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Strict row diagonal dominance guarantees Jacobi and Gauss–Seidel convergence

Statement

Let $n \geq 1$ and $A = (a_{i j}) \in M_{n} (R)$ satisfy strict row diagonal dominance:

$∣ a_{i i} ∣ > \sum_{j \neq i} ∣ a_{i j} ∣ (i = 1, \dots, n) .$

Then the Jacobi and fixed-order Gauss–Seidel iterations defined by Finite-budget Jacobi stationary iteration and Finite-budget Gauss–Seidel stationary iteration converge in exact arithmetic to the unique solution of $A x = b$ for every $b, x^{(0)} \in R^{n}$ .

Facts & Assumptions

Given: $n \geq 1$ , a real matrix $A = (a_{i j})$ satisfying the displayed strict row diagonal-dominance inequalities, any $b, x^{(0)} \in R^{n}$ , and the fixed order $1, \dots, n$ .

[A1]

For each row $i$ , $∣ a_{i i} ∣ > \sum_{j \neq i} ∣ a_{i j} ∣$ .

[L1]

The diagonal/lower/upper convention has $A = D - L - U$ , with $L$ strictly lower and $U$ strictly upper (Diagonal, lower, and upper parts in the stationary-iteration convention).

[F1]

Jacobi uses the stationary split $M_{J} = D$ , $N_{J} = L + U$ and solves $D x^{(k + 1)} = (L + U) x^{(k)} + b$ (Finite-budget Jacobi stationary iteration).

[F2]

Fixed-order Gauss–Seidel uses $M_{G} = D - L$ , $N_{G} = U$ and solves $(D - L) x^{(k + 1)} = U x^{(k)} + b$ (Finite-budget Gauss–Seidel stationary iteration).

[L2]

If a finite-dimensional triangular system has nonzero diagonal, forward or backward substitution gives its unique solution (Forward and backward substitution are correct, unique, and quadratic in scalar operations).

[L3]

For a stationary split $A = M - N$ with invertible $M$ , the iteration matrix is $G = M^{- 1} N$ and the update is $x^{(k + 1)} = G x^{(k)} + M^{- 1} b$ (Stationary iteration from a matrix splitting $A = M - N$ ).

[L4]

Such an iteration converges to the unique solution for every right-hand side and every start if and only if $ρ (G) < 1$ (A stationary splitting converges for every start if and only if its iteration matrix has spectral radius below $1$ ).

Proof

1.1A1F1F2L2algebra

By [A1], every $a_{i i} \neq 0$ . Thus $D$ and the lower-triangular matrix $D - L$ have nonzero diagonal, so the solves in [F1] and [F2] are uniquely defined by [L2].

2.1F1F2L3step 1.1algebra

By [F1]–[F2] and [L3], the iteration matrices are $G_{J} = D^{- 1} (L + U)$ and $G_{G} = (D - L)^{- 1} U$ .

3.1A1L1step 2.1algebra

For each row $i$ , set $q_{i} = \sum_{j \neq i} ∣ a_{i j} ∣ / ∣ a_{i i} ∣$ and $q_{J} = \max_{i} q_{i} < 1$ . Since $L_{i j} = - a_{i j}$ below the diagonal and $U_{i j} = - a_{i j}$ above it, $\sum_{j} ∣ (G_{J})_{i j} ∣ = q_{i}$ . Hence for any eigenpair $G_{J} z = λ z$ with $z \neq 0$ , choose $i$ with $∣ z_{i} ∣ = \max_{j} ∣ z_{j} ∣$ ; the row equation gives $∣ λ ∣ ∣ z_{i} ∣ \leq q_{i} ∣ z_{i} ∣ \leq q_{J} ∣ z_{i} ∣$ , so $∣ λ ∣ < 1$ and $ρ (G_{J}) < 1$ .

3.2A1L1step 2.1algebra

Let $G_{G} z = λ z$ with $z \neq 0$ , and choose $i$ with $∣ z_{i} ∣ = \max_{j} ∣ z_{j} ∣ = : R$ . The equation $U z = λ (D - L) z$ in row $i$ is $\sum_{j > i} U_{i j} z_{j} = λ (a_{i i} z_{i} - \sum_{j < i} L_{i j} z_{j})$ , so $∣ λ ∣ ∣ a_{i i} z_{i} - \sum_{j < i} L_{i j} z_{j} ∣ \leq \sum_{j > i} ∣ U_{i j} ∣ ∣ z_{j} ∣$ . By the reverse triangle inequality and [A1], the left factor is at least $∣ λ ∣ (∣ a_{i i} ∣ - \sum_{j < i} ∣ a_{i j} ∣) R$ , while the right side is at most $\sum_{j > i} ∣ a_{i j} ∣ R$ . Therefore $∣ λ ∣ \leq \max_{i} \frac{\sum_{j > i} ∣ a_{i j} ∣}{∣ a_{i i} ∣ - \sum_{j < i} ∣ a_{i j} ∣} < 1$ ; the denominators are positive and each strict inequality follows from [A1]. Thus $ρ (G_{G}) < 1$ .

4.1L4step 3.1step 3.2∎

The bounds in steps 3.1 and 3.2 and [L4] imply that both methods converge to the unique solution of $A x = b$ for every $b$ and every initial vector.

Depends on

Used by

Jacobi iteration on a strictly row-diagonally-dominant system Example

Dependency tree · two levels

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Sources

Yousef Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., §4.2.3 (standard reference, not scraped)
David Bindel, Matrix Computations (CS 6210), Cornell University, Fall 2012, Lecture 31 (standard reference, not scraped)
Rutgers University MATH 574, Lecture Notes, §3 Matrix iterative methods (standard reference, not scraped)