A computer routine for quadratic and linear programming problems

n~O with increasing z and increasing b. Pochhammer's symbol means (a),, = P(a+n)/r(a). More precisely we see (l/B(a, b))(z~/a) 2F1 (a, l-b; a+l;z) = ~.. (a+~l) ((a),, (l-b),~/(a+l),O z'~+'~/n! n-0 From this expression it is clear that the rate of convergence of the expansion by and large is dominated by the values of b and z. This explains why in test (a) for b = 7 the results remained accurate to 7D for increasing z, and also it explains why in test (b) for b = 17 the results became worse for increasing z. list and (a) after the second DIMENSION statement, insert (b) Replace 0.0 by TOL in the statement labeled 1 and in the statement two lines before that labeled 4. (c) Replace the label 6, occurring two lines before the statement labeled 2, by 9. A comparison of the primal simplex and complementary pivot methods for linear programming. Algorithm 431 is a Fortran implementation of Lemke's complementary pivot algorithm [l]. This algorithm has recently received a considerable amount of attention in the literature; in particular, there is some evidence that the algorithm is an attractive means of solving linear programs [2, 3] and can readily be modified to find stationary points of nonconvex quadratic programs [4]. Eaves [5] has shown that, in principle, degeneracy causes no problems in Lemke's algorithm and that it will always be possible to pivot the artificial variable out of the basis. In the presence of rounding error, however, this may no longer be true, and further pivoting may not be possible despite the presence of the artificial variable with a value close to zero. In such a case Algorithm 431 may incorrectly arrive at the conclusion that the problem has no complementary solution because it only recognizes a complementary solution when the artificial variable leaves the basis. The difficulty can be avoided by: (a) testing whether the value assumed by the artificial variable is acceptably "small" if no further pivoting is possible; and (b) not pivoting on "small" elements. The problem of deciding what is meant by "small" in this context is one for which there is no adequate theory. Clasen [6] has, however, proposed some empirical rules for dealing with similar problems in the revised simplex algorithm, and an adaptation of these has proved satisfactory. The modifications of Algorithm 431 given below incorporate Clasen's pivot tolerance to deal with point (b) …