It was demonstrated in class that the number bits associated with the mantissa i

Question

It was demonstrated in class that the number bits associated with the mantissa in Excel floating point numbers is 50. The text gives an equation relating the number of mantissa bits t to machine epsilon as = 2^. Develop a similar spreadsheet to successively add 1/2^n to e(e^x, x= 1) to estimate the smallest number which can be added to e and have the result distinguishable from e. Calculate the relative error in describing e and compare it to. Explain why the smaller of the two relative errors should be so. Problem 2 (Like Class Text 3.6(. Use a spreadsheet to evaluate e^-3x using 25 terms of the inverse of the series for e^x e^-x=1/e^x= and 25 terms of the series for e^-x= Compare the results lo your spreadsheet's result and explain what makes one more accurate than the other.

Explanation / Answer

We observe that the value of e + 2-n (column 3) remains constant for all n > 35 whereas the value of epsilonm (column 2) keeps decreasing even beyond 2-40.

44.70118449

0.022370772

The last two columns show the value of e3.8 and e-3.8.

We take inverse of e3.8.

(e3.8)-1 = 0.022370772

Incidentally, using Excel the value of 1/3.8 and the value of -3.8 using power series expansion coincide.

n 2-n e + 2-n 0 1 3.718281828 -1 0.5 3.218281828 -2 0.25 2.968281828 -3 0.125 2.843281828 -4 0.0625 2.780781828 -5 0.03125 2.749531828 -6 0.015625 2.733906828 -7 0.0078125 2.726094328 -8 0.00390625 2.722188078 -9 0.00195313 2.720234953 -10 0.00097656 2.719258391 -11 0.00048828 2.71877011 -12 0.00024414 2.718525969 -13 0.00012207 2.718403899 -14 6.1035E-05 2.718342864 -15 3.0518E-05 2.718312346 -16 1.5259E-05 2.718297087 -17 7.6294E-06 2.718289458 -18 3.8147E-06 2.718285643 -19 1.9073E-06 2.718283736 -20 9.5367E-07 2.718282782 -21 4.7684E-07 2.718282305 -22 2.3842E-07 2.718282067 -23 1.1921E-07 2.718281948 -24 5.9605E-08 2.718281888 -25 2.9802E-08 2.718281858 -26 1.4901E-08 2.718281843 -27 7.4506E-09 2.718281836 -28 3.7253E-09 2.718281832 -29 1.8626E-09 2.71828183 -30 9.3132E-10 2.718281829 -31 4.6566E-10 2.718281829 -32 2.3283E-10 2.718281829 -33 1.1642E-10 2.718281829 -34 5.8208E-11 2.718281829 -35 2.9104E-11 2.718281828 -36 1.4552E-11 2.718281828 -37 7.276E-12 2.718281828 -38 3.638E-12 2.718281828 -39 1.819E-12 2.718281828 -40 9.0949E-13 2.718281828

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