34. MONTE CARLO TECHNIQUES

Monte Carlo techniques are often the only practical way to evaluate difficult integrals or to sample random variables governed by complicated probability density functions. Here we describe an assortment of methods for sampling some commonly occurring probability density functions. 34.1. Sampling the uniform distribution Most Monte Carlo sampling or integration techniques assume a “random number generator, ” which generates uniform statistically independent values on the half open interval [0, 1); for reviews see, e.g.,[1, 2]. Uniform random number generators are available in software libraries such as CERNLIB [3], CLHEP [4], and ROOT [5]. For example, in addition to a basic congruential generator TRandom (see below), ROOT provides three more sophisticated routines: TRandom1 implements the RANLUX generator [6] based on the method by Lüscher, and allows the user to select different quality levels, trading off quality with speed; TRandom2 is based on the maximally equidistributed combined Tausworthe generator by L’Ecuyer [7]; the TRandom3 generator implements the Mersenne twister algorithm of Matsumoto an

37. MONTE CARLO TECHNIQUES

Monte Carlo techniques are often the only practical way to evaluate difficult integrals or to sample random variables governed by complicated probability density functions. Here we describe an assortment of methods for sampling some commonly occurring probability density functions. 37.1. Sampling the uniform distribution Most Monte Carlo sampling or integration techniques assume a “random number generator, ” which generates uniform statistically independent values on the half open interval [0, 1); for reviews see, e.g., [1,2]. Uniform random number generators are available in software libraries such as CERNLIB [3], CLHEP [4], and ROOT [5]. For example, in addition to a basic congruential generator TRandom (see below), ROOT provides three more sophisticated routines: TRandom1 implements the RANLUX generator [6] based on the method by Lüscher, and allows the user to select different quality levels, trading off quality with speed; TRandom2 is based on the maximally equidistributed combined Tausworthe generator by L’Ecuyer [7]; the TRandom3 generator implements the Mersenne twister algorithm of Matsumoto an

29. MONTE CARLO TECHNIQUES

Monte Carlo techniques are often the only practical way to evaluate difficult integrals or to sample random variables governed by complicated probability density functions. Here we describe an assortment of methods for sampling some commonly occurring probability density functions. 29.1. Sampling the uniform distribution Most Monte Carlo sampling or integration techniques assume a “random number generator ” which generates uniform statistically independent values on the half open interval [0,1). There is a long history of problems with various generators on a finite digital computer, but recently, the RANLUX generator [1] has emerged with a solid theoretical basis in chaos theory. Based on the method of Lüscher, it allows the user to select different quality levels, trading off with speed. Other generators are also available which pass extensive batteries of tests for statistical independence and which have periods which are so long that, for practical purposes, values from these generators can be considered to be uniform and statistically independent. I

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The technical infrastructure is of crucial importance for modern society and electric power supplies are of particular importance. Society's dependence on electrical energy, communication

32. MONTE CARLO TECHNIQUES

Monte Carlo techniques are often the only practical way to evaluate difficult integrals or to sample random variables governed by complicated probability density functions. Here we describe an assortment of methods for sampling some commonly occurring probability density functions. 32.1. Sampling the uniform distribution Most Monte Carlo sampling or integration techniques assume a “random number generator ” which generates uniform statistically independent values on the half open interval [0, 1). There is a long history of problems with various generators on a finite digital computer, but recently, the RANLUX generator [1] has emerged with a solid theoretical basis in chaos theory. Based on the method of Lüscher, it allows the user to select different quality levels, trading off quality with speed. Other generators are also available which pass extensive batteries of tests for statistical independence and which have periods which are so long that, for practical purposes, values from these generators can be considered to be uniform and statistically independent. I