Scientific computing i-approximations and round-off errors

 
   

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1.Evaluate

e power−5

using two approaches

e power−x=1–x+xpower2/2−xpower3/3!+……

and

epower−x=1/e power x=1/(1+x+(xpower2/2)+(xpower3/3!))+……

and compare with the true value of

6.737947×10power−3.

Use 20 terms to evaluate each series and compute true and approximate relative errors as terms are added.

2.The derivative of f(x)=1/(1-3x power2) is given by

6x/(1=3x power 2)whole power2

Do you expect to have difficulties evaluating this function at x =0.577? Try it using 3- and 4-digit arithmetic with chopping.

3.(a)Evaluate the polynomial

y=xpower 3-5xpower2+6x+0.55

at x =1.37. Use 3-digit arithmetic with chopping. Evaluate thepercent relative error.

(b)Repeat(a) but express y as

y=((x-5)x+6)x=0.55

Evaluate the error and compare with part(a).

 

4.Use 5-digit arithmetic with chopping to determine the roots of the following equation with Eqs. (3.12) and (3.13)

xpower2-5000.002x+10

Compute percent relative errors for your results.

5.The “divide and average” method, an old-time method for approximating the square root of any positive number

a, can be formulated as 

x=(x+(a/x))/2

Write a well-structured function to implement this algorithm basedon the algorithm outlined in Fig. 3.3