Gauss Elimination, Gauss-Jordan Elimination and Gauss Seidel Iteration Method: Performance Comparison using MAT LAB

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1 Itertiol Jourl of Egieerig Tehology Siee d Reserh ISSN Volume 4, Issue 0 Otober 07 Guss Elimitio, Guss-Jord Elimitio d Guss Seidel Itertio Method: Performe Compriso usig MAT LAB *Dr. Udi Bh Trivedi, **Mr. Stosh Kumr Shrm, ***Mr. Vishok Kumr Sigh *PSIT ollege of Higher Edutio, Kpur ** PSIT ollege of Higher Edutio, Kpur ***IMS Uiso Uiversity, Dehrdu ABSTRACT: This Pper ompres the Eeutio time tke by vrious methods for solvig lier systems i.e. Guss Elimitio, Guss Jord Elimitio Methods d Guss Seidel itertive Methods. The pper uses MAT LAB R0 softwre to solve the problems of lier equtios with these methods. The Vrious Lier Equtios with two, three, four, five d si vrible hs bee solved with these methods usig omputer progrm developed i MAT LAB lguge. Totl Time tke by these lgorithms ivolved hve lulted d ompred. This pper lso epliitly revels tht i some situtios the Guss Seidel Itertio Method is ot overget. KEYWORDS: Guss Elimitio, Guss Jord Elimitio, Guss Seidel Itertio Method, MAT LAB Progrmmig, Lier system. I. RELATED RESERCH WORKS Adeeg d his bth mte from mthemtis deprtmet, Adeyemi College of Edutio,Odo Quoted i their reserh pper tht Guss Jord Method is more effiiet method th Gussi Elimitio methods beuse it voids the proess of bkwrd substitutio. It ws ws lso observed tht Guss Jord Method d Gussi Elimitio methods gives the sme swer for eh worked emple with pivotig d prtil pivotig proess. [3]. Fdugb Sudy Emmuel from Deprtmet of Mthemtil Siees, Ekiti Stte Uiversity, Ado Ekiti, Nigeri study o his pper o Some Itertive Methods for Solvig Systems of Lier Equtios shows tht the modified Jobi method is more effiiet i term of storge th the Jobi method. Thus, the modified Jobi method is proeed through less umbers of itertios d the rte of its overgee is fster th Jobi method. They lso Quotes tht System with digolly domit oeffiiet mtri will lwys overget. Digolly domit: [A] i [A] [X] = [C] is digolly domit if the oeffiiet o the digol must stisfied the oditio below for ll vlue of i d where * is the dimesio of ugmeted mtri [A] [6]. ii i i For ll i d ii i for t lest oe i i Dr.P.V.Uble, G.S.Si,Arts,Comm. College, Khmgo o his rtile Compriso of solutio of 33 system of lier equtio i terms of Cost oludes tht Guss Jord method is very epesive s ompred to the Guss elimitio method i order to fid out solutio from system of lier Equtios. [7]. 753 Dr. Udi Bh Trivedi, Mr. Stosh Kumr Shrm,Mr. Vishok Kumr Sigh

