By Eduard L. Stiefel (Auth.)
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Extra resources for An Introduction to Numerical Mathematics
In the more advanced development of approximation theory, such conditions are permitted (approximation with "side conditions"), but we will have to forbid them in our own elementary considerations. Second, the error equations have to be set up correctly. 7). It is forbidden, for example, to add two error equations since the absolute term of the resulting equation will certainly be less accurate than the absolute terms of the original equations. Other algebraic manipulations with the error equations are likewise not permitted.
The analogous notation applies to η equations with η unknowns. 54 3. LEAST-SQUARES APPROXIMATION AND D E F I N I T E PROBLEMS Row sums can be used to check the correctness of the normal equations. Let st = an + ai2 + c% be the row sum of the ith error equation. Then we can easily ascertain through calculation that for the row sums in both normal equations (αϊ, α 2) + (αϊ, c) = (αϊ, (α2, ai) + (α2, α 2) + (α2, c) = (α2, Here s is the column of the Si. 11 αχ) + s) s). The Mean Residual After solving the normal equations (94), we can substitute the resulting values of the unknowns in the error equations (90).
To minimize these contradictions, the Chebyshev principle which we just described is not used in most cases, but rather the Gauss method of least-squares. There are two decisive reasons for this. First of all, from the point of computational technique, the Gauss method is easier to handle, and secondly, in case the measurement errors are randomly distributed according to the laws of probability, only the latter method is really appropriate for the approximation problem. The Chebyshev principle retains its importance, however, in another type of approximation problem to be discussed later in the book.