Chpter 04.08 Guss-Sedel Method After redg ths hpter, you should be ble to:. solve set of equtos usg the Guss-Sedel method,. reogze the dvtges d ptflls of the Guss-Sedel method, d. determe uder wht odtos the Guss-Sedel method lwys overges. Why do we eed other method to solve set of smulteous ler equtos? I ert ses, suh s whe system of equtos s lrge, tertve methods of solvg equtos re more dvtgeous. Elmto methods, suh s Guss elmto, re proe to lrge roud-off errors for lrge set of equtos. Itertve methods, suh s the Guss-Sedel method, gve the user otrol of the roud-off error. Also, f the physs of the problem re well kow, tl guesses eeded tertve methods be mde more udously ledg to fster overgee. Wht s the lgorthm for the Guss-Sedel method? Gve geerl set of equtos d ukows, we hve + + +... + + + +... +...... + + +... + If the dgol elemets re o-zero, eh equto s rewrtte for the orrespodg ukow, tht s, the frst equto s rewrtte wth o the left hd sde, the seod equto s rewrtte wth o the left hd sde d so o s follows 04.08.
04.08. Chpter 04.08,,,,,, These equtos be rewrtte summto form s...,, Hee for y row,.,,,, Now to fd s, oe ssumes tl guess for the s d the uses the rewrtte equtos to lulte the ew estmtes. Remember, oe lwys uses the most reet estmtes to lulte the et estmtes,. At the ed of eh terto, oe lultes the bsolute reltve ppromte error for eh s
Guss-Sedel Method 04.08. where ew ew ew old 00 s the reetly obted vlue of, d old s the prevous vlue of. Whe the bsolute reltve ppromte error for eh s less th the pre-spefed tolere, the tertos re stopped. Emple The upwrd veloty of roket s gve t three dfferet tmes the followg tble Tble Veloty vs. tme dt. Tme, t (s) Veloty, v (m/s) 5 06.8 8 77. 79. The veloty dt s ppromted by polyoml s v ( t) t + t +, 5 t Fd the vlues of,, d usg the Guss-Sedel method. Assume tl guess of the soluto s 5 d odut two tertos. Soluto The polyoml s gog through three dt pots (, v ), ( t, v ), d ( t v ) t, where from the bove tble t 5, v 06.8 t 8, v 77. t, v 79. Requrg tht v ( t) t + t + psses through the three dt pots gves v ( t ) v t + t + v ( t ) v t + t + v ( t ) v t + t + Substtutg the dt ( t, v ), ( t, v ), d ( t, v ) gves ( 5 ) + ( 5) + 06. 8 ( 8 ) + ( 8) + 77. ( ) + ( ) + 79.
04.08.4 Chpter 04.08 or 5 + 5 + 06.8 64 + 8 + 77. 44 + + 79. The oeffets,, d for the bove epresso re gve by 5 5 06.8 64 8 77. 44 79. Rewrtg the equtos gves 06.8 5 5 77. 64 8 79. 44 Iterto # Gve the tl guess of the soluto vetor s 5 we get 06.8 5() (5) 5.670 77. 64(.670) ( 5) 8 7.850 79. 44(.670) ( 7.850) 55.6 The bsolute reltve ppromte error for eh the s.670 00.670 7.76% 7.850 00 7.850 5.47% 55.6 5 00 55.6 0.%
Guss-Sedel Method 04.08.5 At the ed of the frst terto, the estmte of the soluto vetor s.670 7.850 55.6 d the mmum bsolute reltve ppromte error s 5.47%. Iterto # The estmte of the soluto vetor t the ed of Iterto # s.670 7.850 55.6 Now we get 06.8 5( 7.850) ( 55.6) 5.056 77. 64.056 ( 55.6) ( ) 8 54.88 79. 44(.056) ( 54.88) 798. 4 The bsolute reltve ppromte error for eh the s.056.670 00.056 69.54% 54.88 ( 7.850 ) 00 54.88 85.695% 798.4 ( 55.6 ) 00 798.4 80.540% At the ed of the seod terto the estmte of the soluto vetor s.056 54.88 798.54 d the mmum bsolute reltve ppromte error s 85.695%. Codutg more tertos gves the followg vlues for the soluto vetor d the orrespodg bsolute reltve ppromte errors.
