SOLUTION OF SYSTEM OF LINEAR EQUATIONS Lecture 4: () Jcobi's method. method (geerl). (b) Guss Seidel method.
Jcobi s Method: Crl Gustv Jcob Jcobi (804-85) gve idirect method for fidig the solutio of system of lier equtios, which is bsed o the successive better pproimtios of the vlues of the ukows, usig itertive procedure. The sufficiet coditio for the covergece of Guss Jcobi method to solve A b is tht the coefficiet mtri A is strictly digolly row domit, tht is, if A, the ii j ji ij It should be oted tht this method mkes two ssumptios. First, the system of lier equtios to be solved, must hve uique solutio d secod, there should ot be y zeros o the mi digol of the coefficiet mtri A. I cse, there eist zeros o its mi digol, the rows must be iterchged to obti coefficiet mtri tht does ot hve zero etries o the mi digol. Cosider system of lier equtios i ukows, which re strictly digolly row domit, s follows:... b,... b,...... b, Sice the system is strictly digolly row domit, 0. ii
Therefore, the system of equtios is rewritte s b 0...., b 0....,... b,... 0. We the cosider rbitrry iitil guess of the solutio s 0 0 0,,...,, which re row substituted to the right hd side of the rewritte equtios to obti the first pproimtio s b... 0 0 0 0 3 3 b... 0 0 0 0 3 3, b... 0 0 0, This process is repeted by substitutig the first pproimte solutio,,, to the r.h.s of the rewritte equtios. By repeted itertio, we get the required solutio up to the desired level of the ccurcy.
Emple 6. Solve the system of lier equtios by Jcobi s method. 4 9 3 8 3 0 3 4 33 3 Solutio: The give system of equtios is ot digolly row domit s 3. Therefore, we re-rrge the system s 8 3 0 3 4 33 3 4 9 3 Here, 8 3, 4 d 4. Thus, the system is digolly row domit. We ow re-write the system s 0 3 8 33 4 3 9 4 3 3 Let the iitil guess be the solutio is give by 3 0 0 0,, 0.The, the first pproimtio to 3 (0 3 0).875 8 33 4 0.636 9.75 4
Secod pproimtio 3 3 0 3.636.75 3.05 8 33 4.875.75.4 9.875.636 0.87 4 Third pproimtio 3 3 3 3 3.075.969 0.959 Fourth pproimtio 4 4 4 3.999.969 0.989 Fifth pproimtio 5 3 5 5.99.999.008 Sith pproimtio 6.997 6.004 6 3.00 Therefore, 3.0,.0 d 3.0, correct to two sigifict figures.
Guss Seidel Method Guss Seidel itertio method for solvig system of -lier equtios i - ukows is modified Jcobi s method. Therefore, ll the coditios tht is true for Jcobi s method, lso holds for Guss Seidel method. As before, the system of lier equtios re rewritte s If b 0...., b 0....,... b 0 0 0,... 0.,,..., be the iitil guess of the solutio, which is rbitrry, the the first pproimtio to the solutio is obtied s b... 0 0 0 0 3 3, b... 0 0 0 3 3, b... 0 0 0 3 3 3 34 4 3, 3...... b, () Plese ote, while clcultig, the vlue of () is replced by, ot by (0).. This is the bsic differece of Guss Seidel with Jocobi s method.
The successive itertio s re geerted by the scheme clled itertio formule of Guss-Seidel method, which is s follows: b... 3 3 b... 3 3... b...,,,, b..., The umber of itertios required depeds upo the desired degree of ccurcy.
Emple 6. Solve the system of lier equtios by Guss Seidel method. 4 9 3 8 3 0 3 4 33 3 Solutio: The give system of equtios is ot digolly row domit s 3. Therefore, we re-rrge the system s 8 3 0 3 4 33 3 4 9 3 Here, 8 3, 4 d 4. Thus, the system is digolly row domit. We ow re-write the system s 0 3 3 8 33 4 3 3 9 4 0 0 0 Let the iitil guess be,, 3 0. The, the first pproimtio to the solutio is give by 0 3 0.875 8 0 33 4 3 33 4.875 0.955 3 9 9.875.955.043 4 4
d pproimtio 0 3 3 0 3.955.043.97 8 8 3 33 4 33 4.97.043.04 3 9 3 9.97.04.004 4 4 3 rd pproimtio 3 0 3 3 0 3.04.004 3.004 8 8 3 3 33 4 3 33 4 3.004.004.999 3 3 3 3 9 9 3.004.999 0.999 4 4 4 th pproimtio 4 4 4 3 3.00.00.00 Therefore, 3.0,.0 d 3.0, correct to two sigifict figure.
Eercises () Use Jcobi s method to solve the followig system of equtios, with 0,, T s iitil pproimtio, correct to sigifict figures. 0y 3z 39 0 y 5z 6 4 5y 0z 47 Wht is the miimum umber of itertios required to get 5 sigifict digit ccurcy, if 5 digit rithmetic is used. (As: True solutio 3, 3, T () Do three itertios of Jcobi s method to solve 3y 0z 0 y z 9 0y z ; umber of itertio required=36) 0 with,, T s strtig vector. Wht is the miimum umber of itertios required, so tht the solutio is correct to 4 deciml plces. (As: True solutio ; umber of itertio required =7),,3 T (3) Solve, by Guss-Seidl itertio method, the system of lier equtios 3 9 y z 4 y 3z 4 4 y z 8 correct up to four sigifict figures. (As:.43, y.3, z.956)
(4) Compute the solutio of the system of lier equtios by Guss-Seidl itertio method 6.7.y.z 0.5 3. 9.4 y.5z.9..5y 8.4z 8.8 correct up to 3-sigifict figures. (As:.50, y.50, z 3.50 ) (5) Do five itertios of ech Jcobi s d Guss Seidel method to solve 3y 7z 6 3 y z 6 5y 3z 0 with strtig iitil guess s (, y, z) = (,,). Wht is the miimum umber of itertios required, so tht the solutios correct to 8 sigifict figures? (As: True solutio:., y 0.8, z.6 )