Chapter 1: Round-off Errors and Computer Arithmetic PDF

The error that is produced when a calculator or computer is used to perform real number calculations is called round-off error. It occurs because the arithmetic performed in a machine involves numbers with only a finite number of digits, with the result that calculations are performed with only appr...

The error that is produced when a calculator or computer is used to perform real number calculations is called round-off error. It occurs because the arithmetic performed in a machine involves numbers with only a finite number of digits, with the result that calculations are performed with only approximate representations of the actual numbers. Any positive real number within the numerical range of the machine can be normalized to the form ,1≤ d1 ≤ 9, and 0 ≤ di ≤ 9, The error that results from The floating-point form of y, denoted fl(y) fl(y) Chopping Rounding Example 1 The following definition describes two methods for measuring approximation errors. Definition 1.15 Determine the five-digit (a) chopping and (b) rounding values of x=0.00325996 How many significant digits that fl(x) to x Finite-Digit Arithmetic Chopping 5-digits Rounding 3-digits  Stability of an algorithm Stable Conditionally Stable Small changes in the initial unstable Stable only for certain data produce small Small changes in the initial choices of initial data changes in the final result data produce large changes in the final result  1−cosℎ 𝑓(ℎ)= ℎ