By Eduard L. Stiefel (Auth.)
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Extra resources for An Introduction to Numerical Mathematics
Example text
In this way we obtained a linear program with the independent variables Δ#ι, Δ#2, where the objective function is simply equal to the last variable r. Even though we could apply our methods directly to this program, it is more convenient to introduce the new variables Δχι ξι = , ζ2 = Δχ2 , , 1 Ç3 (83) = and to divide all inequalities by r (as upper bound of a set of absolute values r is certainly positive). 1& - 0 . 1& + 0 . 7. CHEBYSHEV APPROXIMATION ζ = ξ ζ has to be made maximal. 1 -1 1 1 1 1 1 1 1 1 1 0 The given error equations (80) appear in the y-columns while they'-columns are obtained from the obvious relation yi + (85) yi This same relation can also be applied in the following to simplify the cal culations.
L I N E A R PROGRAMMING The condition y \ > 0, y 2 > 0 therefore implies that the objective function ζ is always < 5400. This upper bound is attained for the choice of y \ = 0 , y 2 = 0. This choice is permissible since (52) then yields xi = 60, 2 = 25, x y3 = 15. (53) These values turn out to be positive due to the fact that our first require ment was satisfied in scheme (52) as well. Therefore, all five constraints xi > 0, X2 > 0, yi > 0, y 2 > 0, y 3 > 0 are satisfied. 11. This discussion of table (52) provides a fairly general understanding of the two requirements formulated above.
For example, such a system can consist of all the con straints of a linear program. ) To solve this problem, we introduce the extended system \ UikXk + Zn+1 + Ci > (60b) 0 where we have added a new unknown xn+v I t is easy to find a set of values χι, x2, ... , xn+i which satisfy (60b). In fact, we only have to choose xn+i sufficiently large. Using this set of starting values, we now solve that linear program which requires xn+i to be minimal, subject to the constraints (60b). (In other words, the objective function is the last variable xn+v) Let ξι, ξ2, ...