2 Itertiol Jourl of Egieerig Tehology Siee d Reserh ISSN Volume 4, Issue 0 Otober 07 R. B. Srivstv d Viod Kumr emied the ompriso of umeril effiieies of Gussi Elimitio d Guss -Jord Elimitio methods for the solutios of lier simulteous equtios. I his reserh work they solved twety simulteous equtios with the help of omputer progrm whih ws developed i C++ Progrmmig Lguge. They fid tht the effiiey of Gussi elimitio method depeds o the umber of lultio ivolved i the solutio of lier simulteous equtio. As the umber of lultios ireses; the effiiey of Gussi elimitio method dereses d vie-vers [4]. II. BACKGROUND THEORATICAL CONCEPTS A lier system of vribles,... is lier equtios of the form = b; The vlue of,... d b re y ostt rel / omple umbers. The ostt i is lled the oeffiiet of i, d b is lled the ostt term of the equtio. A system of lier equtios (or lier system) is fiite olletio of lier equtios i sme vribles =, =, = 3, =,+ A solutio of lier system is tuple (v,v...v ) of umbers tht stisfied eh lier equtio whe the vlues v,v...v re substituted for,,... respetively. The set of ll solutios of lier system is lled the solutio set of the system. A lier system hve: () No possible solutio. (b) Uique solutio () My solutios. A lier system is osistet if it hs t lest oe solutio otherwise iosistet. I geerl, system with less umber of equtio th ukow hve my possible solutio d tht system is kow s u determiisti system. A system with the equl umber of equtio d ukows hs uique solutio. A system with more equtios th ukows hs o solutio. There re my methods of solvig lier systems. Amog them, Guss, Guss Jord elimitio d Guss Seidel Itertio methods shll be osidered. So, they re disussed i detils i the followig setios [8]. A) GAUSS ELIMINATION METHOD Gussi elimitio method is bsed o method whih pplies vrious row opertios i mtri i suh mer whih filly overt Augmeted Mtri [A] i the form of upper trigulr Mtri. Cosider the system of lier equtio =, =, = 3, =,+ This be represeted i mtri form s follows + +, , == 3, , Dr. Udi Bh Trivedi, Mr. Stosh Kumr Shrm,Mr. Vishok Kumr Sigh

3 Itertiol Jourl of Egieerig Tehology Siee d Reserh ISSN Volume 4, Issue 0 Otober 07 The lgorithms osists of followig three mor steps. Red the Augmet mtri A. Redue the mtri i upper trigulr form 3. Use bkwrd substitutio to get the solutio If 0 is loted o the digol, swith the rows util o zero is i tht ple. If shiftig of row ot possible the system hs either ifiite or o solutio. The Mtlb Futio GAUSS_ELIM (A,b) ode of bove proedure is listed below, This futio ept iput (i) ugmeted mtri(a) (ii) d Costt Vetor (b) d retur solutio to the lier system AX = b s vetor [] futio = GAUSS_ELIM(A, b) = size(a, ); r = zeros(, ); for i = : : r(i) = i; ed = zeros(, ); for k = : : % Compre eh elemet i r(k)th olum for the m m = bs(a(r(k), r(k))); m_pos = k; for l = k : : if bs(a(r(l), r(k))) > m m = bs(a(r(l), r(k))); m_pos = l; ed ed temp_r = r; r(k) = temp_r(m_pos); r(m_pos) = temp_r(k); for i = : : if i ~= k zet = A(r(i), k) / A(r(k), k); for = k : : A(r(i), ) = A(r(i), ) - A(r(k), ) * zet; ed b(r(i)) = b(r(i)) - b(r(k)) * zet; ed ed ed for i = : : (i) = b(r(i)) / A(r(i), i); ed ed B) GAUSS JORDAN ELIMINATION METHOD I the Guss elimitio method the oeffiiet mtri ws redued to upper trigulr mtri d the bkwrd substitutio ws pplied. I Guss Jord elimitio method the mtri is redued ito digol 755 Dr. Udi Bh Trivedi, Mr. Stosh Kumr Shrm,Mr. Vishok Kumr Sigh

