Home > Matrix & Vector calculators > Solving systems of linear equations using Relaxation method example

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

2. Example `10x-2y-2z=6,-x+10y-2z=7,-x-y+10z=8`
(Previous example)
4. Example `10x-2y-3z=205,-2x+10y-2z=154,-2x-y+10z=120`
(Next example)

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





Solve Equations 9x-2y+z=50,x+5y-3z=18,-2x+2y+7z=19 using Relaxation method

Solution:
Total Equations are `3`

`9x-2y+z=50`

`x+5y-3z=18`

`-2x+2y+7z=19`


The residuals from equations, we get
`R_1=50-9x+2y-z`

`R_2=18-x-5y+3z`

`R_3=19+2x-2y-7z`

The table for operation is
`R_1``R_2``R_3`
`deltax`-9-12
`deltay`2-5-2
`deltaz`-13-7


Solution steps are
`1^(st)` Approximation

`R_1=50-0=50`

`R_2=18-0=18`

`R_3=19-0=19`

Maximum is `R_1=50`

`deltax=50/9=5.5556`

`2^(nd)` Approximation

`R_1=50-9*5.5556=50-50=0`

`R_2=18-1*5.5556=18-5.5556=12.4444`

`R_3=19-(-2)*5.5556=19--11.1111=30.1111`

Maximum is `R_3=30.1111`

`deltaz=30.1111/7=4.3016`

`3^(rd)` Approximation

`R_1=0-1*4.3016=0-4.3016=-4.3016`

`R_2=12.4444-(-3)*4.3016=12.4444--12.9048=25.3492`

`R_3=30.1111-7*4.3016=30.1111-30.1111=0`

Maximum is `R_2=25.3492`

`deltay=25.3492/5=5.0698`

`4^(th)` Approximation

`R_1=-4.3016-(-2)*5.0698=-4.3016--10.1397=5.8381`

`R_2=25.3492-5*5.0698=25.3492-25.3492=0`

`R_3=0-2*5.0698=0-10.1397=-10.1397`

Maximum is `R_3=-10.1397`

`deltaz=-10.1397/7=-1.4485`

`5^(th)` Approximation

`R_1=5.8381-1*-1.4485=5.8381--1.4485=7.2866`

`R_2=0-(-3)*-1.4485=0-4.3456=-4.3456`

`R_3=-10.1397-7*-1.4485=-10.1397--10.1397=0`

Maximum is `R_1=7.2866`

`deltax=7.2866/9=0.8096`

`6^(th)` Approximation

`R_1=7.2866-9*0.8096=7.2866-7.2866=0`

`R_2=-4.3456-1*0.8096=-4.3456-0.8096=-5.1552`

`R_3=0-(-2)*0.8096=0--1.6192=1.6192`

Maximum is `R_2=-5.1552`

`deltay=-5.1552/5=-1.031`

`7^(th)` Approximation

`R_1=0-(-2)*-1.031=0-2.0621=-2.0621`

`R_2=-5.1552-5*-1.031=-5.1552--5.1552=0`

`R_3=1.6192-2*-1.031=1.6192--2.0621=3.6813`

Maximum is `R_3=3.6813`

`deltaz=3.6813/7=0.5259`

`8^(th)` Approximation

`R_1=-2.0621-1*0.5259=-2.0621-0.5259=-2.588`

`R_2=0-(-3)*0.5259=0--1.5777=1.5777`

`R_3=3.6813-7*0.5259=3.6813-3.6813=0`

Maximum is `R_1=-2.588`

`deltax=-2.588/9=-0.2876`

`9^(th)` Approximation

`R_1=-2.588-9*-0.2876=-2.588--2.588=0`

`R_2=1.5777-1*-0.2876=1.5777--0.2876=1.8653`

`R_3=0-(-2)*-0.2876=0-0.5751=-0.5751`

Maximum is `R_2=1.8653`

`deltay=1.8653/5=0.3731`

`10^(th)` Approximation

`R_1=0-(-2)*0.3731=0--0.7461=0.7461`

`R_2=1.8653-5*0.3731=1.8653-1.8653=0`

`R_3=-0.5751-2*0.3731=-0.5751-0.7461=-1.3212`

Maximum is `R_3=-1.3212`

`deltaz=-1.3212/7=-0.1887`

`11^(th)` Approximation

`R_1=0.7461-1*-0.1887=0.7461--0.1887=0.9349`

`R_2=0-(-3)*-0.1887=0-0.5662=-0.5662`

`R_3=-1.3212-7*-0.1887=-1.3212--1.3212=0`

Maximum is `R_1=0.9349`

`deltax=0.9349/9=0.1039`

`12^(th)` Approximation

`R_1=0.9349-9*0.1039=0.9349-0.9349=0`

`R_2=-0.5662-1*0.1039=-0.5662-0.1039=-0.6701`

`R_3=0-(-2)*0.1039=0--0.2077=0.2077`

Maximum is `R_2=-0.6701`

`deltay=-0.6701/5=-0.134`

`13^(th)` Approximation

`R_1=0-(-2)*-0.134=0-0.268=-0.268`

