BACHMANN-LANDAU NOTATIONS. Lecturer: Dr. Jomar F. Rabajante IMSP, UPLB MATH 174: Numerical Analysis I 1 st Sem AY
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1 BACHMANN-LANDAU NOTATIONS Lecturer: Dr. Jomar F. Rabajate IMSP, UPLB MATH 174: Numerical Aalysis I 1 st Sem AY
2 RANKING OF FUNCTIONS Name Big-Oh Eamples Costat O(1 10 Logarithmic O(log log, log( Liear O(, 510 Liearithmic O( log log, log! Quadratic O( 7 9
3 RANKING OF FUNCTIONS Name Big-Oh Eamples Cubic O( Polyomial O(log poly(, log, 10
4 RANKING OF FUNCTIONS Name Big-Oh Eamples Epoetial O( 1.1, 10 Epoetial poly(, ^ Factorial O(!!3
5 ORDER OF APPROXIMATION Now, suppose 0 INFINITESIMAL ASYMPTOTICS
6 Or cosider 1 : m m RECIPROCAL OF INFINITE ASYMPTOTICS
7 Eample: f ( : 0 æ 1 ö f ç : m è m ø
8 Ifiitesimal Asymptotics 0 3? 3? ( 3 ( 3?
9 Ifiitesimal Asymptotics 0 3 O( 3 O( ( 3 ( 3 O(
10 Ifiitesimal Asymptotics 0 3? 3? ( 3 ( 3?
11 Ifiitesimal Asymptotics 0 3 O( 3 O( ( 3 ( 3 O( 3
12 Ifiitesimal Asymptotics 0 ( 3 1( ( ( 3 ( 1 ( ( 3 ( O k O O k O k is a costat
13 ERRORS
14 Types of Errors Program Error Machie Error Propagated Error Math Trucatio Error Noise ad Outliers etc
15 ERRORS 1. BUGS or PROGRAM ERROR Compile-time errors Ru-time/Eecutio-time errors Logic errors
16 Three types of program error: ØCompile-time errors. These are errors that are detected by the compiler whe it is attemptig to traslate from the source laguage to the target laguage. Eamples are syta errors such as failig to match up paretheses, mistypig the word PRINT as PINT, ad so o.
17 Three types of program error: To miimize this error, compile at frequet itervals, rather tha typig i a vast program ad oly compilig oce the whole thig has bee typed i.
18 Three types of program error: ØRu-time/Eecutio-time errors. Ru-time errors arise durig eecutio of the program. They caot be detected at compile-time. Whe detected durig eecutio, they cause the program to crash. They usually show up as a umber that is too large (overflow or as a illegal umber (such as requestig the iverse log of a egative umber. Sice these errors caot be detected i advace by the compiler, the covetioal way of discoverig them is through program testig.
19 Three types of program error: ØLogic errors. Logic errors are errors which do ot result i error messages from the compiler or the ru-time eviromet. We are ot otified of them at all. They are coceptual errors: our algorithm or program does ot match the problem specificatio. The actual results are ot the same as the epected results.
20 ERRORS. MACHINE ERRORS Roud-off Error Trucatio/choppig Error
21 MACHINE ERROR ISSUES v Iteger mode ad Floatig poit mode v Loss of sigificace
22 v Floatig poit represetatio What ca you say about these operatios? Ø 5 7 Ø 4 * 3 1 Ø (sqrt (3^?
23 v Floatig poit represetatio What ca you say about these operatios? Ø 5 7 sqrt ( Ø 4 * 3 1 ( ^ Ø (sqrt (3^?
24 v Floatig poit represetatio Most are i biary floatig poit mode, but we will discuss the decimal versio Related to scietific otatio i real umbers (a computer has a way of approimatig irratioals ad the like, ifiity is ot uderstood by computers Sometimes we ecouter uderflow (too small ad overflow (too large
25 v Floatig poit represetatio Let us cosider the ormalized decimal floatig poit mode: ± 0. d 1 d... d k 10 where1 d 1 9 ad 0 d i 9, i,3,..., k
26 v Floatig poit represetatio Covert y to decimal floatig poit mode y what if k5? What is fl(y? Øcomputers use roudig-off
27 v Floatig poit represetatio Commo levels of precisio: sigle, double ad eteded precisio We eed to check for the machie epsilo (eps, which is the maimum relative error of the chose roudig procedure We may also check the Istitute of Electrical ad Electroics Egieers (IEEE Stadards
28 CONVERT TO DECIMAL BINARY: DECIMAL: BINARY: DECIMAL:
29 Eercise: CONVERT 5 10 to biary to decimal to biary to decimal to decimal to biary
30 Sigificat Digits The umber is said to approimate to m sigificat digits if m is the largest oegative iteger for which - ˆ < m E < -4 < 5 10 ^
31 v Loss of sigificace/ subtractive cacellatio Cosider p ad q We kow that: p q 0 1 How may sigificat digits? What is m? If k6, what is 1/(p q?
32 ERRORS 3. PROPAGATED or ITERATION ERROR Stable (error à0 or costat Ustable (error blows-up Eample: y!! y y!! e ( y!! e ye ( e y y e e y
33 PROPAGATED ERROR e be iitial error e ( be growth of the error after steps e (» e, liear growth of error e (» K e, epoetial growth of error K > 1, e ( as 0 < K < 1, e ( 0 as
34 PROPAGATED ERROR Cosider! p p e What is the growth of error whe p ( p! e?
35 PROPAGATED ERROR p ( p! e Eact Appro Error UNSTABLE!
36 PROPAGATED ERROR Iter Error e b - a 1 Error of Bisectio Method STABLE!
37 Aother eample of INSTABILITY Cosider f ( e 0 f f ( 3» ( 3.01» f ( 3 - f (3.01»
38 For Polyomials we ca use Nested Multiplicatio or Horer s Method (((( ( a a a a a a a a a a a a P *Horer s Method for Polyomial Evaluatio (related to sythetic divisio
39
40 RECALL: ABSOLUTE & RELATIVE ERROR AE TV - AV TV - AV AV RE 1- TV TV If Error10, the AV approimates TV to decimal places/ sigificat digits
41 RECALL: ABSOLUTE & RELATIVE ERROR AE TV - AV TV - AV AV RE 1- TV TV Use relative chage whe there s varyig magitudes but TV 0.
42 Well-posed ad Ill-posed A ill-posed problem is idicated by a large coditio umber. If the problem is well-posed, the it maybe solved usig a stable algorithm. If it is illposed, re-formulatio might help.
43 Coditio Number (for fuctios The word coditio is used to describe the sesitivity of the fuctio value f( to chages i the argumet. (Perturbatio Iformal Formula: C ma é ê ê ê ê ë f ( - f ( f ( * - * ù ú ú ú ú û» f '( f ( where - * issufficietly small
44 Coditio Number (for fuctios Whe C>1, the the relative chage i the f value grows faster tha the relative chage i the value. As C icreases, the fuctio becomes more ustable. Whe C decreases ad C<1, the fuctio is stabilizig.
45 It follows that the chage i f( is Cosider * ( '( * ( ( f f f -» ( ad (, ( h e g f - Coditio Number (for fuctios
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