By Eduard L. Stiefel (Auth.)

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**Example text**

In this way, the chain of sides A, B, C in Fig. 3 will be found, which are now outside of the convex polygon and reach the polygon only in their end point. To understand the duality phenomenon better, we consider two linear programs which have exactly the same scheme; they shall differ only in that this scheme, in one case, is written in the normal way and, in the other case, is transposed. For instance, Χχ X2 χ* 1 yi = αχχ «12 «13 Cl yi = «21 «22 «23 c2 z = αχ a2 «3 c and νχ = V 2= Vs = w— Ux «11 «12 «13 Cl U2 «21 «22 «23 c2 1 αχ «2 «3 C (63) 38 2.

Consequently, the Chebyshev approxima tion principle is equivalent to minimizing the maximal absolute value of the residuals. It should further be pointed out that with the help of the relations (85) the computational technique can easily be refined in such a way that only the functions y ι have to be carried in the tables, and not the functions y[ as well. The general approximation problem of this type has the following form. Consider m error equations in η unknowns OikXk + Ci = r u i = 1, 2, .

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. 2. Example. The three angles of a triangle have been measured and the results of these measurements are αϊ, a 2, a 3 (in degrees). We introduce the first two angles x\, x2 of the triangle as unknowns. The third angle cannot be taken as another unknown since the three angles must satisfy the law for the sum of the angles in a triangle.