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Generation of random orthogonal matrices. (English) Zbl 0637.65004

The paper begins with a review of various methods to generate random orthogonal matrices distributed according to the Haar measure over the orthogonal group. Starting from a matrix X of independent standard normal random variables, an orthogonal random matrix with the prescribed distribution can be obtained from the QR factorization, from the eigenvalue decomposition, or from the singular value decomposition. Two other methods are mentioned, based on random skew-symmetric matrices, or on ordered sets of orthogonal vectors over the surface of a unit sphere. A more efficient well-known method is based on Householder transformation \(H_ 1,...,H_{n-1}\), related to independent normal random vectors, such that \(Q=H_ 1...H_{n-1}D\) is the required orthogonal random matrix; here D is diagonal with random entries \(\pm 1.\)
The authors propose another method, formally equivalent to the previous one, but different from the point of view of the generation of random variables. The new method gives a random orthogonal matrix as the product of n(n-1)/2 Givens transformations \(G_{ij}\), times D. The generation of \(G_{ij}\) consists of generating angles according to certain distributions, and the cost of this operation is efficiently reduced using properties of chi-square random variables and beta densities.
Reviewer: F.Flandoli

MSC:

65C10 Random number generation in numerical analysis
65C05 Monte Carlo methods
65F30 Other matrix algorithms (MSC2010)
15B52 Random matrices (algebraic aspects)
43A05 Measures on groups and semigroups, etc.
65C99 Probabilistic methods, stochastic differential equations
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