04.08.6 Chpter 04.08 Iterto %.670 7.767.056 69.54 47.8 74.447 4 9. 75.595 5 800.5 75.850 6.6 75.906 % 7.850 54.88 55.5 09.4 4577. 9049 5.47 85.695 78.5 76.6 76. 75.97 % 55.6 798.4 448.9 4440 6007 49580 0. 80.540 76.85 76.6 75.96 75.9 As see the bove tble, the soluto estmtes re ot overgg to the true soluto of 0.9048 9.690.0857 The bove system of equtos does ot seem to overge. Why? Well, ptfll of most tertve methods s tht they my or my ot overge. However, the soluto to ert lsses of systems of smulteous equtos does lwys overge usg the Guss-Sedel method. Ths lss of system of equtos s where the oeffet mtr [A] [ A ][ X ] [ C] s dgolly domt, tht s > for ll for t lest oe If system of equtos hs oeffet mtr tht s ot dgolly domt, t my or my ot overge. Fortutely, my physl systems tht result smulteous ler equtos hve dgolly domt oeffet mtr, whh the ssures overgee for tertve methods suh s the Guss-Sedel method of solvg smulteous ler equtos. Emple Fd the soluto to the followg system of equtos usg the Guss-Sedel method. + 5 + 5 + 8 + 7 + 76 Use 0 s the tl guess d odut two tertos. Soluto The oeffet mtr
Guss-Sedel Method 04.08.7 5 [ A ] 5 7 s dgolly domt s + + 5 8 5 5 + + 4 + + 7 0 d the equlty s strtly greter th for t lest oe row. Hee, the soluto should overge usg the Guss-Sedel method. Rewrtg the equtos, we get + 5 8 5 76 7 Assumg tl guess of 0 Iterto # ( 0) + 5( ) 0.50000 8 ( 0.50000) ( ) 5 4.9000 76 ( 0.50000) 7( 4.9000).09 The bsolute reltve ppromte error t the ed of the frst terto s 0.50000 00 0.50000 00.00% 4.9000 0 00 4.9000 00.00%.09 00.09 67.66%
04.08.8 Chpter 04.08 The mmum bsolute reltve ppromte error s 00.00% Iterto # ( 4.9000) + 5(.09) 0.4679 8 ( 0.4679) (.09) 5.75 76 ( 0.4679) 7(.75).88 At the ed of seod terto, the bsolute reltve ppromte error s 0.4679 0.50000 00 0.4679 40.6%.75 4.9000 00.75.889%.88.09 00.88 8.874% The mmum bsolute reltve ppromte error s 40.6%. Ths s greter th the vlue of 00.00% we obted the frst terto. Is the soluto dvergg? No, s you odut more tertos, the soluto overges s follows. Iterto % 0.50000 00.00 0.4679 40.6 0.7475 80.6 4 0.94675.546 5 0.9977 4.59 6 0.9999 0.7407 Ths s lose to the et soluto vetor of 4 Emple Gve the system of equtos + 7 + 76 % 4.9000.75.644.08.004.000 00.00.889 7.408 4.4996 0.8499 0.0856 %.09.88.9708.997 4.000 4.000 67.66 8.874 4.0064 0.6577 0.0748 0.000
Guss-Sedel Method 04.08.9 + 5 + 8 + - 5 fd the soluto usg the Guss-Sedel method. Use 0 s the tl guess. Soluto Rewrtg the equtos, we get 76 7 8 5 5 Assumg tl guess of 0 the et s tertve vlues re gve the tble below. Iterto %.000 95.8 96.5 0.7 995.0 09.8 4 049 09.90 5.064 0 5 09.89 6.0579 0 6 09.89 % 0.80000 4.4 6.0 04.6 40.7 0 5 00.00 94.45.4 09.6 09.9 09.89 % 50.680 46.0 478. 4766 4.844 0 5 4.865 0 6 98.07 0.96 09.80 09.90 09.89 09.89 You see tht ths soluto s ot overgg d the oeffet mtr s ot dgolly domt. The oeffet mtr 7 [ A ] 5 5 s ot dgolly domt s + 7 + 0 Hee, the Guss-Sedel method my or my ot overge. However, t s the sme set of equtos s the prevous emple d tht overged. The oly dfferee s tht we ehged frst d the thrd equto wth eh other d tht mde the oeffet mtr ot dgolly domt.
04.08.0 Chpter 04.08 Therefore, t s possble tht system of equtos be mde dgolly domt f oe ehges the equtos wth eh other. However, t s ot possble for ll ses. For emple, the followg set of equtos + + + + 4 9 + 7 + 9 ot be rewrtte to mke the oeffet mtr dgolly domt. Key Terms: Guss-Sedel method Covergee of Guss-Sedel method Dgolly domt mtr