23 Error Source A Finite Precision Arithmetic Consider a fak

2.3 Error Source A: Finite Precision Arithmetic Consider a (fake) 4-significant-figure computer that stores numbers in the form d.ddd x 10 (d-5) 5 pts where d represents any digit between 0-9. Assume also that the exponent reserves d J 9 to mean INFINITY, and d 0 to mean denormal numbers (where the first d in the mantissa is forced to be zero, e. the number now has less than 4 significant figures) a) What is the value of REALMAX, the largest possible positive number? b) What is the value of REALMIN, the smallest possible positive number that is not denormal (i.e. still maintains four significant figures)? c) What is the value of machine precision, e, defined here as the difference between 1 and the next largest valid storable number (slightly different than the \"official\" definition in class)? b aC Consider the quadratic equation x 0.1 which can be solved using x 2a to get the exact roots x 54.318158995 and x 0.00 184100049576 d) Use the 4-sig fig computer to solve the quadratic equation for the roots x1 and x2. Be sure, after every calculation between two numbers, to round the result back into a valid storable number of the form above before moving on to the next calculations. Show your work! That is, show the values of a interim calculations: b2, 4ac, b2-4ac, b2 4ac 4ac, 2a, and x 1,2 e) What is the (absolute or true) error, and the percent relative error, for both roots calculated by the computer? Comment on how well the computer did solving for the roots!

Solution

The smallest number is 0.000*10(-5) = 0;

Denormalized numer 1.111*10(-4) ;

The greatest number is INFINITY ;

Denormalized number 9.999*10(4);