TABLE OF CONTENTS 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. [15]
CARL FRIEDRICH GAUSS, 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.
GEORGE E. FORSYTHE,
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.
HERBERT B. KELLER,
On the Solution of Singular and Semidefinite Linear Systems by Iteration (1965)
[15]Gauss refers here to his relaxation method for solving the normal equations. The translation is taken from Forsythe [422, 1951].
Iterative methods for solving linear systems have a long history, going back at least to Gauss. Table 17.1 shows the dates of publication of selected methods. It is perhaps surprising, then, that rounding error analysis for iterative methods is not well developed. There are two main reasons for the paucity of error analysis. One is that in many applications accuracy requirements are modest and are satisfied without difficulty, resulting in little demand for error analysis. Certainly there is no point in computing an answer to greater accuracy than that determined...
