Numerical Methods (CENG 2002) CHAPTER -III LINEAR ALGEBRAIC EQUATIONS. In this chapter, we will deal with the case of determining the values of x 1

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1 Numericl Methods (CENG 00) CHAPTER -III LINEAR ALGEBRAIC EQUATIONS. Itroductio I this chpter, we will del with the cse of determiig the vlues of,,..., tht simulteously stisfy the set of equtios: f f... f (,,..., ) (,,..., ) 0 0 (,,..., ) 0 I prticulr we will cosider lier lgebric equtios which re of the form: b b b (.) (.) Where the 's re costt coefficiets, the b 's re costts, d is the umber of equtios. The bove system of lier equtios my lso be writte i mtri form s: [ ]{ X } { B} A (.) Where [ A ] is by squre mtri of coefficiets (clled the coefficiet mtri), [ A] { B } is by colum vector of costts, T { B } b b... b Acdemic yer 007/8 Istructor: Workmw Pulos

2 Numericl Methods (CENG 00) d { X } is by colum vector of ukows: { X }.... No - Computer methods There re some o-computer methods which re used to solve smll ( ) sets of simulteous equtios tht do ot require computer... The Grphicl Method Plottig the grphs (stright lies) d fidig the poit of itersectio of the grphs... Crmer's Rule Crmer's rule sttes tht the solutio of set of lier equtios give i Eq.. c be give s: Di i (.4) D where D is the determit of the coefficiet mtri [ A ], d D i is the determit of the mtri obtied by replcig the coefficiets of the ukow i i the coefficiet mtri by the costts b, b,..., b. For emple, c be obtied s: b b b D For more th three equtios, Crmer's rule becomes imprcticl becuse, s the umber of equtios icrese, the determits re time-cosumig to evlute by hd (or by computer). Cosequetly, more efficiet ltertives re used. Acdemic yer 007/8 Istructor: Workmw Pulos

3 Numericl Methods (CENG 00).. Elimitio of Ukows The bsic strtegy is to multiply the equtios by costts so tht oe of the ukows will be elimited whe the equtios re combied. The result is sigle equtio tht c be solved for the remiig ukow. This c the be substituted ito either of the origil equtios to compute the other vrible. The elimitio of ukows c be eteded to systems with more th two or three equtios. However, the umerous clcultios tht re required for lrger systems mke the method etremely tedious to implemet by hd. However, the techique c be formlized d redily progrmmed for the computer.. Guss Elimitio.. Descriptio of the method The pproch is desiged to solve geerl set of equtios: b + (.5) b + (.5b) b (.5c) The ive Guss elimitio method cosists of two phses:. Forwrd Elimitio: The first step is desiged to reduce the set of equtios to upper trigulr system. The iitil step will be to elimite the first ukow, from the secod through the to give: th equtios. To do this, multiply Eq. (.5) by Acdemic yer 007/8 Istructor: Workmw Pulos

4 Numericl Methods (CENG 00) + (.6) b Now this equtio c be subtrcted from Eq..5b to give:... b b + + or b Where the prime idictes tht the elemets hve bee chged from their origil vlues. The procedure is the repeted for the remiig equtios. For istce, Eq. (.5) c be multiplied by d the result subtrcted from the third equtio. Repetig the procedure for the remiig equtios results i the followig modified system: b + (.7) (.7b) b (.7c) b b (.7d) For the foregoig steps, Eq.(.5) is clled the pivot equtio, d is the pivot coefficiet or elemet. Now repet the bove to elimite the secod ukow from Eq. (.7c) through Eq. (.7d). To this multiply Eq. (.7b) by, d subtrct the result from Eq. (.7c). Perform similr elimitio for the remiig equtios to yield Acdemic yer 007/8 Istructor: Workmw Pulos 4

5 Numericl Methods (CENG 00) b b b b Where the double prime idictes tht the elemets hve bee modified twice. The procedure c be cotiued usig the remiig pivot equtios. The fil mipultio i the sequece is to use the ( )th equtio to elimite the term from the th equtio. At this poit, the system will hve bee trsformed to upper trigulr system: (.8) b (.8b) b (.8c) b... ( ) ( ) (.8d) b. Bck Substitutio: Eq. (.8d) c ow be solved for : ( ) b (.9) ( ) This result c be bck substituted ito the ( )th equtio to solve for. The procedure, which is repeted to evlute the remiig 's, c be represeted by the followig formul: Acdemic yer 007/8 Istructor: Workmw Pulos 5

