2006 Jamie Trahan, Autar Kaw, Kevin Martin University of South Florida United States of America

Transcription

1 SOLUTION OF SYSTEMS OF SIMULTANEOUS LINEAR EQUATIONS Gauss-Sedel Method 006 Jame Traha, Autar Kaw, Kev Mart Uversty of South Florda Uted States of Amerca Itroducto Ths worksheet demostrates the use of Mathcad to llustrate Gauss-Sedel Method, a teratve techque used solvg a system of smultaeous lear equatos. Gauss-Sedel method s used to solve a set of smultaeous lear equatos, [A] [X] = [RHS], where [A] x s the square coeffcet matrx, [X] x s the soluto vector, ad [RHS] x s the rght had sde array. The equatos ca be rewrtte as x rhs a (, j ) x j, j = := Equato (.) a, I certa cases, such as whe a system of equatos s large, teratve methods of solvg equatos such as Gauss-Sedel method are more advatageous. Elmato methods, such as Gaussa Elmato, are proe to roud-off errors a large set of equatos whereas teratve methods, such as Gauss-Sedel method, allow the user to cotrol roud-off error. Also f the physcs of the problem are well kow, tal guesses eeded teratve methods ca be made more judcously faster covergece. The steps to apply Gauss-Sedel method are: ) Make a tal guess the soluto vector [X]. Ths ca be based o the physcs of the problem. ) All proper values are plugged to Equato (.). The ew x value that s calculated wll replace the prevous guess, x, the soluto vector. [X] wll the be used to calculate x. Ths wll be doe each x from x to x utl a ew soluto vector s complete. At ths pot, the frst terato s doe. 3) The absolute relatve approxmate error s calculated by comparg each ew guess x wth the prevous guess. The maxmum of these errors s the absolute relatve approxmate error at the ed of the terato. 4) The ew soluto vector becomes the old soluto vector ad Steps -3 are repeated utl ether the maxmum umber of teratos have bee coducted or the pre-specfed tolerace has bee met.

2 Complete detals of how Equato (.) s derved as well as the ptfalls of the method ca be foud here. A example demostratg Gauss-Sedel method follows. Secto : Iput The followg are the put parameters to beg the smulato. Ths s the oly secto that requres user put. The user ca chage those values that are hghlghted ad the worksheet wll use four teratos to calculate a approxmate soluto to the system of equatos. ORIGIN := Number of equatos. := 4 The x coeffcet matrx [A]. Note that f the coeffcet matrx s dagoally domat, covergece of the soluto s esured. Otherwse, the soluto may or may ot coverge. A := x rght had sde array. RHS := x tal guess of the soluto vector. Xt :=

3 Secto : Gauss-Sedel Iteratos Four teratos wll be coducted usg Gauss-Sedel method. For each terato, the followg three values wll be calculated: ) New estmate of the soluto vector ) Absolute relatve approxmate error (abs_ea) each x 3) Maxmum relatve approxmate error (Max_abs_ea) the gve terato Iterato ) For the frst terato, the tal guess values of the soluto vector [Xt], as well as the proper elemets of coeffcet matrx [A] ad the rght had sde vector [RHS] are substtuted to Equato (.) to calculate a ew, approxmate soluto vector, deoted as [Xew] the frst terato. The followg procedure calculates [Xew] usg Equato (.). Xew := Defg row elemets sum 0 Italzg the seres sum to 0 X j sum sum + A Xt f j, j j RHS sum Xt A, X Xt Defg colum elemets Geeratg the summato term by oly addg jterms Applyg equato (.) Replacg old guess wth ew guess Returg ew, approxmate [X] vector The ew approxmate soluto vector s Xew =

4 ) The absolute relatve percetage approxmate error each Xew s calculated by the followg procedure abs_ea := Xew Xt absea 00.0 Xew absea The absolute relatve percetage approxmate error s abs_ea = ) The maxmum of these errors s the absolute relatve approxmate error at the ed of the frst terato. Max_abs_ea := max_absea 0 max_absea abs_ea f abs_ea > max_absea max_absea The maxmum absolute relatve percetage approxmate error at the ed of terato s Max_abs_ea =

