What is diagonal dominance and why must equations be rearranged before Jacobi or Seidel iteration?

In one line

A system is (strictly) diagonally dominant when, in every row, the size of the diagonal coefficient is greater than the sum of the sizes of the other coefficients; this guarantees that Jacobi and Seidel iterations converge.

Explained simply

In iteration, each unknown is mostly decided by its own equation, with small corrections from the others. If the "own" coefficient is big, those corrections are small and errors shrink each round. If it is small, errors grow and the iteration blows up, like a rumour that gets bigger each time it is passed on.

Technically, row \(i\) is diagonally dominant if \(|a_{ii}|\ge\sum_{j\ne i}|a_{ij}|\). If the inequality is strict in every row (strict diagonal dominance, the case used in exam problems), both Gauss–Jacobi and Gauss–Seidel are guaranteed to converge, whatever the starting values. The condition is sufficient, not necessary: some non-dominant systems still converge, but you cannot rely on it.

How it works

1. For each unknown, find the equation in which its coefficient is largest.
2. Reorder the equations so those coefficients lie on the diagonal.
3. Check each row: diagonal size \(>\) sum of other sizes.
4. Only then make each unknown the subject and iterate.

Key formula

Strict diagonal dominance: \(|a_{ii}|>\displaystyle\sum_{j\ne i}|a_{ij}|\) for every row \(i\).

Quick example

Given: \(x+y+54z=110\), \(27x+6y-z=85\), \(6x+15y+2z=72\).

As written, row 1 has diagonal 1 but others \(1+54\): not dominant. The largest \(x\)-coefficient is in the second equation (27), largest \(y\)-coefficient in the third (15), largest \(z\)-coefficient in the first (54). Reorder:

\(27x+6y-z=85\) (\(27>6+1\)), \(\ 6x+15y+2z=72\) (\(15>6+2\)), \(\ x+y+54z=110\) (\(54>1+1\)). Now it is diagonally dominant and either method will converge (to about \(x=2.425,\ y=3.573,\ z=1.926\)).

Common mistakes

• Iterating the equations in the order given without checking.
• Comparing signed coefficients instead of absolute values.
• Thinking dominance is required for the answer to exist; it is a condition for the iteration to converge.

How MU asks it

Not usually a separate question, but examiners often give equations in a non-dominant order to test you. A line such as "Rearranging for diagonal dominance: ..." with the inequality checks earns marks in any 5/6-mark Jacobi or Seidel question. It may also appear as a 3-mark theory part: "State the condition for convergence of Gauss–Seidel method".