4 Itertiol Jourl of Egieerig Tehology Siee d Reserh ISSN Volume 4, Issue 0 Otober 07 mtri. At ll steps of Guss elimitio method the elimitio is doe ot oly for the lower digol etries but lso the upper digol etries. Cosider the system of lier equtio =, =, = 3, =,+ Where i d i,+ re kow ostt d i s re ukows. The system is equivlet to AX = b. Where A is ugmeted mtri d X is olum vetor of ukow vrible d b is Colum Vetor of Costt kow s Costt vetor The geerl proedure for Guss Jord elimitio be summrized i the followig steps: The lgorithms osists of followig three mor steps. Red the Augmet mtri A. Redue the ugmeted mtri [A/b] to the trsform A ito digol form (pivotig). 3. Divide right-hd side s elemets s well s digol elemets by the digol elemets i the row, whih will mke eh digol elemet equl to oe If 0 is loted o the digol, swith the rows util o-zero is i tht ple. If you re uble to do so, stop; the system hs either ifiite or o solutio. The MAT LAB Futio guss_ord () ode of bove proedure is listed below, This futio ept iput (i) otetio of ugmeted mtri(a) d Costt Vetor (b) d retur solutio to the lier system AX = b s vetor [y] futio [y] = guss_ord () for = :(legth()-) A = (,:); A = A/A(); (,:) = A; for k = :(legth()-) if ~=k (k,:) = A*(-*(k,)) + (k,:); ed ed ed y = (:,legth())'; ed C) THE GAUSS-SEIDEL METHOD The Guss-Seidel method, med fter Crl Friedrih Guss ( ) d Philipp L. Seidel (8 896). With the Guss-Seidel method, lgorithm uses the ew vlues of eh i s soo s they re kow. Tht is, first we hve to determied from the first equtio d the vlue of is used i the seod equtio to 756 Dr. Udi Bh Trivedi, Mr. Stosh Kumr Shrm,Mr. Vishok Kumr Sigh

757 Dr. Udi Bh Trivedi, Mr. Stosh Kumr Shrm,Mr. Vishok Kumr Sigh Itertiol Jourl of Egieerig Tehology Siee d Reserh www.ietsr.om ISSN 394 3386 Volume 4, Issue 0 Otober 07 fid the ew vlue of.

5 757 Dr. Udi Bh Trivedi, Mr. Stosh Kumr Shrm,Mr. Vishok Kumr Sigh Itertiol Jourl of Egieerig Tehology Siee d Reserh ISSN Volume 4, Issue 0 Otober 07 fid the ew vlue of. Similrly, the ew d will be used i third equtio to fid the vlue of 3 d so o. fter every itertio, the lgorithm lultes the bsolute reltive error for eh of i s e = ( i(ew)- i(old))/ i(ew) * 00 Whe the bsolute reltive pproimte error for eh i is less th the pre-speified tolere, the itertios re stopped [8] Cosider the system of lier equtio =, =, = 3, =,+ Rewritig eh equtio From Equtio () From equtio () d so o from equtio (-) d from equtio () Geerl Form of eh equtio ,,,,,, 3 3,,

6 Itertiol Jourl of Egieerig Tehology Siee d Reserh ISSN Volume 4, Issue 0 Otober 07 The lgorithm strt with iitil guess = =... =0 the iitil guess lso be stored i other vrible y =y =...y =0 whe these vlues re substituted i equtio () tht will produe the ew vlue for d whe the ew vlues of, together with old vlues of, 3,... re substituted i equtio() we get the ew vlue of. Ad i sme mer the lgorithm lulte the ew vlue of,.... The old vlues re vilble i y, y...y. If the old vlues d ew vlues re beome idetil the proess will stop, otherwise trsfer the ew vlues to the vrible y,y,...y d otiuig the sme proedure to get fresh better vlue of,,.... The proedure will stop whe old vlues d ew vlues re lmost sme The MAT LAB Futio guss_ord iterguss( A,b, ) ode of bove proedure is listed below, This futio ept iput (i) ugmeted mtri(a) (ii) Costt Vetor (b) (iii) umber of itertio() d solutio to the lier system AX = b s vetor [] futio [ ] = iterguss( A,b, ) %%hek if guss-seidel method is pplible [si,s]=size(a); if si~=s disp('error mtri is ot squre'); retur ed if prod(dig(a))==0 disp('guss-seidel itertive method is ot pplible'); retur ed E=tril(A,-); F=triu(A,); D=A-E-F B=-iv(D)*(E+F); rho=m(bs(eig(b))); %spetrl rdius if rho>= disp('guss-seidel method do ot overge'); retur ed y=zeros(si,); ti for i=[::] y(:,i+)=iv(d+e)*(b-f*y(:,i)); ed =y(:,); eeutio_time=to ed 758 Dr. Udi Bh Trivedi, Mr. Stosh Kumr Shrm,Mr. Vishok Kumr Sigh

