# Scientific Computing I-APPROXIMATIONS AND ROUND-OFF ERRORS

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

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