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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

3. Example `9x-2y+z=50,x+5y-3z=18,-2x+2y+7z=19`
(Previous example)

4. Example `10x-2y-3z=205,-2x+10y-2z=154,-2x-y+10z=120`





Solve Equations 10x-2y-3z=205,-2x+10y-2z=154,-2x-y+10z=120 using Relaxation method

Solution:
Total Equations are `3`

`10x-2y-3z=205`

`-2x+10y-2z=154`

`-2x-y+10z=120`


The residuals from equations, we get
`R_1=205-10x+2y+3z`

`R_2=154+2x-10y+2z`

`R_3=120+2x+y-10z`

The table for operation is
`R_1``R_2``R_3`
`deltax`-1022
`deltay`2-101
`deltaz`32-10


Solution steps are
`1^(st)` Approximation

`R_1=205-0=205`

`R_2=154-0=154`

`R_3=120-0=120`

Maximum is `R_1=205`

`deltax=205/10=20.5`

`2^(nd)` Approximation

`R_1=205-10*20.5=205-205=0`

`R_2=154-(-2)*20.5=154--41=195`

`R_3=120-(-2)*20.5=120--41=161`

Maximum is `R_2=195`

`deltay=195/10=19.5`

`3^(rd)` Approximation

`R_1=0-(-2)*19.5=0--39=39`

`R_2=195-10*19.5=195-195=0`

`R_3=161-(-1)*19.5=161--19.5=180.5`

Maximum is `R_3=180.5`

`deltaz=180.5/10=18.05`

`4^(th)` Approximation

`R_1=39-(-3)*18.05=39--54.15=93.15`

`R_2=0-(-2)*18.05=0--36.1=36.1`

`R_3=180.5-10*18.05=180.5-180.5=0`

Maximum is `R_1=93.15`

`deltax=93.15/10=9.315`

`5^(th)` Approximation

`R_1=93.15-10*9.315=93.15-93.15=0`

`R_2=36.1-(-2)*9.315=36.1--18.63=54.73`

`R_3=0-(-2)*9.315=0--18.63=18.63`

Maximum is `R_2=54.73`

`deltay=54.73/10=5.473`

`6^(th)` Approximation

`R_1=0-(-2)*5.473=0--10.946=10.946`

`R_2=54.73-10*5.473=54.73-54.73=0`

`R_3=18.63-(-1)*5.473=18.63--5.473=24.103`

Maximum is `R_3=24.103`

`deltaz=24.103/10=2.4103`

`7^(th)` Approximation

`R_1=10.946-(-3)*2.4103=10.946--7.2309=18.1769`

`R_2=0-(-2)*2.4103=0--4.8206=4.8206`

`R_3=24.103-10*2.4103=24.103-24.103=0`

Maximum is `R_1=18.1769`

`deltax=18.1769/10=1.8177`

`8^(th)` Approximation

`R_1=18.1769-10*1.8177=18.1769-18.1769=0`

`R_2=4.8206-(-2)*1.8177=4.8206--3.6354=8.456`

`R_3=0-(-2)*1.8177=0--3.6354=3.6354`

Maximum is `R_2=8.456`

`deltay=8.456/10=0.8456`

`9^(th)` Approximation

`R_1=0-(-2)*0.8456=0--1.6912=1.6912`

`R_2=8.456-10*0.8456=8.456-8.456=0`

`R_3=3.6354-(-1)*0.8456=3.6354--0.8456=4.481`

Maximum is `R_3=4.481`

`deltaz=4.481/10=0.4481`

`10^(th)` Approximation

`R_1=1.6912-(-3)*0.4481=1.6912--1.3443=3.0355`

`R_2=0-(-2)*0.4481=0--0.8962=0.8962`

`R_3=4.481-10*0.4481=4.481-4.481=0`

Maximum is `R_1=3.0355`

`deltax=3.0355/10=0.3035`

`11^(th)` Approximation

`R_1=3.0355-10*0.3035=3.0355-3.0355=0`

`R_2=0.8962-(-2)*0.3035=0.8962--0.6071=1.5033`

`R_3=0-(-2)*0.3035=0--0.6071=0.6071`

Maximum is `R_2=1.5033`

`deltay=1.5033/10=0.1503`

`12^(th)` Approximation

`R_1=0-(-2)*0.1503=0--0.3007=0.3007`

`R_2=1.5033-10*0.1503=1.5033-1.5033=0`

`R_3=0.6071-(-1)*0.1503=0.6071--0.1503=0.7574`

Maximum is `R_3=0.7574`

`deltaz=0.7574/10=0.0757`

`13^(th)` Approximation

`R_1=0.3007-(-3)*0.0757=0.3007--0.2272=0.5279`

`R_2=0-(-2)*0.0757=0--0.1515=0.1515`

