Overview

Here is a summary of some of the most important ideas in this book.

• Floating point representation of numbers is ubiquitous in numerical computing, since fixed point numbers have very restricted ranges. Floating point uses exponential notation, storing a sign, an exponent, and a significand in each floating point word.

• The range of possible values for IEEE single format floating point numbers is from tiny (approximately 10 ^?38) to huge (approximately 10 ³⁸). In addition, there are the corresponding range of negative numbers, the subnormal numbers, and the special numbers 0 and ?. NaN is used for the result of invalid operations. Double format numbers have a much greater finite range.

• Floating point numbers are inherently accurate only to a certain number of bits or digits. In the case of the IEEE single format, numbers have 24-bit significands, corresponding to approximately 7 significant decimal digits, and in the case of the double format, 53-bit significands, corresponding to approximately 16 significant decimal digits. Theorem 5.1 says that, when p-bit significands are in use, the rounded value of a number x satisfies

  
  

  with ? < 2 ^{? p} when the rounding mode is round to nearest. The quantity ? is called the relative rounding error and its size depends only on the precision p of the floating point system and not on the size of x. The absolute rounding error x ? round( x) does depend...