# 17. Stationary Iterative Methods

## Abstract

*I recommend this method to you for imitation. You will hardly ever again eliminate directly at least not when you have more than 2 unknowns. The indirect*[

*iterative*]

*procedure can be done while half asleep, or while thinking about other things*.

*Letter to C. L. Gerling*(1823)

*The iterative method is commonly called the “Seidel process,” or the “Gauss-Seidel process.” But, as Ostrowski*(1952)

*points out, Seidel*(1874)

*mentions the process but advocates not using it. Gauss nowhere mentions it*.

*Solving Linear Algebraic Equations Can Be Interesting*(1953)

*The spurious contributions in*null(

*A*)

*grow at worst linearly and if the rounding errors are small the scheme can be quite effective*.

*On the Solution of Singular and Semidefinite Linear Systems by Iteration*(1965)

*k*= 1,2, …?” Without an answer to this question we cannot be sure that a convergence test of the form ‖

*b*−

*A*$\hat{x}$

_{k}‖ ≤ ϵ (say) will ever be satisfied, for any given value of ϵ < ‖

*b*−

*Ax*

_{0}‖!

*A*is the 100 × 100 lower bidiagonal matrix with

*a*

_{ii}= 1.5 and

*a*

_{i, i-1}≡ 1, and

*b*

_{i}≡ 2.5. The successive overrelaxation (SOR) method is applied in MATLAB with parameter ω = 1.5, starting with the rounded version of the exact solution x, given by

*x*

_{i}= 1 − (−2/3)

^{i}. The forward errors ‖ $\hat{x}$

_{k}−

*x*‖

_{∞}/‖

*x*‖

_{∞}and the ∞-norm backward errors η

_{A,b}( $\hat{x}$

_{k}) are plotted in Figure 17.1. The SOR method converges in exact arithmetic, since the iteration matrix has spectral radius 1/2, but in the presence of rounding errors it diverges. The iterate $\hat{x}$

_{238}has a largest element of order 10

^{13}, $\hat{x}$

_{k +2}≡ $\hat{x}$

_{k}for

*k*≥ 238, and for

*k*> 100, $\hat{x}$

_{k}(60: 100) ≈ (−1)

^{k}$\hat{x}$

_{100}(60: 100). The divergence is not a result of ill conditioning of

*A*, since κ

_{∞}(

*A*) ≈ 5. The reason for the initial rapid growth of the errors in this example is that the iteration matrix is far from normal; this allows the norms of its powers to become very large before they ultimately decay by a factor ≈ 1/2 with each successive power.

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## Information & Authors

### Information

#### Published In

**ISBN (Print)**: 0898715210

**ISBN (Print)**: 978-0-89871-521-7

**ISBN (Online)**: 978-0-89871-802-7

#### Copyright

#### History

**Published online**: 23 March 2012

#### Keywords

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- QA297 .H53 2002Numerical analysis—Data processing, Computer algorithms

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