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13. Relaxation method example ( Enter your problem )
  1. Example `8x+y+z=8,2x+4y+z=4,x+3y+5z=5`
  2. Example `10x-2y-2z=6,-x+10y-2z=7,-x-y+10z=8`
  3. Example `9x-2y+z=50,x+5y-3z=18,-2x+2y+7z=19`
  4. Example `10x-2y-3z=205,-2x+10y-2z=154,-2x-y+10z=120`
Other related methods
  1. Inverse Matrix method
  2. Cramer's Rule method
  3. Gauss-Jordan Elimination method
  4. Gauss Elimination Back Substitution method
  5. Gauss Seidel method
  6. Gauss Jacobi method
  7. Elimination method
  8. LU decomposition using Gauss Elimination method
  9. LU decomposition using Doolittle's method
  10. LU decomposition using Crout's method
  11. Cholesky decomposition method
  12. SOR (Successive over-relaxation) method
  13. Relaxation method

12. SOR (Successive over-relaxation) method
(Previous method)
2. Example `10x-2y-2z=6,-x+10y-2z=7,-x-y+10z=8`
(Next example)

1. Example `8x+y+z=8,2x+4y+z=4,x+3y+5z=5`





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
IterationOperation`deltax`
(8)
`deltay`
(4)
`deltaz`
(5)
`R_1``R_2``R_3`
1`x=y=z=0`000845
2`deltax=8/8=1`100024
3`deltaz=4/5=0.8`000.8-0.81.20
4`deltay=1.2/4=0.3`00.30-1.10-0.9
5`deltax=-1.1/8=-0.1375`-0.13750000.275-0.7625
6`deltaz=-0.7625/5=-0.1525`00-0.15250.15250.42750
7`deltay=0.4275/4=0.1069`00.106900.04560-0.3206
8`deltaz=-0.3206/5=-0.0641`00-0.06410.10980.06410
9`deltax=0.1098/8=0.0137`0.01370000.0367-0.0137
10`deltay=0.0367/4=0.0092`00.00920-0.00920-0.0412
11`deltaz=-0.0412/5=-0.0082`00-0.0082-0.00090.00820
12`deltay=0.0082/4=0.0021`00.00210-0.0030-0.0062
13`deltaz=-0.0062/5=-0.0012`00-0.0012-0.00170.00120
Total0.8760.41810.5739



This material is intended as a summary. Use your textbook for detail explanation.
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12. SOR (Successive over-relaxation) method
(Previous method)
2. Example `10x-2y-2z=6,-x+10y-2z=7,-x-y+10z=8`
(Next example)





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