7 Itertiol Jourl of Egieerig Tehology Siee d Reserh ISSN Volume 4, Issue 0 Otober 07 III. DESIGN AND IMPLEMENTATION OF THE COMPARATIVE SYSTEM I The omprtive study of Eeutio time mog Guss Elimitio method, Guss Jord method d Guss Seidel Methods Whih hve bee implemeted by MAT LAB. GAUSS_ELIM(A, b), guss_ord () d iterguss( A,b, ) respetively. These futios hs bee lled betwee ti..to syt i order to get the eeutio time of the lgorithm. Three Sripts file mes s gelim.m, gor.m d gsei.m hs bee reted to ll these futios i MAT LAB eviromet. Si iputs hve bee submitted to these methods d reorded eeutio time for eh iput. Figure : The Proposed System Desig START Guss Elimitio or Guss Jord or Guss Seidel Itertio Sed iput problem Sed iput problem Sed iput problem Apply GAUSS_ELIM(A, b) method of Guss Elimitio Apply guss_ord () method of Guss Jord Apply iterguss( A,b, ) method of Guss Seidel Show result d eeutio time Show result d eeutio time Show result d eeutio time Compre the result END 759 Dr. Udi Bh Trivedi, Mr. Stosh Kumr Shrm,Mr. Vishok Kumr Sigh

Itertiol Jourl of Egieerig Tehology Siee d Reserh www.ietsr.

8 Itertiol Jourl of Egieerig Tehology Siee d Reserh ISSN Volume 4, Issue 0 Otober 07 Figure : Sp Shot of MAT LAB Softwre The tble listed below is shows the eeutio time tke by eh futio i se of, 3, 4, 5, 6 Vrible. The sript hs bee fired five times i eh se d the verge of these times is tke s eeutio time for the lgorithm The vlue of eeutio time betwee the Guss Elimitio, Guss Jord Elimitio d Guss Seidel Algorithm is metioed i Tble d summtive tble. 760 Dr. Udi Bh Trivedi, Mr. Stosh Kumr Shrm,Mr. Vishok Kumr Sigh

9 Itertiol Jourl of Egieerig Tehology Siee d Reserh ISSN Volume 4, Issue 0 Otober 07 Numbers of Vribles Averge 63 Averge 4 Averge 5 Averge 6 Averge Eeutio Time for Guss Elimitio method (seods) Eeutio Time for Guss-Jord Elimitio method (seods) TABLE : EXECUTION TIME Eeutio Time for Guss-seidel Itertio method (seods) guss-seidel method do ot overge guss-seidel method do ot overge guss-seidel method do ot overge guss-seidel method do ot overge guss-seidel method do ot overge guss-seidel method do ot overge guss-seidel method do ot overge guss-seidel method do ot overge guss-seidel method do ot overge guss-seidel method do ot overge guss-seidel method do ot overge guss-seidel method do ot overge 3-=7 +3= Lier Equtio 0++3=9 +0-3= = = = = = = = = = = = = = = = = -7 Beuse the stdrd devitio of ll the bove observtio i the tble is very low for respetive methods we olude tht the verge be represet the etire lss. 76 Dr. Udi Bh Trivedi, Mr. Stosh Kumr Shrm,Mr. Vishok Kumr Sigh