6 Numericl Methods (CENG 00) b ( i ) ( i ) i ij j i+ i ( i ) ii j for i,,..., (.0).. Pitflls of Guss Elimitio Wheres there re my systems of equtios tht c be solved with ive Guss elimitio, there re some pitflls tht must be eplored before writig geerl computer progrm to implemet the method. i, Divisio by Zero The primry reso tht the foregoig techique is clled "ive" is tht durig both elimitio d bck-substitutio phses, it is possible tht divisio by zero c occur. Problems lso rise whe the coefficiet is very close to zero. The techique of pivotig (to be discussed lter) hs bee developed to prtilly void these problems. ii, Roud-off Errors The problem of roud-off errors c become prticulrly importt whe lrge umbers of equtios re to be solved. A rough rule of thumb is tht roud-off errors my be importt whe delig with 00 or more equtios. I y evet, oe should lwys substitute the swers bck ito the origil equtios to check whether substtil error hs occurred. iii, Ill coditioed Systems Ill-coditioed systems re those where smll chge i coefficiets i lrge chges i the solutio. A ltertive iterprettio of ill-coditioig is tht wide rge of swers c pproimtely stisfy the equtios. A ill-coditioed system is oe with determit of the coefficiet mtri close to zero. It is difficult to specify how close to zero the determit must be to idicte illcoditioig. This is complicted by the fct tht the determit c be chged by Acdemic yer 007/8 Istructor: Workmw Pulos 6

7 Numericl Methods (CENG 00) multiplyig oe or more of the equtios by scle fctor without chgig the solutio. Oe wy to void this difficulty is to scle the equtios so tht the mimum elemet i y row is equl to (This process is clled sclig). iv, Sigulr Systems The system is sigulr whe t lest two of the equtios re ideticl. I such cses, we would lose oe degree of freedom, d would be delig with impossible cse of equtios i ukows. Such cses might ot be obvious prticulrly whe delig with lrge equtio sets. Cosequetly, it would be ice to hve some wy of utomticlly detectig sigulrity. The swer to this problem is etly offered by the fct tht the determit of sigulr system is zero. This ide c, i tur, be coected to Guss elimitio by recogizig tht fter the elimitio step, the determit c be evluted s the product of the digol elemets. Thus, computer lgorithm c test to discer whether zero digol elemet is creted durig the elimitio stge... Techiques for Improvig Solutios. Use of more sigifict figures.. Pivotig: c be prtil or complete. Prtil Pivotig: Determie the lrgest vilble coefficiet i the colum below the pivot elemet. The rows re the switched so tht the lrgest elemet is the pivot elemet. Complete Pivotig: Whe colums s well s rows re switched.. Sclig: Sclig is the process by which the mimum elemet i row is mde to be by dividig the equtio by the lrgest coefficiet. Guss-Jord Elimitio Guss-Jord is vritio of the Guss elimitio. The mjor differece is tht whe ukow is elimited i the Guss-Jord method, it is elimited from ll other equtios rther th just the subsequet oes. I dditio, ll rows re ormlized by Acdemic yer 007/8 Istructor: Workmw Pulos 7

8 Numericl Methods (CENG 00) dividig them by their pivot elemets. Thus, the elimitio step results i idetity mtri rther th trigulr mtri. Thus, bck-substitutio is ot ecessry. The method is ttributed to Joh Crl Friedrich Guss ( ) d Wilhelm Jord (84 to 899). to Emple Use the Guss-Jord elimitio method solve the lier system First form the ugmeted mtri M [A, B] The perform Guss-Jord elimitio. Hece, the solutio is LU-Decompositio Guss elimitio is soud wy to solve systems of lgebric equtios of the form Acdemic yer 007/8 Istructor: Workmw Pulos 8

9 Numericl Methods (CENG 00) [ ]{ X } { B} A (.) However, it becomes iefficiet whe solvig equtios with the sme coefficiets[ A ], but with differet right-hd side costts. LU decompositio methods seprte the time-cosumig elimitio of the mtri [ A ] from the mipultios of the right-hd side{ B }. Thus, oce [ A ] hs bee "decomposed", multiple right-hd side vectors c be evluted i efficiet mer..5. Derivtio of LU Decompositio Method Eq. (.) c be rerrged to give: [ ]{ X } { B} 0 A (.) Suppose tht Eq. (.) could be epressed s upper trigulr system: u 0 0 u u 0 u u u d d d (.) Eq. (.) c lso be epressed i mtri ottio d rerrged to give: [ ]{ X } { D} 0 U (.4) Now, ssume tht there is lower digol mtri with 's o the digol, 0 0 L l 0 (.5) l l [ ] tht hs the property tht whe Eq. (.4) is pre multiplied by it, Eq. (.) is the result. Tht is, Acdemic yer 007/8 Istructor: Workmw Pulos 9

