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Jacobi and Gauss-Seidel splittings can be compared by the spectral radii of their iteration matrices

Example

For

$A = (\begin{array}{cc} 4 & 1 \\ 2 & 3 \end{array}), b = (\binom{1}{1}), x_{0} = (\binom{0}{0}),$

compare the Jacobi splitting with

$M_{J} = (\begin{array}{cc} 4 & 0 \\ 0 & 3 \end{array}), N_{J} = (\begin{array}{cc} 0 & - 1 \\ - 2 & 0 \end{array}),$

and the Gauss-Seidel splitting with

$M_{G} = (\begin{array}{cc} 4 & 0 \\ 2 & 3 \end{array}), N_{G} = (\begin{array}{cc} 0 & - 1 \\ 0 & 0 \end{array}) .$

The Gauss-Seidel iteration matrix has the smaller spectral radius and the faster visible error decay on this system.

Facts & Assumptions

Given: The two displayed splittings of the same system.

[F1]

A stationary splitting has iteration matrix $B = M^{- 1} N$ (Stationary iteration from a matrix splitting $A = M - N$ ).

[L1]

A stationary splitting converges for every start exactly when $ρ (B) < 1$ (A stationary splitting converges for every start if and only if its iteration matrix has spectral radius below $1$ ).

Verification

technique · direct calculation

1.1F1L1algebra

By [F1], $B_{J} = M_{J}^{- 1} N_{J} = (\begin{array}{cc} 0 & - 1 / 4 \\ - 2 / 3 & 0 \end{array}), B_{G} = M_{G}^{- 1} N_{G} = (\begin{array}{cc} 0 & - 1 / 4 \\ 0 & 1 / 6 \end{array}) .$ The eigenvalues of $B_{J}$ are $\pm 1 / \sqrt{6}$ , so $ρ (B_{J}) = 1 / \sqrt{6}$ , while the eigenvalues of $B_{G}$ are $0$ and $1 / 6$ , so $ρ (B_{G}) = 1 / 6$ . Both are below $1$ , and Gauss-Seidel has the smaller spectral radius.

2.1L1step 1.1algebra∎

The exact solution is $x_{*} = (1 / 5, 1 / 5)^{T}$ . Jacobi gives $x_{1}^{(J)} = (\binom{1 / 4}{1 / 3}), x_{2}^{(J)} = (\binom{1 / 6}{1 / 6}),$ so $∥ x_{1}^{(J)} - x_{*} ∥_{\infty} = \frac{2}{15}, ∥ x_{2}^{(J)} - x_{*} ∥_{\infty} = \frac{1}{30} .$ Gauss-Seidel gives $x_{1}^{(G)} = (\binom{1 / 4}{1 / 6}), x_{2}^{(G)} = (\binom{5 / 24}{7 / 36}),$ so $∥ x_{1}^{(G)} - x_{*} ∥_{\infty} = \frac{1}{20}, ∥ x_{2}^{(G)} - x_{*} ∥_{\infty} = \frac{1}{120} .$ Thus the splitting with smaller spectral radius also shows faster observed error decay here, consistent with [L1].

Depends on

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Sources

Gilbert Strang, 18.086 Mathematical Methods for Engineers II, Section 6.2 Iterative Methods (standard reference, not scraped)