10 Numbers of Vribles Eeutio Time for Guss Elimitio method (Mro Seod) Itertiol Jourl of Egieerig Tehology Siee d Reserh ISSN Volume 4, Issue 0 Otober 07 TABLE : SUMMATIVE EXECUTION TIME Eeutio Time for Guss-Jord Elimitio method (Mro Seod) Eeutio Time for Guss-seidel Itertio method (Mro Seod) guss-seidel method do ot overge guss-seidel method do ot overge Figure 3: Lie Grph Amog time tke by differet Algorithm i MAT LAB IV. CONCLUSION After lysig the bove tble, olusio be drw tht geerlly Guss Elimitio Method is fster th the Guss Jord Elimitio method d Guss Seidel Itertive Method. The effiiey of Gussi elimitio method depeds o the umber of lultio ivolved i the solutio of lier simulteous equtio. As the umber of lultios ireses; the effiiey of Gussi elimitio method dereses d vie-vers due to more time hs bee tke by lgorithm i bkwrd substitutio. The time tke by these lgorithms implemeted i MAT LAB softwre is fr less th if sme would be implemeted by usig C++, Jv or y high progrmmig lguge, due to support of Vetor Proessig(prllel proessig) while delig with Mtri, by MAT LAB. The Guss Seidel Itertive Method is the slowest mog ll lgorithms d suffers with problem of overgee (i our five d si vrible se). Guss Seidel Algorithm whih is itertive lgorithm fid the solutio for lier system where the ugmeted mtri [A] hs digolly domit oeffiiet [6]. 76 Dr. Udi Bh Trivedi, Mr. Stosh Kumr Shrm,Mr. Vishok Kumr Sigh

11 Itertiol Jourl of Egieerig Tehology Siee d Reserh ISSN Volume 4, Issue 0 Otober 07 V. REFERENCES []. Luke Smith, Jo Powell. A Altertive Method to Guss-Jord Elimitio: Miimizig Frtio Arithmeti, the Mthemtis Edutor, 0. []. K. Rlkshmi. Prllel Algorithm for Solvig Lrge System of Simulteous Lier Equtios, IJCSNS Itertiol Jourl of Computer Siee d Network Seurity. [3]. Adeeg, Kehide Emmuel, Aluko. Tope Moses:Guss d Guss-Jord elimitio methods for solvig system of lier equtios: omprisos d pplitios, Jourl of Siee d Siee Edutio, Odo. [4]. R.B Srivstv, Viod Kumr. Compriso of Numeril Effiieies of Gussi Elimitio d Guss-Jord Elimitio methods for the Solutios of lier Simulteous Equtios, Deprtmet of Mthemtis M.L.K.P.G. College Blrmpur. U.P., Idi. [5]. T.J. Dekker, W. Hoffm. Rehbilittio of the Guss- Jord lgorithm, Deprtmet of Computer Systems, Uiversity of Amsterdm, Kruisl [6] Fdugb Sudy Emmuel. Some Itertive Methods for Solvig Systems of Lier Equtios. Deprtmet of Mthemtil Siees, Ekiti Stte Uiversity, Ado Ekiti, Nigeri [7] Dr.P.V.Uble. Compriso of solutio of 33 system of lier equtio i terms of Cost G.S.Si,Arts,Comm. College, Khmgo [8] C.vier. Fortr 77 d Numeril Methods. Derprtmet of Computer Siee, Mduri Kmr Uiversity. [9] Jeff Christese. A Brief History of Lier Algebr. Fil Proet Mth 70 Grt Gustfso, Uiversity of Uth [0] Tuker, Al. (993). The Growig Importe of Lier Algebr i Udergrdute Mthemtis. The College Mthemtis Jourl,, Dr. Udi Bh Trivedi, Mr. Stosh Kumr Shrm,Mr. Vishok Kumr Sigh

Addendum. Addendum. Vector Review. Department of Computer Science and Engineering 1-1

Addendum. Addendum. Vector Review. Department of Computer Science and Engineering 1-1 Addedum Addedum Vetor Review Deprtmet of Computer Siee d Egieerig - Coordite Systems Right hded oordite system Addedum y z Deprtmet of Computer Siee d Egieerig - -3 Deprtmet of Computer Siee d Egieerig