![]() This takes as a first argument a range in which to pick entries, passed to the function as a list, e.g. The code is essentially comprised of two functions: one which randomly generates invertible matrices with pre-specified entries and dimensions, and one which for every stage uses these matrices to create linear transformations, and subsequently checks whether these have the correct number of distinct entries (providing an isomorphism between the original specification and the new variables).įirst, define a function which creates a list of invertible matrices with integer entries. We here thus follow an approach which employs random transformations on the stage matrices to overcome this computational obstacle and still provide an idea of whether a given staged tree might have toric structure. Even for medium sized examples, it is computationally infeasible to try out all such possible reparametrisations. The Wolfram Languages matrix operations handle both numeric and symbolic matrices, automatically accessing large numbers of highly efficient algorithms. However, sometimes a given parametrisation does not immediately give rise to binomials and these can only be found after an adequate reparametrisation. Whenever these minors are binomials, the staged tree has toric structure. Support for circular ensembles (COE, CUE. Support for Gaussian ensembles (GOE, GUE. Efficient sampling from matrix distributions and their derived properties. This section describes what happens if the input matrix contains a mixture of different types of entries.As presented in our paper, a staged tree statistical model can be characterised as the \(2\times 2\)-minors of its stage matrices. Random matrices have uses in a surprising variety of fields, including statistics, physics, pure mathematics, biology, and finance, among others. ![]() This meets the design goal of integrating symbolic and numerical computation. Matrix computations involving exact numbers and general symbolic techniques are carried out with computer algebra techniques.Īll computations provided for numerical matrices are also available for symbolic matrices. More information can be found in the section " Arbitrary-Precision Matrices". These libraries are adapted from standard libraries so they can work for arbitrary-precision computations. In Mathematica, matrix computations involving arbitrary-precision Real and arbitrary-precision approximate Complex numbers are carried out with special numerical libraries. In the case of linear algebra computations, Mathematica makes use of a considerable amount of sophisticated technology, some of which is described under " Performance of Linear Algebra Computation". This is in keeping with the design goals of Mathematica, as described under " Design Principles of Mathematica". ![]() In many cases computations involve optimized libraries, many of which are described in "Software References".Īn important goal for many of these computations is to match and surpass the performance of any software package that is dedicated to machine-precision numbers. In Mathematica, matrix computations involving machine-precision Real and machine-precision approximate Complex numbers are carried out with standard numerical techniques. These three different categories are briefly reviewed. is a great resource for calculating and exploring the properties of vectors and matrices. ![]() ![]() Arbitrary ‐precision numerical techniquesĭifferent types of matrices in Mathematica. The most popular versions of the Wolfram Mathematica 12. ![]()
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