The Islamic University of Gaza Faculty of Engineering Civil Engineering Department Numerical Analysis ECIV 3306 Chapter 6 Open Methods

Transcription

1 The Islamc Unerst o Gaza Faclt o Engneerng Cl Engneerng Department Nmercal Analss ECIV 3306 Chapter 6 Open Methods

2 Open Methods Bracketng methods are based on assmng an nteral o the ncton whch brackets the root. The bracketng methods alwas conerge to the root. Open methods are based on ormlas that reqre onl a sngle startng ale o or two startng ales that do not necessarl bracket the root. These method sometmes derge rom the tre root.

3 Open Methods Conergence and Dergence Concepts Dergng ncrements Conergng ncrements

4 . Smple FedPont Iteraton Rearrange the ncton so that s on the let sde o the eqaton: 0 Þ g g Bracketng methods are conergent. Fedpont methods ma sometme derge, dependng on the statng pont ntal gess and how the ncton behaes.

5 Smple FedPont Iteraton Eamples:.. 3 è g 3/ 3. sn è g sn 3. e è g e g or g or g 0 >

6 Smple FedPont Iteraton Conergence g can be epressed as a par o eqatons: g. component eqatons Plot them separatel.

7 Smple FedPont Iteraton Conergence Fedpont teraton conerges : g < slope o the lne When the method conerges, the error s roghl proportonal to or less than the error o the preos step, thereore t s called lnearl conergent.

8 Smple FedPont IteratonConergence

9 Eample: Smple FedPont Iteraton. s manplated so that we get gè g e. Ths, the ormla predctng the new ale o s: 3. Gess o 0 e e e Root 4. The teratons contnes tll the appro. error reaches a certan lmtng ale g e

10 Eample: Smple FedPont Iteraton e e g e a %

11 Eample; Eample: Smple FedPont Iteraton Use Smple FedPont Iteraton method wth ntal o to nd the root o Contne the teratons ntl the appromate error alls below a stoppng crtera e s 0.5%

12 . The NewtonRaphson Method Most wdel sed method. Based on Talor seres epanson: 0 Rearrangng, 0 when the ale o s root The...! D D Sole or NewtonRaphson ormla

13 The NewtonRaphson Method A tangent to at the ntal pont s etended tll t meets the as at the mproed estmate o the root. The teratons contnes tll the appro. error reaches a certan lmtng ale. Slope / Root / 0 /

14 Eample: The Newton Raphson Method Use the NewtonRaphson method to nd the root o e 0 è e and ` e ; ths / e e Iter. e t % <0 8 e e

15 Ptalls o The Newton Raphson Method

16 3. The Secant Method The derate / s replaced b a backward nte dded derence Ths, the ormla predctng the s: /

17 The Secant Method Reqres two ntal estmates o, e.g, o,. Howeer, becase s not reqred to change sgns between estmates, t s not classed as a bracketng method. The scant method has the same propertes as Newton s method. Conergence s not garanteed or all o,,.

18 Secant Method: Eample Use the Secant method to nd the root o e 0; e and 0, 0 to get o the rst teraton sng: Iter e t %

19 Comparson o conergence o False Poston and Secant Methods Use the alseposton and secant method to nd the root o ln. Start comptaton wth l 0.5, 5.. False poston method Iter l r Secant method Iter

20 False Poston and Secant Methods Althogh the secant method ma be dergent, when t conerges t sall does so at a qcker rate than the alse poston method l See the net gre

21 Comparson o the tre percent relate Errors E t or the methods to the determne the root o e

22 Mltple Roots Doble roots 3 trple roots 3

23 Mltple Roots Mltple root corresponds to a pont where a ncton s tangent to the as. Dcltes Fncton does not change sgn wth doble or een nmber o mltple root, thereore, cannot se bracketng methods. Both and 0, dson b zero wth Newton s and Secant methods whch ma derge arond ths root.

24 4. The Moded Newton Raphson Method Another s ntrodced sch that / / ; Gettng the roots o sng Newton Raphson technqe: [ ] ] [ // / / / // / / / / Ths ncton has roots at all the same locatons as the orgnal ncton

25 Moded Newton Raphson Method: Eample Usng the Newton Raphson and Moded Newton Raphson ealate the mltple roots o wth an ntal gess o 0 0 [ ] // / / / Newton Raphson ormla: Moded Newton Raphson ormla:

26 Moded Newton Raphson Method: Eample Newton Raphson Moded NewtonRaphson Iter e t % ter e t % Newton Raphson technqe s lnearl conergng towards the tre ale o.0 whle the Moded Newton Raphson s qadratcall conergng. For smple roots, moded Newton Raphson s less ecent and reqres more comptatonal eort than the standard Newton Raphson method

27 Sstems o Nonlnear Eqatons Roots o a set o smltaneos eqatons:,,., n 0,,., n 0 n,,., n 0 The solton s a set o ales that smltaneosl get the eqatons to zero.

28 Sstems o Nonlnear Eqatons Eample: 0 & 3 57, 0 0, The solton wll be the ale o and whch makes,0 and,0 These are and 3 Nmercal methods sed are etenson o the open methods or solng sngle eqaton; Fed pont teraton and NewtonRaphson.

29 Sstems o Nonlnear Eqatons:.Fed Pont Iteraton. Use an ntal gess.5 and 3.5. The teraton ormlae: 0 / and Frst teraton, 0.5 / Second teraton: 0.49 / Solton s dergng so tr another teraton ormla

30 Sstems o Nonlnear Eqatons:.Fed Pont Iteraton. Usng teraton ormla: 0 and Frst gess:.5 and 3.5. st teraton: / / / nd teraton: / / / The approach s conergng to the tre root, and

31 Sstems o Nonlnear Eqatons:.Fed Pont Iteraton The scent condton or conergence or the twoeqaton case,0 and,0 are: < and <

32 Talor seres epanson o a ncton o more than one arable The root o the eqaton occrs at the ale o and where and eqal to zero. Sstems o Nonlnear Eqatons:. Newton Raphson Method

33 A set o two lnear eqatons wth two nknowns that can be soled or. Sstems o Nonlnear Eqatons:. Newton Raphson Method

34 Determnant o the Jacoban o the sstem. Sstems o Nonlnear Eqatons:. Newton Raphson Method

35 0 and 3 57 are two nonlnear smltaneos eqatons wth two nknown and the can be epressed n the orm:, o Sstems o Nonlnear Eqatons:. Newton Raphson Method 3, 6.5 o Use an ntal gess.5 and 3.5 o o o o

36 e a, e a, Sstems o Nonlnear Eqatons:. Newton Raphson Method