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