1. Solve Equations 8x+y+z=8,2x+4y+z=4,x+3y+5z=5 using Relaxation method

Solution:
Total Equations are `3`

`8x+y+z=8`

`2x+4y+z=4`

`x+3y+5z=5`


The residuals from equations, we get
`R_1=8-8x-y-z`

`R_2=4-2x-4y-z`

`R_3=5-x-3y-5z`

The table for operation is

	`R_1`	`R_2`	`R_3`
`deltax`	-8	-2	-1
`deltay`	-1	-4	-3
`deltaz`	-1	-1	-5

Solution steps are
`1^(st)` Approximation

`R_1=8-0=8`

`R_2=4-0=4`

`R_3=5-0=5`

Maximum is `R_1=8`

`deltax=8/8=1`

`2^(nd)` Approximation

`R_1=8-8*1=8-8=0`

`R_2=4-2*1=4-2=2`

`R_3=5-1*1=5-1=4`

Maximum is `R_3=4`

`deltaz=4/5=0.8`

`3^(rd)` Approximation

`R_1=0-1*0.8=0-0.8=-0.8`

`R_2=2-1*0.8=2-0.8=1.2`

`R_3=4-5*0.8=4-4=0`

Maximum is `R_2=1.2`

`deltay=1.2/4=0.3`

`4^(th)` Approximation

`R_1=-0.8-1*0.3=-0.8-0.3=-1.1`

`R_2=1.2-4*0.3=1.2-1.2=0`

`R_3=0-3*0.3=0-0.9=-0.9`

Maximum is `R_1=-1.1`

`deltax=-1.1/8=-0.1375`

`5^(th)` Approximation

`R_1=-1.1-8*-0.1375=-1.1--1.1=0`

`R_2=0-2*-0.1375=0--0.275=0.275`

`R_3=-0.9-1*-0.1375=-0.9--0.1375=-0.7625`

Maximum is `R_3=-0.7625`

`deltaz=-0.7625/5=-0.1525`

`6^(th)` Approximation

`R_1=0-1*-0.1525=0--0.1525=0.1525`

`R_2=0.275-1*-0.1525=0.275--0.1525=0.4275`

`R_3=-0.7625-5*-0.1525=-0.7625--0.7625=0`

Maximum is `R_2=0.4275`

`deltay=0.4275/4=0.1069`

`7^(th)` Approximation

`R_1=0.1525-1*0.1069=0.1525-0.1069=0.0456`

`R_2=0.4275-4*0.1069=0.4275-0.4275=0`

`R_3=0-3*0.1069=0-0.3206=-0.3206`

Maximum is `R_3=-0.3206`

`deltaz=-0.3206/5=-0.0641`

`8^(th)` Approximation

`R_1=0.0456-1*-0.0641=0.0456--0.0641=0.1098`

`R_2=0-1*-0.0641=0--0.0641=0.0641`

`R_3=-0.3206-5*-0.0641=-0.3206--0.3206=0`

Maximum is `R_1=0.1098`

`deltax=0.1098/8=0.0137`

`9^(th)` Approximation

`R_1=0.1098-8*0.0137=0.1098-0.1098=0`

`R_2=0.0641-2*0.0137=0.0641-0.0274=0.0367`

`R_3=0-1*0.0137=0-0.0137=-0.0137`

Maximum is `R_2=0.0367`

`deltay=0.0367/4=0.0092`

`10^(th)` Approximation

`R_1=0-1*0.0092=0-0.0092=-0.0092`

`R_2=0.0367-4*0.0092=0.0367-0.0367=0`

`R_3=-0.0137-3*0.0092=-0.0137-0.0275=-0.0412`

Maximum is `R_3=-0.0412`

`deltaz=-0.0412/5=-0.0082`

`11^(th)` Approximation

`R_1=-0.0092-1*-0.0082=-0.0092--0.0082=-0.0009`

`R_2=0-1*-0.0082=0--0.0082=0.0082`

`R_3=-0.0412-5*-0.0082=-0.0412--0.0412=0`

Maximum is `R_2=0.0082`

`deltay=0.0082/4=0.0021`

`12^(th)` Approximation

`R_1=-0.0009-1*0.0021=-0.0009-0.0021=-0.003`

`R_2=0.0082-4*0.0021=0.0082-0.0082=0`

`R_3=0-3*0.0021=0-0.0062=-0.0062`

Maximum is `R_3=-0.0062`

`deltaz=-0.0062/5=-0.0012`

`13^(th)` Approximation

`R_1=-0.003-1*-0.0012=-0.003--0.0012=-0.0017`

`R_2=0-1*-0.0012=0--0.0012=0.0012`

`R_3=-0.0062-5*-0.0012=-0.0062--0.0062=0`

Maximum is `R_1=-0.0017`

`deltax=-0.0017/8=-0.0002`


Solution By Relaxation Method.
`x=sum deltax=0.876~=0.88`

`y=sum deltay=0.4181~=0.42`

`z=sum deltaz=0.5739~=0.57`

Iterations are tabulated as below

Iteration	Operation	`deltax` (8)	`deltay` (4)	`deltaz` (5)	`R_1`	`R_2`	`R_3`
1	`x=y=z=0`	0	0	0	8	4	5
2	`deltax=8/8=1`	1	0	0	0	2	4
3	`deltaz=4/5=0.8`	0	0	0.8	-0.8	1.2	0
4	`deltay=1.2/4=0.3`	0	0.3	0	-1.1	0	-0.9
5	`deltax=-1.1/8=-0.1375`	-0.1375	0	0	0	0.275	-0.7625
6	`deltaz=-0.7625/5=-0.1525`	0	0	-0.1525	0.1525	0.4275	0
7	`deltay=0.4275/4=0.1069`	0	0.1069	0	0.0456	0	-0.3206
8	`deltaz=-0.3206/5=-0.0641`	0	0	-0.0641	0.1098	0.0641	0
9	`deltax=0.1098/8=0.0137`	0.0137	0	0	0	0.0367	-0.0137
10	`deltay=0.0367/4=0.0092`	0	0.0092	0	-0.0092	0	-0.0412
11	`deltaz=-0.0412/5=-0.0082`	0	0	-0.0082	-0.0009	0.0082	0
12	`deltay=0.0082/4=0.0021`	0	0.0021	0	-0.003	0	-0.0062
13	`deltaz=-0.0062/5=-0.0012`	0	0	-0.0012	-0.0017	0.0012	0
	Total	0.876	0.4181	0.5739

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