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 | | | |
This material is intended as a summary. Use your textbook for detail explanation.
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