`R_2=-0.6701-5*-0.134=-0.6701--0.6701=0`

`R_3=0.2077-2*-0.134=0.2077--0.268=0.4758`

Maximum is `R_3=0.4758`

`deltaz=0.4758/7=0.068`

`14^(th)` Approximation

`R_1=-0.268-1*0.068=-0.268-0.068=-0.336`

`R_2=0-(-3)*0.068=0--0.2039=0.2039`

`R_3=0.4758-7*0.068=0.4758-0.4758=0`

Maximum is `R_1=-0.336`

`deltax=-0.336/9=-0.0373`

`15^(th)` Approximation

`R_1=-0.336-9*-0.0373=-0.336--0.336=0`

`R_2=0.2039-1*-0.0373=0.2039--0.0373=0.2412`

`R_3=0-(-2)*-0.0373=0-0.0747=-0.0747`

Maximum is `R_2=0.2412`

`deltay=0.2412/5=0.0482`

`16^(th)` Approximation

`R_1=0-(-2)*0.0482=0--0.0965=0.0965`

`R_2=0.2412-5*0.0482=0.2412-0.2412=0`

`R_3=-0.0747-2*0.0482=-0.0747-0.0965=-0.1712`

Maximum is `R_3=-0.1712`

`deltaz=-0.1712/7=-0.0245`

`17^(th)` Approximation

`R_1=0.0965-1*-0.0245=0.0965--0.0245=0.1209`

`R_2=0-(-3)*-0.0245=0-0.0734=-0.0734`

`R_3=-0.1712-7*-0.0245=-0.1712--0.1712=0`

Maximum is `R_1=0.1209`

`deltax=0.1209/9=0.0134`

`18^(th)` Approximation

`R_1=0.1209-9*0.0134=0.1209-0.1209=0`

`R_2=-0.0734-1*0.0134=-0.0734-0.0134=-0.0868`

`R_3=0-(-2)*0.0134=0--0.0269=0.0269`

Maximum is `R_2=-0.0868`

`deltay=-0.0868/5=-0.0174`

`19^(th)` Approximation

`R_1=0-(-2)*-0.0174=0-0.0347=-0.0347`

`R_2=-0.0868-5*-0.0174=-0.0868--0.0868=0`

`R_3=0.0269-2*-0.0174=0.0269--0.0347=0.0616`

Maximum is `R_3=0.0616`

`deltaz=0.0616/7=0.0088`

`20^(th)` Approximation

`R_1=-0.0347-1*0.0088=-0.0347-0.0088=-0.0435`

`R_2=0-(-3)*0.0088=0--0.0264=0.0264`

`R_3=0.0616-7*0.0088=0.0616-0.0616=0`

Maximum is `R_1=-0.0435`

`deltax=-0.0435/9=-0.0048`


Solution By Relaxation Method.
`x=sum deltax=6.1528~=6.15`

`y=sum deltay=4.3087~=4.31`

`z=sum deltaz=3.2425~=3.24`

Iterations are tabulated as below
IterationOperation`deltax`
(9)
`deltay`
(5)
`deltaz`
(7)
`R_1``R_2``R_3`
1`x=y=z=0`000501819
2`deltax=50/9=5.5556`5.555600012.444430.1111
3`deltaz=30.1111/7=4.3016`004.3016-4.301625.34920
4`deltay=25.3492/5=5.0698`05.069805.83810-10.1397
5`deltaz=-10.1397/7=-1.4485`00-1.44857.2866-4.34560
6`deltax=7.2866/9=0.8096`0.8096000-5.15521.6192
7`deltay=-5.1552/5=-1.031`0-1.0310-2.062103.6813
8`deltaz=3.6813/7=0.5259`000.5259-2.5881.57770
9`deltax=-2.588/9=-0.2876`-0.28760001.8653-0.5751
10`deltay=1.8653/5=0.3731`00.373100.74610-1.3212
11`deltaz=-1.3212/7=-0.1887`00-0.18870.9349-0.56620
12`deltax=0.9349/9=0.1039`0.1039000-0.67010.2077
13`deltay=-0.6701/5=-0.134`0-0.1340-0.26800.4758
14`deltaz=0.4758/7=0.068`000.068-0.3360.20390
15`deltax=-0.336/9=-0.0373`-0.03730000.2412-0.0747
16`deltay=0.2412/5=0.0482`00.048200.09650-0.1712
17`deltaz=-0.1712/7=-0.0245`00-0.02450.1209-0.07340
18`deltax=0.1209/9=0.0134`0.0134000-0.08680.0269
19`deltay=-0.0868/5=-0.0174`0-0.01740-0.034700.0616
20`deltaz=0.0616/7=0.0088`000.0088-0.04350.02640
Total6.15284.30873.2425



This material is intended as a summary. Use your textbook for detail explanation.
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2. Example `10x-2y-2z=6,-x+10y-2z=7,-x-y+10z=8`
(Previous example)
4. Example `10x-2y-3z=205,-2x+10y-2z=154,-2x-y+10z=120`
(Next example)





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