By Walter Freiberger, Ulf Grenander (auth.)

This publication arose out of a few assorted contexts, and diverse folks have contributed to its notion and improvement. It had its starting place in a undertaking initiated together with the IBM Cambridge Scien tific middle, relatively with Dr. Rhett Tsao, then of that middle. we're thankful to Mr. Norman Rasmussen, supervisor of the IBM medical middle complicated, for his preliminary help. The paintings is being carried on at Brown collage with beneficiant aid from the place of work of Computing actions of the nationwide technological know-how origin (grants GJ-174 and GJ-7l0); we're thankful to Dr. John Lehmann of this place of work for his curiosity and encouragement. Professors Donald McClure and Richard Vitale of the department of utilized arithmetic at Brown college contributed vastly to the venture and taught classes in its spirit. we're indebted to them and to Dr. Tore Dalenius of the collage of Stockholm for important criticisms of the manuscript. the ultimate stimulus to the book's of completion got here from an invLtation to coach a path on the IBM eu platforms study Institute at Geneva. we're thankful to Dr. J.F. Blackburn, Director of the Institute, for his invitation, and to him and his spouse Beverley for his or her hospitality. we're significantly indebted to Mrs. Katrina Avery for her fantastic secretarial and editorial paintings at the manuscript.

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I=l p. 1. on . 4] Processing time required: if P is of size nxn, then i) for each level of iteration we require n 2 multiplications and (n 2_n) additions; ii) for completely column reducing an nxn matrix we reguire (n 3+n 2 -2n)/2 multiplications and (n 3-n)/2 additions and n divisions; 22 iii) for (n+l)/2 levels of iteration, the exact solution requires n fewer multiplications, the same number of additions and n additional divisions compared to iteration; iv) for greater than (n+l)/2 levels of iteration, the exact solution is faster as far as processing is concerned.

B) The statistical precision is often very low; more about this later. 2 we shall indicate that this observation may be less relevant than one would think at first sight). We shall below use Monte Carlo only to study problems presented in a stochastic form. It should be mentioned, however, that the method can be applicable in situa- tions where no randomness appears in the original presentation of the problem. We shall not deal with such cases in what follows. Before going ahead to a more detailed discussion of some Monte Carlo refinements, let us point out the following.

J 1 < x < i I 1 P. J where x is the pseudo-random number from R(O,l). tributed according to F. continuous. 5) z = where F- l is the inverse function of F. In both cases z will have the distribution function F. As an example, assume that we want to generate exponentially distributed numbers such as, say, in simulation of a Markovian queueing problem. z = -log(l-x), which has the same distribution as z' = -log For F(x) x, x = l_e- x = R(O,l). 6) e -x x > ° This procedure is simple enough and if only a moderate-size Monte Carlo experiment is planned it will be acceptable.