ITERATIVE SOLUTION REFINEMENT

Transcription

1 Numericl nlysis f ngineers Germn Jdnin University ITRTIV SOLUTION RFINMNT Numericl solution of systems of liner lgeric equtions using direct methods such s Mtri Inverse, Guss limintion, Guss-Jdn limintion, nd LU-Decomposition my include lrge round-off errs, especilly f ill-conditioned systems. To improve the ccurcy of the solution, n itertive solution refinement cn e used. Consider the following system of equtions: Let the pproimte solution otined y direct method is given s,, T Sustituting in the iginl system of equtions, new right-hnd-side vect is otined Sutrct equtions from equtions to get sites.google.com/site/ziydmsoud/numericl

2 Numericl nlysis f ngineers Germn Jdnin University sites.google.com/site/ziydmsoud/numericl 4 Let nd let Then, the system of equtions ecomes 4 Δ The refinement procedure is done s follows:. Otin n pproimte solution of the system of equtions using direct method.. Sustitute the pproimte solution in the sme system of equtions to otin new right-hnd-side vect, s in system.

3 Numericl nlysis f ngineers Germn Jdnin University. Clculte the vect nd sustitute in the system of equtions 4 nd solve f the crections vect Δ. 4. Otin new pproimte solution s Δ. 5. Clculte the pproimte err the reltive pproimte err nd compre to preset tolernce. 6. Repet steps 5 until the required tolernce is met. Since system 4 hs to e solved severl times f different right-hnd-side vects, the most suitle method f otining n pproimte solution is either the Mtri Inverse method the LU-Decomposition method. mple Using 4 significnt figures clcultions, determine the solution of the following system of liner lgeric equtions to the est possile 4 significnt figures The est 4 significnt figures mens tht the reltive err in the pproimte solution must ecome zero. The reltive errs vect is Δ which is the crections vect. This mens tht itertions hve to e perfmed until there ecome no crections to e mde. In this emple, we will use the direct Mtri Inverse method, tht is The inverse of the mtri cn e otined using severl methods including the Guss- Jdn limintion nd the LU-Decomposition methods. The inverse of the mtri is sites.google.com/site/ziydmsoud/numericl 5

4 Numericl nlysis f ngineers Germn Jdnin University The infinity nms of the mtri nd its inverse re m n in j m n ij in j The condition numer of the mtri is ij cond The mtri condition numer cn e rounded to the nerest power of 0 numer s cond ssuming minimum reltive err in the nm of the mtri, which is 4 0 using 4 significnt figures clcultions, leds to minimum reltive err in the nm of the solution vect s cond This mens tht the minimum mesurle reltive err in the solution vect is 0 which indictes tht the solution cn e trusted up to significnt figure only. Therefe itertive solution refinement is needed to rech the 4 significnt figures ccurcy. sites.google.com/site/ziydmsoud/numericl 6

5 Numericl nlysis f ngineers Germn Jdnin University Using the direct mtri inverse method, the first pproimtion of the solution cn e otined s Sustituting this pproimtion ck in the iginl system new right-hnd-side vect s , we cn determine the Notice the ig difference etween the new right-hnd-side vect [0.6,.78,.85] T nd the ect right-hnd-side vect [,, ] T The errs vect is Use the errs vect to determine the crections vect s Δ Δ sites.google.com/site/ziydmsoud/numericl 7

6 Numericl nlysis f ngineers Germn Jdnin University Δ The new crected pproimtion ecomes Δ The reltive err in this itertion is the Δ vect which is still not zero. Therefe me itertions re needed. Sustitute the new pproimtion ck in the iginl system right-hnd-side vect s to determine the new Notice tht the difference etween the new right-hnd-side vect [0.9,.94,.97] T nd the ect right-hnd-side vect [,, ] T is decresing. The errs vect is Use the errs vect to determine the crections vect s sites.google.com/site/ziydmsoud/numericl 8

7 Numericl nlysis f ngineers Germn Jdnin University Δ The new crected pproimtion ecomes Δ Δ The reltive err in this itertion is the Δ vect which is still not zero. Therefe me itertions re still needed. Crrying out me itertions, the reltive errs vect crections vect Δ ecomes zero in the th itertion. The solution vect of tht itertion is Using 4 significnt figures clcultions, the ove solution is considered ect since it produces the ect right-hnd-side vect. However, using infinite numer of digits, the ect solution of the system is sites.google.com/site/ziydmsoud/numericl 9