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).

Braxton Pugh 2020-10-18 Answered
a) Evaluate the polynomial y=x37x2+8x0.35
at x=1.37 . Use 3-digit arithmetic with chopping. Evaluate the percent relative error. b) Repeat (a) but express y as y=((x7)x+8)x0.35 Evaluate the error and compare with part (a).
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pattererX
Answered 2020-10-19 Author has 95 answers

(a)First determine the powers of x needed for evaluating the polynomial. x=1.377x2=13.1383 Chopping to 3 digit arithmetic yields x2=13.18x=10.96 Chopping to 3 digit arithmetic yields 8x=10.9x3=2.571353 Chopping to 3 digit arithmetic yields x3=2.57 Insert these values in the given polynomial to find y(1.37).y(1.37)=2.5713.1+10.90.35=0.02 Whereas the exact result is, y(1.37)=2.57135313.1383+10.960.35=0.043053 Hence, the percent relative error is, ξ=0.0430530.020.043053×100%=53.54% (b) y=((x7)x+8)x0.35 Now, when x = 1.37,
(x - 7)x = (1.37 - 7)(1.37)
=-7.7131 Chopping to 3 digit arithmetic yields, (x7)x=7.71((x7)x+8)x=(7.71+8)1.37=0.047 Here, the percent relative error is, ξ=0.0430530.0470.043053 ×100%=9.16% The percent relative error is smaller when compared to part a.

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Jeffrey Jordon
Answered 2021-10-13 Author has 2313 answers

First, we can calculate the true value of the polynomial

p(x):=x37x2+8x0.35

at point x=1.37

yexast:=p(1.37)=1.37371.372+81.370.35=0.043053

Using 3-digits with chopping we have to calculate separately 1.372,71.372 and 81.37

1.373=2.571353chopping2.5771.372=13.1383chopping13.0

81.37=10.96chopping10.9

p(1.37)2.5713.0+10.90.35=0.12

The relative percent error is equal to

|0.0430530.120.043053|=1.787=178.7

b)

Polynomial p can be expressed in a different way:

p(x)=((x7)x+8)x0.35

Now we calculate

(1.377)1.37=7.7131chopping7.71

(7.71+8)1.37=0.3973chopping0.397

0.3970.35=0.047

The relative percent error is equal to

|0.0430530.0470.043053|=0.092=9.2

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

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Jeffrey Jordon
Answered 2022-01-23 Author has 2313 answers

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