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

(a)First determine the powers of x needed for evaluating the polynomial. $x=1.377{x}^{2}=13.1383$ Chopping to 3 digit arithmetic yields ${x}^{2}=13.18x=10.96$ Chopping to 3 digit arithmetic yields $8x=10.9{x}^{3}=2.571353$ Chopping to 3 digit arithmetic yields ${x}^{3}=2.57$ Insert these values in the given polynomial to find $y\left(1.37\right).y\left(1.37\right)=2.57-13.1+10.9-0.35=0.02$ Whereas the exact result is, $y\left(1.37\right)=2.571353-13.1383+10.96-0.35=0.043053$ Hence, the percent relative error is, $\xi =\frac{0.043053-0.02}{0.043053}×100\mathrm{%}=53.54\mathrm{%}$ (b) $y=\left(\left(x-7\right)x+8\right)x-0.35$ Now, when x = 1.37,
(x - 7)x = (1.37 - 7)(1.37)
=-7.7131 Chopping to 3 digit arithmetic yields, $\left(x-7\right)x=-7.71\left(\left(x-7\right)x+8\right)x=\left(-7.71+8\right)1.37=0.047$ Here, the percent relative error is, The percent relative error is smaller when compared to part a.

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Jeffrey Jordon

First, we can calculate the true value of the polynomial

$p\left(x\right):={x}^{3}-7{x}^{2}+8x-0.35$

at point x=1.37

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

Using 3-digits with chopping we have to calculate separately

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

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

$⇒p\left(1.37\right)\approx 2.57-13.0+10.9-0.35=0.12$

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:

$p\left(x\right)=\left(\left(x-7\right)x+8\right)x-0.35$

Now we calculate

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

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

$0.397-0.35=0.047$

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.

Jeffrey Jordon