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

(a)First determine the powers of x needed for evaluating the polynomial. ${\displaystyle x=1.377x^2=13.1383}$ Chopping to 3 digit arithmetic yields ${\displaystyle x^2=13.18x=10.96}$ Chopping to 3 digit arithmetic yields ${\displaystyle 8x=10.9x^3=2.571353}$ 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, ${\displaystyle \xi =\frac{0.043053-0.02}{0.043053}\times 100\mathrm{\%}=53.54\mathrm{\%}}$ (b) ${\displaystyle y=((x-7)x+8)x-0.35}$ Now, when x = 1.37,
(x - 7)x = (1.37 - 7)(1.37)
=-7.7131 Chopping to 3 digit arithmetic yields, ${\displaystyle (x-7)x=-7.71((x-7)x+8)x=(-7.71+8)1.37=0.047}$ Here, the percent relative error is, ${\displaystyle \xi =\frac{0.043053-0.047}{0.043053}\times 100\mathrm{\%}=-9.16\mathrm{\%}}$ The percent relative error is smaller when compared to part a.

First, we can calculate the true value of the polynomial

$p(x):=x^3-7x^2+8x-0.35$

at point x=1.37

$y_{exast}:=p(1.37)={1.37}^3-7\cdot {1.37}^2+8\cdot 1.37-0.35=0.043053$

Using 3-digits with chopping we have to calculate separately ${1.37}^2,-7\cdot {1.37}^{2}and8\cdot 1.37$

${1.37}^3=2.571353\stackrel{chopping}{\to }2.57-7\cdot {1.37}^2=-13.1383\stackrel{chopping}{\to }-13.0$

$8\cdot 1.37=10.96\stackrel{chopping}{\to }10.9$

$\Rightarrow p(1.37)\approx 2.57-13.0+10.9-0.35=0.12$

The relative percent error is equal to

$|\frac{0.043053-0.12}{0.043053}|=1.787=178.7$

b)

Polynomial p can be expressed in a different way:

$p(x)=((x-7)x+8)x-0.35$

Now we calculate

$(1.37-7)\cdot 1.37=-7.7131\stackrel{chopping}{\to }-7.71$

$(-7.71+8)\cdot 1.37=0.3973\stackrel{chopping}{\to }0.397$

$0.397-0.35=0.047$

The relative percent error is equal to

$|\frac{0.043053-0.047}{0.043053}|=0.092=9.2$

Therefore, the second form of polynomial is better for evaluation.

Answer is given below (on video)