MAT360 Section Summary:

1.2: Roundoff Errors and Computer Arithmetic

1. Definitions
   • long real: 8 byte real (64 bits):
     • first bit for the sign (positive or negative);
     • 11 bits for the characteristic (exponent); and the remaining
     • 52 bits for the mantissa, which is the rational representation of the number in the interval from 0 to 1.
     ``To save storage and provide a unique representation for each floating-point number, a normalization is imposed'', so that the decimal representation of the binary number is

     

     (where f is the decimal expansion of the mantissa).

     11 bits for exponents gives distinct numbers that can be represented;

     52 bits for mantissa gives distinct numbers (that's pretty many....).

     The largest number that can be represented using this normalized scheme is about , and the smallest about . Calculations resulting in numbers larger than result in overflows, which usually mean ``expect junk'' (if not an impolite crash); numbers smaller than result in underflows, which generally cause no trouble (they're set to zero).

   • k-digit decimal machine numbers:

     

   • chopping to a k-digit decimal number: simply truncating an

     

   • rounding to a k-digit decimal number: add 5 in the k+1 place, then chop.

   • floating-point form: the form fl(y) of a number y that results from chopping or rounding.

   • roundoff error: the error that results from replacing a number with its floating-point form.

   • absolute error:

   • relative error:

     

   • is said to approximate p to t significant digits (or figures) if t is the largest non-negative integers for which

     

2. Properties/Tricks/Hints/Etc.

   Relative errors for floating-point form:

   • chopping:
   • rounding:

3. Summary

   Machine numbers are the approximations we may use for all real numbers. Each is generally stored as a binary number, including information about sign, exponent, and mantissa (with a fixed number of digits dedicated to distinguishing adjacent numbers).

   By replacing the infinite number of numbers within the interval of and by the finite number of machine numbers between those values, we're obviously making some errors. Those errors get compounded as we perform arithmetic operations. Two very dangerous operations are

   • The subtraction of nearly equal numbers, resulting in the cancellation significant digits;
   • Division by very small numbers (or multiplication by very large numbers).
   These two problems can be seen clearly in two standard mathematical computations:
   • The quadratic formula (e.g. example 5) and
   • Polynomial evaluation (e.g. example 6).