Chapter 1 PDF

92112 error Lily ask.it sisnitiantdis gffstem 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 perfo...

92112 error Lily ask.it sisnitiantdis gffstem 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. sinx i was so 96 Sis sink was If f Ose ya writtenasrational numbercan'tbe 5 3 201 I getp 6g 0 3333 g I 3 1 3 3313 I 3 9 1 99  Any positive real number within the numerical range of SII the machine can be normalized to the form gg.i.mn 0 EE ,1≤ d1 ≤ 9, and 0 ≤ di ≤ 9, The error that results from The floating-point form of y, 0 denoted fl(y) a'T fl(y) Chopping Rounding 5 S JÑia IT 2 Example 1 516 to 7 0 t tormalizatie o 0.31415 10 notifieda 0 314159 26 5 410 0 3 14 16 10 u i 0 0832  I b b The following definition describes two methods for measuring approximation errors. Definition 1.15 700 I absoluteffe relative euror error 38 1 10.0003 703 0.00031 3000 3100 a b c 10 0 I 881 Absolute relative 0.333 0.333 It 3537 8 0 so  3 SET 5 10 0 loss jigger Determine the five-digit (a) chopping and (b) rounding values of x=0.00325996 How many significant digits that fl(x) to x F1 x 0.325996 t d i f Pdf by rounding 0 326 152 then t f Finite-Digit Arithmetic Chopping 5-digits Rounding 3-digits  Stability of an algorithm IF.EE 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 0 for ex 0  b b b Bn Bn jj Ian x ̅ K Bn 2 2βn nF n  4Bn nF u  1−cosℎ 𝑓(ℎ)= ℎ  1−cosℎ 𝑓(ℎ)= ℎ