10 Numericl Methods (CENG 00) [ L] {[ U ]{ X } { D } [ A]{ X } [ B] (.6) If this equtio holds, it follows tht [ ][ U ] [ A] L (.7) d [ ]{ D} { B} L (.8) A two-step strtegy (see Fig..) for obtiig solutios c be bsed o Eqs. (.4), (.7) d (.8):. LU decompositio step. [ A ] is fctored or decomposed ito the lower [ L ] d upper [ U ] trigulr mtrices.. Substitutio step. [ L ] d [ U ] re used to determie solutio { X } for right-hd side { B }. This step itself cosists of two steps. First, Eq. (.8) is used to geerte itermedite vector by forwrd substitutio. The, the result is substituted ito Eq. (.4) which c be solved by bck substitutio for{ X }. Acdemic yer 007/8 Istructor: Workmw Pulos 0

11 Numericl Methods (CENG 00) () Decompositio [ U ] [ L ] [ A ] { X } { B} [ L ] { D} { B} [ D ] (b) Forwrd [ U ] { X } { D} Substitutio [ X ] (c) Bckwrd Fig.. Steps i LU Decompositio Emple Give Fid mtrices L d U so tht LU A. Acdemic yer 007/8 Istructor: Workmw Pulos

12 Numericl Methods (CENG 00) Hece, Guss-Seidel Method Itertio is populr techique fidig roots of equtios. Geerliztio of fied poit itertio c be pplied to systems of lier equtios to produce ccurte results. The Guss-Seidel method is the most commo itertive method d is ttributed to Philipp Ludwig vo Seidel (8-896). Cosider tht the squre mtri A is split ito three prts, the mi digol D, below digol L d bove digol U. We hve A D - L - U. A Acdemic yer 007/8 Istructor: Workmw Pulos

13 Numericl Methods (CENG 00) D U L The solutio to the lier system AXB c be obtied strtig with P 0, d usig itertio scheme Acdemic yer 007/8 Istructor: Workmw Pulos

14 Numericl Methods (CENG 00) where d. A sufficiet coditio for the method to be pplicble is tht A is strictly digolly domit. Emple Use Guss-Seidel itertio to solve the lier system Try 0 itertios.. The system c be epressed s Usig 0 itertios we hve: Acdemic yer 007/8 Istructor: Workmw Pulos 4

15 Numericl Methods (CENG 00) Hece, For the purpose of hd clcultio let s see set of lier equtios cotiig ukows b b b If the digol elemets re ll ozero, the first equtio c be solved for, the secod for d the third for to yield: b () b (b) b (c) Steps to be followed I. Usig the iitil guess 0.0 solve for from () II. Usig the vlues of from step i d 0.0 solve for from (b) III. Usig the vlue of from step i d tht of from step ii solve for from (c) IV. Usig the vlue of from step ii d tht of from step iii solve for from () V. Usig the vlue of from step iv d tht of from step iii solve for from (b) Acdemic yer 007/8 Istructor: Workmw Pulos 5

16 Numericl Methods (CENG 00) VI. Usig the vlue of from step iv d tht of from step v solve for from(c) VII. Repet the process util the required ccurcy is chieved. Emple Use the Guss-Seidel method to obti the solutio of the followig system lier equtios Solvig for: from eq from eq from eq + 4 Eecutig the bove steps repetitively we will hve the followig result Acdemic yer 007/8 Istructor: Workmw Pulos 6

17 Numericl Methods (CENG 00) As we c see the vlues strt to repet fter the 8 th itertio hece we c stop the clcultio d tke the fil vlues s the solutio of the lier system of equtios. Hece, Acdemic yer 007/8 Istructor: Workmw Pulos 7

Autar Kaw Benjamin Rigsby. Transforming Numerical Methods Education for STEM Undergraduates

Autar Kaw Benjamin Rigsby. Transforming Numerical Methods Education for STEM Undergraduates Autr Kw Bejmi Rigsby http://m.mthforcollege.com Trsformig Numericl Methods Eductio for STEM Udergrdutes http://m.mthforcollege.com . solve set of simulteous lier equtios usig Nïve Guss elimitio,. ler the