5 Iterato ) The ew soluto vector, [Xew], obtaed terato wll become the old soluto vector terato. Defg the old soluto vector as the soluto vector that was obtaed from the prevous terato: Xold := Xew Xold = The ew approxmate soluto vector terato wll ow be calculated wth [Xold], [A], ad [RHS] by substtutg the proper elemets to Equato (.). Xew := Defg row elemets. sum 0 Italzg the seres sum to 0. X j sum sum + A Xold f j, j j RHS sum Xold A, X Xold Defg colum elemets. Geeratg the summato term by oly addg jterms. Applyg Equato (.). Replacg old guess wth ew guess. Returg ew, approxmate [X] vector. At the ed of the d terato, the ew estmate of the soluto vector s Xew =

6 ) The absolute relatve percetage approxmate error the secod terato s calculated by the followg procedure abs_ea := Xew Xold absea 00.0 Xew absea The absolute relatve percetage approxmate error s abs_ea = ) The maxmum absolute relatve approxmate error the secod terato s defed by the followg procedure Max_abs_ea := max_absea 0 max_absea abs_ea f abs_ea > max_absea max_absea The maxmum absolute relatve percetage approxmate error the terato s Max_abs_ea =

7 Iterato 3 ) Aga, the ew estmate of the soluto vector from the prevous terato wll replace the old soluto vector. Xold := Xew Xold = Substtutg the proper [Xold], [A], ad [RHS] elemets to Equato (.) wth the followg procedure wll retur a estmate of the soluto vector after 3 teratos. Xew3 := Defg row elemets. sum 0 Italzg the seres sum to 0. X j sum sum + A Xold f j, j j RHS sum Xold A, X Xold Defg colum elemets. Geeratg the summato term by oly addg jterms. Applyg Equato (.). Replacg old guess wth ew guess. Returg ew estmate of soluto vector [X]. The ew guess soluto vector after the thrd terato s: Xew3 =

8 ) Calculatg the absolute relatve approxmate error the thrd terato: abs_ea3 := Xew3 Xold absea 00.0 Xew3 absea The absolute relatve percetage approxmate error terato 3 s abs_ea3 = ) Defg the maxmum absolute relatve percetage error terato 3: Max_abs_ea3 := max_absea 0 max_absea abs_ea3 f abs_ea3 > max_absea max_absea The maxmum absolute relatve approxmate error the thrd terato s Max_abs_ea3 =

9 Iterato 4 ) The ew soluto vector obtaed at the ed of terato 3 becomes the old guess soluto vector: Xold := Xew3 Xold = Substtutg the proper [Xold], [A], ad [RHS] elemets to equato (.) wth the followg procedure wll yeld the fal estmate of the soluto vector. Xew4 := Defg row elemets sum 0 Italzg the seres sum to 0. X j sum sum + A Xold f j, j j RHS sum Xold A, X Xold Defg colum elemets. Geeratg the summato term by oly addg jterms. Applyg Equato (.). Replacg old guess wth ew guess. Returg ew, approxmate [X] vector. The fal estmate of the soluto vector s Xew4 =

10 ) Calculatg the absolute relatve percetage approxmate error the fourth terato: abs_ea4 := Xew4 Xold absea 00.0 Xew3 absea The absolute relatve percetage approxmate error terato 3 s abs_ea4 = ) Defg the maxmum absolute relatve percetage error terato 4: Max_abs_ea4 := max_absea 0 max_absea abs_ea4 f abs_ea4 > max_absea max_absea The maxmum absolute relatve percetage approxmate error the fourth ad fal terato s Max_abs_ea4 = 8.7

11 Secto 3: Exact Soluto The exact soluto to the system of lear equatos ca be foud by usg Mathcad's bult- tools. exactsol := lsolve( A, RHS) The exact soluto s exactsol = The followg procedure calculates the absolute relatve percetage true error (abs_et). abs_et := exactsol Xew4 abs_et 00.0 exactsol abs_et The absolute relatve percetage true error s abs_et = The maxmum absolute relatve percetage true error s Max_abs_et := max_abset 0 max_abset abs_et f abs_et > max_abset max_abset Max_abs_et =

12 Refereces Autar Kaw, Holstc Numercal Methods Isttute, How does Gauss-Sedel method work? Coclusos Mathcad helped us apply our kowledge of Gauss-Sedel method to solve a system of smultaeous lear equatos. The values obtaed are Fal approxmate soluto vector usg 4 teratos Xew4 = Exact value exactsol = Absolute relatve percetage true error abs_et = Questo : Chage the coeffcet matrx to oe that s ot dagoally domat ad see f Gauss-Sedel method coverges. Questo : See f you ca get a set of equatos wth a coeffcet matrx that s ot dagoally domat to coverge by Gauss-Sedel method.