Razlike između izmjena na stranici "Numerička analiza"

Obrisano 65 bajtova ,  prije 6 godina
m
|-
|
'''DirectDirektni vs iterativeiterativni methodsmetodi'''
 
Consider theRazmotrimo problem of solvingrešavanja
 
:3''x''<sup>3</sup> + 4 = 28
 
forza thenepoznatu unknown quantitypromenljivu ''x''.
 
{| style="margin:auto;"
|+ DirectDirektni methodmetod
|-
| || 3''x''<sup>3</sup> + 4 = 28.
|-
| ''SubtractOduzmi 4'' || 3''x''<sup>3</sup> = 24.
|-
| ''DividePodeli bysa 3'' || ''x''<sup>3</sup> = 8.
|-
| ''TakeUradi cubekubni rootskoren'' || ''x'' = 2.
|}
 
ForZa theiterativni iterative methodmetod, apply theprimeni [[bisectionMetoda methodpolovljenja intervala |bisekcioni metod]] tona ''f''(''x'') = 3''x''<sup>3</sup> &minus; 24. The initialInicijalne valuesvrednosti aresu ''a'' = 0, ''b'' = 3, ''f''(''a'') = &minus;24, ''f''(''b'') = 57.
 
{| style="margin:auto;"
|+ IterativeIterativni methodmetod
|-
! ''a'' !! ''b'' !! mid !! ''f''(mid)
|}
 
WeIz concludeove fromtabele thiszaključujemo tableda thatje therešenje solution is betweenizmeđu 1.,875 andi 2.,0625. TheAlgoritam algorithmmože mightda returnvrati anybilo numberkoji inbroj thatu rangetom withopsegu ansa errorgreškom lessmanjom thanod 0.,2.
 
==== Diskretizacija i numerička integracija ====
 
[[Image:Schumacher (Ferrari) in practice at USGP 2005.jpg|right|125px]]
Tokom dvočasovne trke smo izmerili brzinu kola tri puta, i zapisali smo merenja u sledećoj tabeli.
In a two hour race, we have measured the speed of the car at three instants and recorded them in the following table.
 
{| style="margin:auto;"
! TimeVreme
| 0:20 || 1:00 || 1:40
|-
|}
 
A '''discretizationDiskretizacija''' would bebi tobila sayda thatse thekaže speedda ofje thebrzina carkola wasbila constantkonstantna fromod 0:00 todo 0:40, thenzatim fromod 0:40 todo 1:20 andi finallykonačno fromod 1:20 todo 2:00. ForNa instanceprimer, thetotalno totalrastojanje distancepređeno traveledu in the firstprvih 40 minutesminuta isje approximatelyaproksimativno (2/3h&nbsp;&timesputa;&nbsp;140&nbsp;km/h)&nbsp;=&nbsp;93.3&nbsp;km. ThisTo wouldbi allownam usomogučilo toda estimateprocenimo thetotalno totalpređeno distancerastojanje traveled askao 93.3&nbsp;km + 100&nbsp;km + 120&nbsp;km = 313.3&nbsp;km, whichšto isje an example ofprimer '''numericalnumeričke integrationintegracije''' (seevidite belowispod) using akoristeći [[RiemannRimanova sumsuma|Rimanovu sumu]], becausepošto displacementje is therastojanje [[integral]] of velocitybrzine.
 
'''Ill-conditionedNestabilan problem''': TakeUzmimo the functionfunkciju ''f''(''x'') = 1/(''x''&nbsp;&minus;&nbsp;1). NoteUočimo thatda ''f''(1.,1) = 10 andi ''f''(1.,001) = 1000: a change inpromena ''x'' ofod lessmanje thanod 0.,1 turns into a changeodgovara inpromeni ''f''(''x'') ofod nearlyskoro 1000. EvaluatingEvaluacija ''f''(''x'') nearu blizini ''x'' = 1 is anje ill-conditionednestabilni problem.
 
'''Well-conditionedStabilan problem''': ByU contrast,kontrastu s evaluatingtim theevaluacija sameiste functionfunkcije ''f''(''x'') = 1/(''x''&nbsp;&minus;&nbsp;1) nearu blizini ''x'' = 10 isje a well-conditionedstabilan problem. ForNa instanceprimer, ''f''(10) = 1/9 ≈ 0.,111 andi ''f''(11) = 0.,1: askromna modest change inpromena ''x'' leads to adovodi modestdo changeskromne inpromene ''f''(''x'').
|}
 
DirectDirektni methodsmetodi computeračunaju therešenja solutionproblema tou akonačnom problembroju in a finite number of stepskoraka. These methods would give the precise answer if they were performed in [[Computer numbering formats|infinite precision arithmetic]]. Examples include [[Gaussian elimination]], the [[QR algorithm|QR]] factorization method for solving [[system of linear equations|systems
of linear equations]], and the [[simplex method]] of [[linear programming]]. In practice, [[Floating point|finite precision]] is used and the result is an approximation of the true solution (assuming [[Numerically stable|stability]]).
 
3.736

izmjena