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).

Question

a) Evaluate the polynomial

y = x^{3} - 7 x^{2} + 8 x - 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).

Answer 1

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

Answer 2

First, we can calculate the true value of the polynomial

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

at point x=1.37

$y_{e x a s t} := p (1.37) = {1.37}^{3} - 7 \cdot {1.37}^{2} + 8 \cdot 1.37 - 0.35 = 0.043053$

Using 3-digits with chopping we have to calculate separately ${1.37}^{2}, - 7 \cdot {1.37}^{2} a n d 8 \cdot 1.37$

${1.37}^{3} = 2.571353 \overset{c h o p p i n g}{\to} 2.57 - 7 \cdot {1.37}^{2} = - 13.1383 \overset{c h o p p i n g}{\to} - 13.0$

$8 \cdot 1.37 = 10.96 \overset{c h o p p i n g}{\to} 10.9$

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

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

$(1.37 - 7) \cdot 1.37 = - 7.7131 \overset{c h o p p i n g}{\to} - 7.71$

$(- 7.71 + 8) \cdot 1.37 = 0.3973 \overset{c h o p p i n g}{\to} 0.397$

$0.397 - 0.35 = 0.047$

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 3

Jeffrey Jordon

Answered 2022-01-23 Author has 2313 answers

Answer is given below (on video)

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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).

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