ITERATIVE METHODS FOR SOLVING SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS

Transcription

1 Numercl Alyss for Egeers Germ Jord Uversty ITERATIVE METHODS FOR SOLVING SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS Numercl soluto of lrge systems of ler lgerc equtos usg drect methods such s Mtr Iverse, Guss Elmto, Guss-Jord Elmto, d LU-Decomposto my clude lrge roud-off errors, especlly for ll-codtoed systems. Aother pproch for solvg such systems s usg tertve methods. The pproch s to strt wth tl soluto vector, d tertvely coverge to the true soluto. The tertos re the stopped whe pre-specfed tolerce s reched. Itertve methods re preferred over drect methods some cses such s whe the coeffcet mtr s sprse hs too my zeros). The tertve methods hve my dvtges such s My e more rpd. My e ecoomcl storge d clcultos. Self correctg. Reduce roud-off errors. JACOBI METHOD Cosder the followg system of equtos: stes.google.com/ste/zydmsoud/umercl

2 Numercl Alyss for Egeers Germ Jord Uversty The ove system c e rewrtte s ) Strtg wth tl guess vector, ew ppromto c e clculted y susttutg the tl guess the ove system s ) ) ) After ech terto, the reltve error of the lst ppromto c e determed s k k k % where k s the curret terto, k s the prevous terto, d. The reltve error s the compred wth commo tolerce for ll vrles, or specfc tolerce for ech vrle. Emple Solve the followg system of ler lgerc equtos usg Jco method d strtg wth the tl guess vector,, T. Use tolerce of.% for ll vrles. stes.google.com/ste/zydmsoud/umercl

3 Numercl Alyss for Egeers Germ Jord Uversty The ove system c e rewrtte s Susttutg the tl guess vector the ove equtos, we c ot the frst ppromto of the system soluto The error ths frst ppromto c e clculted s.8 % % %.8.75 % % %.75 %.. % % The secod ppromto of the system soluto c e oted y susttutg the frst ppromto ck the system of equtos s stes.google.com/ste/zydmsoud/umercl

4 Numercl Alyss for Egeers Germ Jord Uversty ) ).) ) The error ths frst ppromto c e clculted s % % %.9.8 %.5% % 9.55% % 76.5%.856 The secod ppromto s the used to determe the thrd d so o. The error s clculted every terto d compred wth the tolerce. Iterto results re show the tle elow. It # stes.google.com/ste/zydmsoud/umercl

5 Numercl Alyss for Egeers Germ Jord Uversty stes.google.com/ste/zydmsoud/umercl GAUSS-SEIDEL METHOD The Guss-Sedel method s very smlr to the Jco method ecept tht ech ew ppromto s used mmedtely the followg clculto. Cosder the followg system of equtos: The ove system c e rewrtte s ) ) Strtg wth tl guess vector, ew ppromto c e clculted y susttutg the tl guess the ove system s

6 Numercl Alyss for Egeers Germ Jord Uversty stes.google.com/ste/zydmsoud/umercl 5 ) ) After ech terto, the reltve error of the lst ppromto c e determed s % k k k where k s the curret terto, k s the prevous terto, d. The reltve error s the compred wth commo tolerce for ll vrles, or specfc tolerce for ech vrle. Emple Solve the followg system of ler lgerc equtos usg Guss-Sedel method d strtg wth the tl guess vector T,,. Use tolerce of.% for ll vrles The ove system c e rewrtte s

7 Numercl Alyss for Egeers Germ Jord Uversty Susttutg the tl guess vector the ove equtos, we c ot the frst ppromto of the system soluto ) ) 5.6 The error ths frst ppromto c e clculted s.8 % % %.8.9 % % %.9 %.6.6 % % The secod ppromto of the system soluto c e oted y susttutg the frst ppromto ck the system of equtos s 5 6.9) ).6) ) 5.5 stes.google.com/ste/zydmsoud/umercl 6

8 Numercl Alyss for Egeers Germ Jord Uversty The error ths frst ppromto c e clculted s % % %.7.8 %.%.7..9 %.65%..5.6 %.6%.5 The secod ppromto s the used to determe the thrd d so o. The error s clculted every terto d compred wth the tolerce. Iterto results re show the tle elow. It # We c oserve tht the Guss-Sedel method s much fster th the Jco method sce ewer d more ccurte ppromtos re lwys used mmedtely ll clcultos. stes.google.com/ste/zydmsoud/umercl 7

9 Numercl Alyss for Egeers Germ Jord Uversty stes.google.com/ste/zydmsoud/umercl 8 CONVERGENCE CRITERION FOR GAUSS-SEIDEL METHOD Recll tht for the Fed-Pot Iterto method ) g, the covergece crtero ws tht ) g The lgorthm of the Guss-Sedel method s As wth the Fed-Pot Iterto method, we c ssume tht g ) It c e prove tht the covergece of the Guss-Sedel method requres tht g g g g g, g,, g The

10 Numercl Alyss for Egeers Germ Jord Uversty or whch mes tht for gurteed covergece, the mgtude of the dgol coeffcet must e lrger th the sum of mgtudes of the off-dgol coeffcets. Emple The order of equtos of the pervous emple s ltered y terchgg equtos ) d ) of the system of equtos s show elow. Solve the system of ler lgerc equtos usg Guss-Sedel method d strtg wth the tl guess vector,, T. Use tolerce of.% for ll vrles. The ove system c e rewrtte s stes.google.com/ste/zydmsoud/umercl 9

11 Numercl Alyss for Egeers Germ Jord Uversty Susttutg the tl guess vector the ove equtos, we c ot the frst ppromto of the system soluto ). 5.5).) The error ths frst ppromto c e clculted s.5 % % %.5. % % %. % % % The secod ppromto of the system soluto c e oted y susttutg the frst ppromto ck the system of equtos s ) 5.5) ) 5.5) ) 68.8) stes.google.com/ste/zydmsoud/umercl 5

12 Numercl Alyss for Egeers Germ Jord Uversty The error ths frst ppromto c e clculted s % % % 5.5.5) % 95.% ) % 9.% ) % 9.95% 78. Crryg out more tertos, the soluto cotues to dverge wy from the system soluto. Ths s due to the fct tht the mgtude dgol coeffcet equtos ) d ) s smller th the sum of mgtudes of the off-dgol coeffcets of ech equto. stes.google.com/ste/zydmsoud/umercl 5