7}\right)}{x}+{8}\right)}{x}-{0.35}\) Now, when x = 1.37,
(x - 7)x = (1.37 - 7)(1.37)
=-7.7131 Chopping to 3 digit arithmetic yields, \(\displaystyle{\left({x}-{7}\right)}{x}=-{7.71}{\left({\left({x}-{7}\right)}{x}+{8}\right)}{x}={\left(-{7.71}+{8}\right)}{1.37}={0.047}\) Here, the percent relative error is, \(\displaystyle\xi={\frac{{{0.043053}-{0.047}}}{{{0.043053}}}}\ \times{100}\%=-{9.16}\%\) The percent relative error is smaller when compared to part a.