This presentation is about the successive over relaxation method. (SOR Method). It is the proof of the Sor Method.

1. Numerical Analysis
Method of Successive Over Relaxation
SOR Method

2. Formula: 𝑥𝑖
𝑘+1
= 1 − 𝑤 𝑥𝑖
𝑘
+
𝑤
𝑎 𝑖𝑖
{𝑏 𝑛 − 𝑗=1
𝑖−1
𝑎𝑖𝑗 𝑥𝑖
𝑘+1
− 𝑗=𝑖+1
𝑛
𝑎𝑖𝑗 𝑥𝑗
𝑘
}
To Prove the Formula of Succesive over relaxation (SOR).

3. Proof: We prove this formula from Gauss
seidal Formula which implies:
𝑥𝑖
𝑘+1
=
1
𝑎𝑖𝑖
{𝑏𝑖 −
𝑗=1
𝑖−1
𝑎𝑖𝑗 𝑥𝑗
𝑘+1
−
𝑗=𝑖+1
𝑛
𝑎𝑖𝑗 𝑥𝑗
𝑘
}

4. 𝑥𝑖
𝑘+1
= 𝑥𝑖
𝑘
+
1
𝑎𝑖𝑖
{𝑏 𝑛 −
𝑗=1
𝑖−1
𝑎𝑖𝑗 𝑥𝑖
𝑘+1
−
𝑗=𝑖+1
𝑛
𝑎𝑖𝑗 𝑥𝑗
𝑘
− 𝑎𝑖𝑖 𝑥𝑖
𝑘
}
Now Add and Subtract 𝑥𝑖
𝑘

5. 𝑥𝑖
𝑘+1 = 𝑤𝑥𝑖
𝑘 +
𝑤
𝑎𝑖𝑖
{𝑏 𝑛 −
𝑗=1
𝑖−1
𝑎𝑖𝑗 𝑥𝑖
𝑘+1 −
𝑗=𝑖+1
𝑛
𝑎𝑖𝑗 𝑥𝑗
𝑘 − 𝑎𝑖𝑖 𝑥𝑖
𝑘}
Now Multiplying the Equation by “W”

6. 𝑥𝑖
𝑘+1
= 𝑤𝑥𝑖
𝑘
+
𝑤
𝑎 𝑖𝑖
{−𝑎𝑖𝑖 𝑥𝑖
𝑘}{𝑏 𝑛 − 𝑗=1
𝑖−1
𝑎𝑖𝑗 𝑥𝑖
𝑘+1 − 𝑗=𝑖+1
𝑛
𝑎𝑖𝑗 𝑥𝑗
𝑘}
Now By further Steps We can write it as

7. 𝑥𝑖
𝑘+1
= 𝑥𝑖
𝑘
− 𝑤𝑥𝑖
𝑘
+
𝑤
𝑎 𝑖𝑖
{𝑏𝑖 − 𝑗=1
𝑖−1
𝑎𝑖𝑗 𝑥𝑗
𝑘+1
− 𝑗=𝑖+1
𝑛
𝑎𝑖𝑗 𝑥𝑗
𝑘
}
Now taking 𝑥𝑖
𝑘
as common
𝑥𝑖
𝑘+1
= 1 − 𝑤 𝑥𝑖
𝑘
+
𝑤
𝑎𝑖𝑖
{𝑏𝑖 −
𝑗=1
𝑖−1
𝑎𝑖𝑗 𝑥𝑗
𝑘+1
−
𝑗=𝑖+1
𝑛
𝑎𝑖𝑗 𝑥𝑗
𝑘
}
Proved.

8. 𝐻𝑒𝑟𝑒 "𝑤" 𝑖𝑠 𝑡ℎ𝑒 𝑅𝑒𝑙𝑎𝑥𝑎𝑡𝑖𝑜𝑛 𝑃𝑒𝑟𝑎𝑚𝑒𝑡𝑒𝑟.
𝑖𝑓 𝑤 = 1 𝑡ℎ𝑒𝑛 𝑡ℎ𝑖𝑠 𝑚𝑒𝑡ℎ𝑜𝑑 𝑖𝑠 𝑟𝑒𝑑𝑢𝑐𝑒𝑑 𝑡𝑜 𝐺𝑎𝑢𝑠𝑠 𝑆𝑒𝑖𝑑𝑎𝑙 𝑀𝑒𝑡ℎ𝑜𝑑.
𝑖𝑓 𝑤 > 1 𝑡ℎ𝑒𝑛 𝑖𝑡 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑜𝑣𝑒𝑟 𝑅𝑒𝑙𝑎𝑥𝑎𝑡𝑖𝑜𝑛.
𝑖𝑓 𝑤 < 1 𝑡ℎ𝑒𝑛 𝑖𝑡 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑈𝑛𝑑𝑒𝑟 𝑅𝑒𝑙𝑎𝑥𝑎𝑡𝑖𝑜𝑛.
𝑇ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 "𝑤" 𝑙𝑖𝑒𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 0 < 𝑤 < 2.
𝑖𝑓 "𝑤"𝑖𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 2 𝑡ℎ𝑒𝑛 𝑡ℎ𝑒 𝑚𝑒𝑡ℎ𝑜𝑑 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑠.
𝑤𝑒 𝑐𝑎𝑛 𝑓𝑖𝑛𝑑 𝑊 𝑏𝑦 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚𝑢𝑙𝑎: 𝑤 =
2
1 + 1 − 𝑝(𝑡 𝑗)2
𝐴𝑠 𝑡𝑗 = 𝐷−1
(𝐿 + 𝑈)

9. Best Of Luck
By: Khushdil Ahmad
BS-Mathematics
Govt: P.G Jahanzeb
College Swat
03428978608