Solve Equations 10x-2y-2z=6,-x+10y-2z=7,-x-y+10z=8 using Relaxation method

Solution:
Total Equations are `3`

`10x-2y-2z=6`

`-x+10y-2z=7`

`-x-y+10z=8`


The residuals from equations, we get
`R_1=6-10x+2y+2z`

`R_2=7+x-10y+2z`

`R_3=8+x+y-10z`

The table for operation is

	`R_1`	`R_2`	`R_3`
`deltax`	-10	1	1
`deltay`	2	-10	1
`deltaz`	2	2	-10

Solution steps are
`1^(st)` Approximation

`R_1=6-0=6`

`R_2=7-0=7`

`R_3=8-0=8`

Maximum is `R_3=8`

`deltaz=8/10=0.8`

`2^(nd)` Approximation

`R_1=6-(-2)*0.8=6--1.6=7.6`

`R_2=7-(-2)*0.8=7--1.6=8.6`

`R_3=8-10*0.8=8-8=0`

Maximum is `R_2=8.6`

`deltay=8.6/10=0.86`

`3^(rd)` Approximation

`R_1=7.6-(-2)*0.86=7.6--1.72=9.32`

`R_2=8.6-10*0.86=8.6-8.6=0`

`R_3=0-(-1)*0.86=0--0.86=0.86`

Maximum is `R_1=9.32`

`deltax=9.32/10=0.932`

`4^(th)` Approximation

`R_1=9.32-10*0.932=9.32-9.32=0`

`R_2=0-(-1)*0.932=0--0.932=0.932`

`R_3=0.86-(-1)*0.932=0.86--0.932=1.792`

Maximum is `R_3=1.792`

`deltaz=1.792/10=0.1792`

`5^(th)` Approximation

`R_1=0-(-2)*0.1792=0--0.3584=0.3584`

`R_2=0.932-(-2)*0.1792=0.932--0.3584=1.2904`

`R_3=1.792-10*0.1792=1.792-1.792=0`

Maximum is `R_2=1.2904`

`deltay=1.2904/10=0.129`

`6^(th)` Approximation

`R_1=0.3584-(-2)*0.129=0.3584--0.2581=0.6165`

`R_2=1.2904-10*0.129=1.2904-1.2904=0`

`R_3=0-(-1)*0.129=0--0.129=0.129`

Maximum is `R_1=0.6165`

`deltax=0.6165/10=0.0616`

`7^(th)` Approximation

`R_1=0.6165-10*0.0616=0.6165-0.6165=0`

`R_2=0-(-1)*0.0616=0--0.0616=0.0616`

`R_3=0.129-(-1)*0.0616=0.129--0.0616=0.1907`

Maximum is `R_3=0.1907`

`deltaz=0.1907/10=0.0191`

`8^(th)` Approximation

`R_1=0-(-2)*0.0191=0--0.0381=0.0381`

`R_2=0.0616-(-2)*0.0191=0.0616--0.0381=0.0998`

`R_3=0.1907-10*0.0191=0.1907-0.1907=0`

Maximum is `R_2=0.0998`

`deltay=0.0998/10=0.01`

`9^(th)` Approximation

`R_1=0.0381-(-2)*0.01=0.0381--0.02=0.0581`

`R_2=0.0998-10*0.01=0.0998-0.0998=0`

`R_3=0-(-1)*0.01=0--0.01=0.01`

Maximum is `R_1=0.0581`

`deltax=0.0581/10=0.0058`

`10^(th)` Approximation

`R_1=0.0581-10*0.0058=0.0581-0.0581=0`

`R_2=0-(-1)*0.0058=0--0.0058=0.0058`

`R_3=0.01-(-1)*0.0058=0.01--0.0058=0.0158`

Maximum is `R_3=0.0158`

`deltaz=0.0158/10=0.0016`

`11^(th)` Approximation

`R_1=0-(-2)*0.0016=0--0.0032=0.0032`

`R_2=0.0058-(-2)*0.0016=0.0058--0.0032=0.009`

`R_3=0.0158-10*0.0016=0.0158-0.0158=0`

Maximum is `R_2=0.009`

`deltay=0.009/10=0.0009`

`12^(th)` Approximation

`R_1=0.0032-(-2)*0.0009=0.0032--0.0018=0.005`

`R_2=0.009-10*0.0009=0.009-0.009=0`

`R_3=0-(-1)*0.0009=0--0.0009=0.0009`

Maximum is `R_1=0.005`

`deltax=0.005/10=0.0005`

`13^(th)` Approximation

`R_1=0.005-10*0.0005=0.005-0.005=0`

`R_2=0-(-1)*0.0005=0--0.0005=0.0005`

`R_3=0.0009-(-1)*0.0005=0.0009--0.0005=0.0014`

Maximum is `R_3=0.0014`

`deltaz=0.0014/10=0.0001`


Solution By Relaxation Method.
`x=sum deltax=1~=1`

`y=sum deltay=0.9999~=1`

`z=sum deltaz=1~=1`

Iterations are tabulated as below

Iteration	Operation	`deltax` (10)	`deltay` (10)	`deltaz` (10)	`R_1`	`R_2`	`R_3`
1	`x=y=z=0`	0	0	0	6	7	8
2	`deltaz=8/10=0.8`	0	0	0.8	7.6	8.6	0
3	`deltay=8.6/10=0.86`	0	0.86	0	9.32	0	0.86
4	`deltax=9.32/10=0.932`	0.932	0	0	0	0.932	1.792
5	`deltaz=1.792/10=0.1792`	0	0	0.1792	0.3584	1.2904	0
6	`deltay=1.2904/10=0.129`	0	0.129	0	0.6165	0	0.129
7	`deltax=0.6165/10=0.0616`	0.0616	0	0	0	0.0616	0.1907
8	`deltaz=0.1907/10=0.0191`	0	0	0.0191	0.0381	0.0998	0
9	`deltay=0.0998/10=0.01`	0	0.01	0	0.0581	0	0.01
10	`deltax=0.0581/10=0.0058`	0.0058	0	0	0	0.0058	0.0158
11	`deltaz=0.0158/10=0.0016`	0	0	0.0016	0.0032	0.009	0
12	`deltay=0.009/10=0.0009`	0	0.0009	0	0.005	0	0.0009
13	`deltax=0.005/10=0.0005`	0.0005	0	0	0	0.0005	0.0014
	Total	1	0.9999	1

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