`R_3=0.7574-10*0.0757=0.7574-0.7574=0`

Maximum is `R_1=0.5279`

`deltax=0.5279/10=0.0528`

`14^(th)` Approximation

`R_1=0.5279-10*0.0528=0.5279-0.5279=0`

`R_2=0.1515-(-2)*0.0528=0.1515--0.1056=0.2571`

`R_3=0-(-2)*0.0528=0--0.1056=0.1056`

Maximum is `R_2=0.2571`

`deltay=0.2571/10=0.0257`

`15^(th)` Approximation

`R_1=0-(-2)*0.0257=0--0.0514=0.0514`

`R_2=0.2571-10*0.0257=0.2571-0.2571=0`

`R_3=0.1056-(-1)*0.0257=0.1056--0.0257=0.1313`

Maximum is `R_3=0.1313`

`deltaz=0.1313/10=0.0131`

`16^(th)` Approximation

`R_1=0.0514-(-3)*0.0131=0.0514--0.0394=0.0908`

`R_2=0-(-2)*0.0131=0--0.0263=0.0263`

`R_3=0.1313-10*0.0131=0.1313-0.1313=0`

Maximum is `R_1=0.0908`

`deltax=0.0908/10=0.0091`

`17^(th)` Approximation

`R_1=0.0908-10*0.0091=0.0908-0.0908=0`

`R_2=0.0263-(-2)*0.0091=0.0263--0.0182=0.0444`

`R_3=0-(-2)*0.0091=0--0.0182=0.0182`

Maximum is `R_2=0.0444`

`deltay=0.0444/10=0.0044`

`18^(th)` Approximation

`R_1=0-(-2)*0.0044=0--0.0089=0.0089`

`R_2=0.0444-10*0.0044=0.0444-0.0444=0`

`R_3=0.0182-(-1)*0.0044=0.0182--0.0044=0.0226`

Maximum is `R_3=0.0226`

`deltaz=0.0226/10=0.0023`

`19^(th)` Approximation

`R_1=0.0089-(-3)*0.0023=0.0089--0.0068=0.0157`

`R_2=0-(-2)*0.0023=0--0.0045=0.0045`

`R_3=0.0226-10*0.0023=0.0226-0.0226=0`

Maximum is `R_1=0.0157`

`deltax=0.0157/10=0.0016`

`20^(th)` Approximation

`R_1=0.0157-10*0.0016=0.0157-0.0157=0`

`R_2=0.0045-(-2)*0.0016=0.0045--0.0031=0.0077`

`R_3=0-(-2)*0.0016=0--0.0031=0.0031`

Maximum is `R_2=0.0077`

`deltay=0.0077/10=0.0008`


Solution By Relaxation Method.
`x=sum deltax=31.9997~=32`

`y=sum deltay=25.9998~=26`

`z=sum deltaz=20.9995~=21`

Iterations are tabulated as below
IterationOperation`deltax`
(10)
`deltay`
(10)
`deltaz`
(10)
`R_1``R_2``R_3`
1`x=y=z=0`000205154120
2`deltax=205/10=20.5`20.5000195161
3`deltay=195/10=19.5`019.50390180.5
4`deltaz=180.5/10=18.05`0018.0593.1536.10
5`deltax=93.15/10=9.315`9.31500054.7318.63
6`deltay=54.73/10=5.473`05.473010.946024.103
7`deltaz=24.103/10=2.4103`002.410318.17694.82060
8`deltax=18.1769/10=1.8177`1.81770008.4563.6354
9`deltay=8.456/10=0.8456`00.845601.691204.481
10`deltaz=4.481/10=0.4481`000.44813.03550.89620
11`deltax=3.0355/10=0.3035`0.30350001.50330.6071
12`deltay=1.5033/10=0.1503`00.150300.300700.7574
13`deltaz=0.7574/10=0.0757`000.07570.52790.15150
14`deltax=0.5279/10=0.0528`0.05280000.25710.1056
15`deltay=0.2571/10=0.0257`00.025700.051400.1313
16`deltaz=0.1313/10=0.0131`000.01310.09080.02630
17`deltax=0.0908/10=0.0091`0.00910000.04440.0182
18`deltay=0.0444/10=0.0044`00.004400.008900.0226
19`deltaz=0.0226/10=0.0023`000.00230.01570.00450
20`deltax=0.0157/10=0.0016`0.00160000.00770.0031
Total31.999725.999820.9995



This material is intended as a summary. Use your textbook for detail explanation.
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3. Example `9x-2y+z=50,x+5y-3z=18,-2x+2y+7z=19`
(Previous example)





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