43,227 research outputs found
On computing complex square roots of real matrices
We present an idea for computing complex square roots of matrices using only real
arithmetic.Fundação para a Ciência e a Tecnologia (FCT) - Project PEst-C/MAT/UI0013/2011, PTDC/MAT/112273/2009National Natural Science Foundation of China - Project PEst-C/MAT/UI0013/2011, PTDC/MAT/112273/2009, Portugal.Major Foundation of Educational Committee of Hunan Province - Grant No. 09A002 [2009]FEDER Funds- "Programa Operacional Factores de Competitividade - COMPETE
Computing the square roots of matrices with central symmetry
For computing square roots of a nonsingular matrix A, which are functions of A, two well known fast and stable algorithms, which are
based on the Schur decomposition of A, were proposed by Bj¨ork and Hammarling
[3], for square roots of general complex matrices, and by Higham [10], for real square roots of real matrices. In this paper we further consider (the computation of) the square roots of matrices with central symmetry.
We first investigate the structure of the square roots of these matrices and
then develop several algorithms for computing the square roots. We show that our algorithms ensure significant savings in computational costs as compared to the use of standard algorithms for arbitrary matrices.Fundação para a Ciência e a Tecnologia (FCT
New Structured Matrix Methods for Real and Complex Polynomial Root-finding
We combine the known methods for univariate polynomial root-finding and for
computations in the Frobenius matrix algebra with our novel techniques to
advance numerical solution of a univariate polynomial equation, and in
particular numerical approximation of the real roots of a polynomial. Our
analysis and experiments show efficiency of the resulting algorithms.Comment: 18 page
Novel Approach to Real Polynomial Root-finding and Matrix Eigen-solving
Univariate polynomial root-finding is both classical and important for modern
computing. Frequently one seeks just the real roots of a polynomial with real
coefficients. They can be approximated at a low computational cost if the
polynomial has no nonreal roots, but typically nonreal roots are much more
numerous than the real ones. We dramatically accelerate the known algorithms in
this case by exploiting the correlation between the computations with matrices
and polynomials, extending the techniques of the matrix sign iteration, and
exploiting the structure of the companion matrix of the input polynomial. We
extend some of the proposed techniques to the approximation of the real
eigenvalues of a real nonsymmetric matrix.Comment: 17 pages, added algorithm
Complex and Hypercomplex Discrete Fourier Transforms Based on Matrix Exponential Form of Euler's Formula
We show that the discrete complex, and numerous hypercomplex, Fourier
transforms defined and used so far by a number of researchers can be unified
into a single framework based on a matrix exponential version of Euler's
formula , and a matrix root of -1
isomorphic to the imaginary root . The transforms thus defined can be
computed using standard matrix multiplications and additions with no
hypercomplex code, the complex or hypercomplex algebra being represented by the
form of the matrix root of -1, so that the matrix multiplications are
equivalent to multiplications in the appropriate algebra. We present examples
from the complex, quaternion and biquaternion algebras, and from Clifford
algebras Cl1,1 and Cl2,0. The significance of this result is both in the
theoretical unification, and also in the scope it affords for insight into the
structure of the various transforms, since the formulation is such a simple
generalization of the classic complex case. It also shows that hypercomplex
discrete Fourier transforms may be computed using standard matrix arithmetic
packages without the need for a hypercomplex library, which is of importance in
providing a reference implementation for verifying implementations based on
hypercomplex code.Comment: The paper has been revised since the second version to make some of
the reasons for the paper clearer, to include reviews of prior hypercomplex
transforms, and to clarify some points in the conclusion
Maximum Likelihood for Matrices with Rank Constraints
Maximum likelihood estimation is a fundamental optimization problem in
statistics. We study this problem on manifolds of matrices with bounded rank.
These represent mixtures of distributions of two independent discrete random
variables. We determine the maximum likelihood degree for a range of
determinantal varieties, and we apply numerical algebraic geometry to compute
all critical points of their likelihood functions. This led to the discovery of
maximum likelihood duality between matrices of complementary ranks, a result
proved subsequently by Draisma and Rodriguez.Comment: 22 pages, 1 figur
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