The Relaxation Method of Solving Simultaneous Linear Equations

The method of relaxation is used to solve systems of linear equations by iteration. Given the system of equations
\[3x+9y-2z=11\]

\[4x+2y+13z=24\]

\[11x-4y+3z=-8\]

Write the equations as
\[3x+9y-2z-11=0\]

\[4x+2y+13z-24=0\]

\[11x-4y+3z+8=0\]

and let  
\[R_1=3x+9y-2z-11\]
,
\[R_2=4x+2y+13z-24\]
  and  
\[R_3=11x-4y+3z+8\]

Now construct the tableau.
\[\Delta x\]
\[\Delta y\]
\[\Delta z\]
\[\Delta R_1\]
\[\Delta R_2\]
\[\Delta R_3\]
1003411
01092-4
001-2133
The  
\[\Delta\]
  symbols indicate an increment so increasing  
\[x\]
  by 1 increases  
\[R_1\]
  by 3.
We take initial solution  
\[x=y=z=0\]
  then  
\[R_1=11, \: R_2=-24 \: R_3=8\]
. We aim for each  
\[R\]
  to equal 0.
If we make  
\[R_1=0\]
  we obtain the tableau below.
\[x=0\]
\[y=0\]
\[z=0\]
\[R_1=-11\]
\[R_2=-24\]
\[R_3=8\]
002-15214
-100-18-23
020-02-5
x=-1y=2z=202-5
To understand this, observe that a change of 1 in  
\[z\]
  produces of -2 in  
\[R_1\]
.
When  
\[z\]
  goes from 0 to 2,  
\[R_1\]
  goes from -11 to -15.A change of 1 in  
\[z\]
  produces a change of 13 in  
\[R_2 \]
  so when  
\[z\]
  changes from 0 to 2,  
\[R_2\]
  changes from -24 to 2. When  
\[x\]
  changes from 0 to -1,  
\[R_1\]
  changes by to -3. Hence  
\[R_1\]
  is now -18. When  
\[y\]
  changes by 2,  
\[R_1\]
  changes by 18 and is now 0. The other columns are similarly changed.
In the next tableau, multiply the top row by 10 - in order to avoid decimals/fractions - to obtain
\[-10\]
\[20\]
\[20\]
\[0\]
\[20\]
\[-50\]
50015405
00-3211-4
0-203-34
x=-5y=18z=173-34
The solution is now  
\[10x=-5, \: 10y=18, \: 10z=17 \rightarrow x=-0.5, \: y=1.8, \: z=1.7\]
.
To obtain a second decimal place multiply the top row - containing the solutions - by 100.
\[-50\]
\[180\]
\[170\]
\[30\]
\[-30\]
\[40\]
-400-18-46-4
004-1068
010-184
00-11-51
-541811731-51
The solution is now  
\[100x=-54, \: 100y=181, \: 100z=173 \rightarrow x=-0.54, \: y=1.81, \: z=1.73\]
.
Continuing in this way gives the solution to 5 decimal place  
\[x=-0.55004, \: y=1.79216, \: z=1.73968\]
.

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