By Eduard L. Stiefel (Auth.)
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On the time of Professor Rademacher's demise early in 1969, there has been on hand an entire manuscript of the current paintings. The editors had in basic terms to provide a couple of bibliographical references and to right a number of misprints and mistakes. No great adjustments have been made within the manu script other than in a single or areas the place references to extra fabric seemed; due to the fact that this fabric used to be no longer present in Rademacher's papers, those references have been deleted.
Ausgehend von der Programmierung moderner Hochleistungsalgorithmen stellen die Autoren das mathematische und programmtechnische Umfeld der Zahl Pi ausführlich dar. So werden zur Berechnung von Pi sowohl die arithmetischen Algorithmen, etwa die FFT-Multiplikation, die super-linear konvergenten Verfahren von Gauß, Brent, Salamin, Borwein, die Formeln von Ramanujan und Borwein-Bailey-Plouffe bis zum neuen Tröpfel-Algorithmus behandelt.
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METHOD OF D U A L S O L U T I O N 37 These values are > 0 because, after the last EX-step, the last column has become positive. Hence, all five constraints *i>0; x2 > 0; yx > 0; y2 > 0; y3 > 0 are satisfied. 5; # 2 = 3 , and the minimal value of the objective function is ζ = χχ -f χ2 = 4,5. Since the values in the lower right-hand corners of the tables are constantly increasing during the simplex algorithm, they con stitute lower bounds for the minimum. The objective function is thus not systematically decreased by the algorithm until the minimal value has been attained, but, on the contrary, it is constantly increased.
The objective function is thus not systematically decreased by the algorithm until the minimal value has been attained, but, on the contrary, it is constantly increased. The procedure described here is called the dual-solution method. It can always be used when a linear objective function of the independent variables xjc has co efficients > 0 and is to be made minimal. , the variables on the upper border of the last table—equal to zero. For a geometric interpretation of the dual solution method, the corresponding variables on the upper border of the previous tables shall also be set equal to zero.
2#i + x2 — 6, (59) = max. 2/3 = ζ— Xl -1 -2 -2 1 X2 -4 -3 -1 1 1 -8 -12 -6 0 Now the last row of the c\ is no longer positive. To proceed in this case, we first Χ look for special values of the independent variables #i, x 2 which satisfy 1 I f this last element is equal to zero, we may arbitrarily regard it as either positive or negative. 34 2. L I N E A R PROGRAMMING the five constraints in (58). " Geometrically (Fig. 3), this means finding a feasible point or, in other words, a point which lies inside the shaded convex polygon.