1. Consider the real number r := e 106 2718281.828459046 . . . Give an upper bound on the absolute relative error and on the absolute error of the floating-point representation fl(r) of this real number r represented in a computer using IEEE double precision arithmetic, i.e., give upper bounds on

(e 10^6 ) fl(e 10^6 ) /(e 10^6) ?

and (e 10^6 ) fl(e 10^6 ) ?

Hint: do not try to find fl(e 106 ) the exact floating-point representation of e 106 in the computer!

2. (a) What is the relative condition number of the function cos(x) at x = 1.5707964?

(b) Compute cos(1.5707964) on a computer using IEEE double precision arithmetic. Assuming that cos(f) is computed exactly for floating-point numbers f F, how many correct significant decimal digits can you at least expect in the numerical result cos(1.5707964) obtained?

3. Show that the two expressions x + 1 x and 1/( x + 1 + x) are mathematically equal. Evaluate these two expressions in MATLAB for x = 1012. Do you obtain the same numerical results? Which result is more accurate and why? The first 32 correct significant decimal digits of the exact result are 4.9999999999999875000000000000625 107 .

4. Compute in MATLAB for x = /4 and h = 2n (n = 1, 2, . . . , 40) the approximation of the derivative of cos(x) at x = /4 given by

cos(x + h) cos(x) /h

and the corresponding absolute error

sin(x) (cos(x + h) cos(x) /h ) .

For which two integer values n = n1, n2 of hn = 2n do you obtain the two minimal absolute errors? Compare the average n := (n1 + n2)/2 of these two values to 53 (which is the number of bits in the mantissa in IEEE double precision arithmetic) by calculating 53/n.

5. Give at least 14 correct significant digits of the two roots of

x ^2 + 3800x + 0.0000002018 = 0.