Gauss-Siedel Method. Major: All Engineering Majors. Authors: Autar Kaw
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1 Guss-Siedel Method Mjor: All Egieerig Mjors Authors: Autr Kw Trsformig Numericl Methods Eductio for STEM Udergrdutes 4//06
2 Guss-Seidel Method
3 Guss-Seidel Method A itertive method. Bsic Procedure: -Algebriclly solve ech lier equtio for i -Assume iitil guess solutio rry -Solve for ech i d repet -Use bsolute reltive pproimte error fter ech itertio to check if error is withi pre-specified tolerce.
4 Guss-Seidel Method Why? The Guss-Seidel Method llows the user to cotrol roud-off error. Elimitio methods such s Gussi Elimitio d LU Decompositio re proe to proe to roud-off error. Also: If the physics of the problem re uderstood, close iitil guess c be mde, decresig the umber of itertios eeded.
5 A set of equtios d ukows: Guss-Seidel Method Algorithm... b... b b Jik: eleme digol buk ol Tulis ulg tip persm, selesik yg tk dikthui cotoh: First equtio, solve for Secod equtio, solve for
6 Guss-Seidel Method Algorithm Rewritig ech equtio c c c c,,,,,, From Equtio From equtio From equtio - From equtio
7 Guss-Seidel Method Algorithm Betuk umum tip persm c j j j j c j j j j,, j j j j c j j j j c
8 Guss-Seidel Method Algorithm Geerl Form for y row i i c i j ji ii ij j, i,,,. How or where c this equtio be used?
9 Guss-Seidel Method Selesik yg tk dikethui Asumsik ili tebk [ - Guk persm yg telh ditulis ulg utuk meyelesik tip ili Xi Petig: igt utuk sellu megguk ili Xi yg berrti megguk ili yg telh dihitug utk perhitug yg sedg berlgsug pd itersi tersebut
10 Guss-Seidel Method Hitug pedekt error reltif multk i ew i ew i old i 00 So whe hs the swer bee foud? The itertios re stopped whe the bsolute reltive pproimte error is less th prespecified tolerce for ll ukows.
11 Guss-Seidel Method: Emple Kelju ik roket diberik dt sbb: Tble Velocity vs. Time dt. Time, t s Velocity v m/s Dt kelju didektk deg poliomil sbb v t t t, t.
12 Guss-Seidel Method: Emple v v v t t t t t t Usig Mtri templte of the form The system of equtios becomes Iitil Guess: Assume iitil guess of
13 Guss-Seidel Method: Emple Rewritig ech equtio
14 Guss-Seidel Method: Emple Applyig the iitil guess d solvig for i 06.8 () () Iitil Guess Whe solvig for, how my of the iitil guess vlues were used?
15 Guss-Seidel Method: Emple Fidig the bsolute reltive pproimte error i ew i ew i old i % At the ed of the first itertio % The mimum bsolute reltive pproimte error is.47% %
16 Guss-Seidel Method: Emple Usig from itertio # Itertio # the vlues of i re foud:
17 Guss-Seidel Method: Emple Fidig the bsolute reltive pproimte error % At the ed of the secod itertio % The mimum bsolute reltive pproimte error is % 8.69%
18 Guss-Seidel Method: Emple Itertio % % % 4 6 Repetig more itertios, the followig vlues re obtied Notice The reltive errors re ot decresig t y sigifict rte Also, the solutio is ot covergig to the true solutio of
19 Guss-Seidel Method: Pitfll Ap yg slh Eve though doe correctly, the swer is ot covergig to the correct swer This emple illustrtes pitfll of the Guss-Siedel method: ot ll systems of equtios will coverge. Apkh bis diperbiki Stu kels sistem k sellu koverge: sistem mtriks deg koefisie digol domi Digolly domit: [A] i [A] [X] = [C] is digolly domit if: ii j ji ij ii ij for ll i d for t lest oe i j ji
20 Guss-Seidel Method: Pitfll Digolly domit: The coefficiet o the digol must be t lest equl to the sum of the other coefficiets i tht row d t lest oe row with digol coefficiet greter th the sum of the other coefficiets i tht row. Which coefficiet mtri is digolly domit? A [B] Most physicl systems do result i simulteous lier equtios tht hve digolly domit coefficiet mtrices.
21 Guss-Seidel Method: Emple Give the system of equtios With iitil guess of The coefficiet mtri is: 7 A Will the solutio coverge usig the Guss-Siedel method?
22 A Guss-Seidel Method: Emple Checkig if the coefficiet mtri is digolly domit The iequlities re ll true d t lest oe row is strictly greter th: Therefore: The solutio should coverge usig the Guss-Siedel Method
23 7 Guss-Seidel Method: Emple Rewritig ech equtio With iitil guess of
24 Guss-Seidel Method: Emple The bsolute reltive pproimte error % % % The mimum bsolute reltive error fter the first itertio is 00%
25 Guss-Seidel Method: Emple After Itertio # Substitutig the vlues ito the equtios After Itertio #
26 Guss-Seidel Method: Emple Itertio # bsolute reltive pproimte error % % % The mimum bsolute reltive error fter the first itertio is 40.6% This is much lrger th the mimum bsolute reltive error obtied i itertio #. Is this problem?
27 Guss-Seidel Method: Emple Ulgi lgi itersiy, didpt ili sbb: % % Itertio % Solusi didpt.000 dekt dg solusi eksk
28 Guss-Seidel Method: Emple Give the system of equtios With iitil guess of 0 Rewritig the equtios
29 Itertio A 4 6 Guss-Seidel Method: Emple Coductig si itertios, the followig vlues re obtied % The vlues re ot covergig. % % Does this me tht the Guss-Seidel method cot be used?
30 Guss-Seidel Method The Guss-Seidel Method c still be used The coefficiet mtri is ot digolly domit But this is the sme set of equtios used i emple #, which did coverge. A A 7 7 If system of lier equtios is ot digolly domit, check to see if rerrgig the equtios c form digolly domit mtri.
31 Guss-Seidel Method Not every system of equtios c be rerrged to hve digolly domit coefficiet mtri. Observe the set of equtios Which equtio(s) prevets this set of equtio from hvig digolly domit coefficiet mtri?
32 Guss-Seidel Method Summry -Advtges of the Guss-Seidel Method -Algorithm for the Guss-Seidel Method -Pitflls of the Guss-Seidel Method
33 Guss-Seidel Method Questios?
34 Additiol Resources For ll resources o this topic such s digitl udiovisul lectures, primers, tetbook chpters, multiple-choice tests, worksheets i MATLAB, MATHEMATICA, MthCd d MAPLE, blogs, relted physicl problems, plese visit el.html
35 THE END
Gauss-Seidel Method. An iterative method. Basic Procedure:
Guss-Siedel Method Guss-Seidel Method A itertive method. Bsic Procedure: -Algebriclly solve ech lier equtio for i -Assume iitil guess solutio rry -Solve for ech i d repet -Use bsolute reltive pproimte
LU Decomposition. Mechanical Engineering Majors. Authors: Autar Kaw
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