Gauss-Seidel Method. An iterative method. Basic Procedure:
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1 Guss-Siedel Method
2 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. Egieerig Mthemtics III
3 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. Egieerig Mthemtics III
4 A set of equtios d ukows: Guss-Seidel Method Algorithm... b... b b If: the digol elemets re o-zero Rewrite ech equtio solvig for the correspodig ukow e: First equtio, solve for Secod equtio, solve for 4 Egieerig Mthemtics III
5 Egieerig Mthemtics III Guss-Seidel Method Algorithm Rewritig ech equtio c c c c,,,,,, From Equtio From equtio From equtio - From equtio 5
6 Egieerig Mthemtics III Guss-Seidel Method Algorithm Geerl Form of ech equtio c c,, c c 6
7 Guss-Seidel Method Algorithm Geerl Form for y row i i c i i ii i, i,,,. How or where c this equtio be used? 7 Egieerig Mthemtics III
8 Guss-Seidel Method Solve for the ukows Assume iitil guess for [X] - Use rewritte equtios to solve for ech vlue of i. Importt: Remember to use the most recet vlue of i. Which mes to pply vlues clculted to the clcultios remiig i the curret itertio. 8 Egieerig Mthemtics III
9 Guss-Seidel Method Clculte the Absolute Reltive Approimte Error 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. 9 Egieerig Mthemtics III
10 Guss-Seidel Method: Emple Give the system of equtios With iitil guess of 0 The coefficiet mtri is: A Will the solutio coverge usig the Guss-Siedel method? 0 Egieerig Mthemtics III
11 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 Egieerig Mthemtics III
12 Guss-Seidel Method: Emple Rewritig ech equtio With iitil guess of Egieerig Mthemtics III
13 Guss-Seidel Method: Emple The bsolute reltive pproimte error % % % The mimum bsolute reltive error fter the first itertio is 00% Egieerig Mthemtics III
14 Guss-Seidel Method: Emple After Itertio # Substitutig the vlues ito the equtios After Itertio # Egieerig Mthemtics III
15 Guss-Seidel Method: Emple Itertio # bsolute reltive pproimte error % % %.88 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? 5 Egieerig Mthemtics III
16 Guss-Seidel Method: Emple Repetig more itertios, the followig vlues re obtied Itertio The solutio obtied is close to the ect solutio of Egieerig Mthemtics III
17 Guss-Seidel Method: Emple The upwrd velocity of rocket is give t three differet times Tble Velocity vs. Time dt. Time, t s Velocity v m/s The velocity dt is pproimted by polyomil s: v t t t, 5 t.
18 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 5
19 Guss-Seidel Method: Emple Rewritig ech equtio
20 Guss-Seidel Method: Emple Applyig the iitil guess d solvig for i () (5) Iitil Guess Whe solvig for, how my of the iitil guess vlues were used?
21 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 5.47% %
22 Guss-Seidel Method: Emple Usig from itertio # Itertio # the vlues of i re foud:
23 Guss-Seidel Method: Emple Fidig the bsolute reltive pproimte error % At the ed of the secod itertio % The mimum bsolute reltive pproimte error is % %
24 Itertio % % % Guss-Seidel Method: Emple 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
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