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
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
Numercl Alyss for Egeers Germ Jord Uversty The ove system c e rewrtte s 6 7 5 5 5 6 7 5 Susttutg the tl guess vector the ove equtos, we c ot the frst ppromto of the system soluto 5 6 6.8 7 5 7.75 5 5. 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
Numercl Alyss for Egeers Germ Jord Uversty 5 6.75). 6.9 7 5.8).) 7.8 5.8.75) 5.856 The error ths frst ppromto c e clculted s % % %.9.8 %.5%.9.8.75 % 9.55%.8.856. % 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 #.8.75..9.8.856.5 9.55 76.5.86.5.8.5.5..5..8. 5.9. 5.995.99..5.79.59 6.997.9986.9955..7.659 7...9988.999.. stes.google.com/ste/zydmsoud/umercl
Numercl Alyss for Egeers Germ Jord Uversty stes.google.com/ste/zydmsoud/umercl 8....5.. 9...... 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
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. 5 5 7 6 The ove system c e rewrtte s
Numercl Alyss for Egeers Germ Jord Uversty 5 6 7 5 Susttutg the tl guess vector the ove equtos, we c ot the frst ppromto of the system soluto 5 6 6.8 7 5.8) 7.9 5.8.9) 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 6.7 7 5.7).6) 7. 5.7.) 5.5 stes.google.com/ste/zydmsoud/umercl 6
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 #.8.9.6.7..5..65.6.999.9959.998.55.65.68....5.595.8 5....998.. 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
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
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 6 7 5 5 stes.google.com/ste/zydmsoud/umercl 9
Numercl Alyss for Egeers Germ Jord Uversty 5 7 6 5 Susttutg the tl guess vector the ove equtos, we c ot the frst ppromto of the system soluto 5 7 6 5.5 6.5). 5.5).) 5 5.5 The error ths frst ppromto c e clculted s.5 % % %.5. % % %. % 5.5 5.5 % % The secod ppromto of the system soluto c e oted y susttutg the frst ppromto ck the system of equtos s 5 7 6 5 7.) 5.5) 5.5 5 6 5.5) 5.5) 68.8 5 5.5) 68.8) 5 78. stes.google.com/ste/zydmsoud/umercl 5
Numercl Alyss for Egeers Germ Jord Uversty The error ths frst ppromto c e clculted s % % % 5.5.5) % 95.% 5.5 68.8.) % 9.% 68.8 78. 5.5) % 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