scholarly journals Multiprecision Algorithms for Computing the Matrix Logarithm

2018 ◽  
Vol 39 (1) ◽  
pp. 472-491 ◽  
Author(s):  
Massimiliano Fasi ◽  
Nicholas J. Higham
Keyword(s):  
Matrix Logarithm ◽  
The Matrix

scholarly journals On Randomized Trace Estimates for Indefinite Matrices with an Application to Determinants

2021 ◽  
Author(s):  
Alice Cortinovis ◽  
Daniel Kressner
Keyword(s):  
Large Scale ◽  
Gaussian Process Regression ◽  
Likelihood Estimation ◽  
Random Vectors ◽  
Symmetric Positive Definite ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Trace Estimation ◽  
Scale Matrix ◽  
Existing Result

AbstractRandomized trace estimation is a popular and well-studied technique that approximates the trace of a large-scale matrix B by computing the average of $$x^T Bx$$ x T B x for many samples of a random vector X. Often, B is symmetric positive definite (SPD) but a number of applications give rise to indefinite B. Most notably, this is the case for log-determinant estimation, a task that features prominently in statistical learning, for instance in maximum likelihood estimation for Gaussian process regression. The analysis of randomized trace estimates, including tail bounds, has mostly focused on the SPD case. In this work, we derive new tail bounds for randomized trace estimates applied to indefinite B with Rademacher or Gaussian random vectors. These bounds significantly improve existing results for indefinite B, reducing the number of required samples by a factor n or even more, where n is the size of B. Even for an SPD matrix, our work improves an existing result by Roosta-Khorasani and Ascher (Found Comput Math, 15(5):1187–1212, 2015) for Rademacher vectors. This work also analyzes the combination of randomized trace estimates with the Lanczos method for approximating the trace of f(B). Particular attention is paid to the matrix logarithm, which is needed for log-determinant estimation. We improve and extend an existing result, to not only cover Rademacher but also Gaussian random vectors.

scholarly journals A dynamical systems view of network centrality

2014 ◽  
Vol 470 (2165) ◽  
pp. 20130835 ◽  
Author(s):  
Peter Grindrod ◽  
Desmond J. Higham
Keyword(s):  
Dynamical Systems ◽  
Large Scale ◽  
Dynamic Networks ◽  
Network Evolution ◽  
Mathematical Framework ◽  
The Novel ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Systems View ◽  
Logarithm Function

To gain insights about dynamic networks, the dominant paradigm is to study discrete snapshots , or timeslices , as the interactions evolve. Here, we develop and test a new mathematical framework where network evolution is handled over continuous time, giving an elegant dynamical systems representation for the important concept of node centrality. The resulting system allows us to track the relative influence of each individual. This new setting is natural in many digital applications, offering both conceptual and computational advantages. The novel differential equations approach is convenient for modelling and analysis of network evolution and gives rise to an interesting application of the matrix logarithm function. From a computational perspective, it avoids the awkward up-front compromises between accuracy, efficiency and redundancy required in the prevalent discrete-time setting. Instead, we can rely on state-of-the-art ODE software, where discretization takes place adaptively in response to the prevailing system dynamics. The new centrality system generalizes the widely used Katz measure, and allows us to identify and track, at any resolution, the most influential nodes in terms of broadcasting and receiving information through time-dependent links. In addition to the classical static network notion of attenuation across edges, the new ODE also allows for attenuation over time, as information becomes stale. This allows ‘running measures’ to be computed, so that networks can be monitored in real time over arbitrarily long intervals. With regard to computational efficiency, we explain why it is cheaper to track good receivers of information than good broadcasters. An important consequence is that the overall broadcast activity in the network can also be monitored efficiently. We use two synthetic examples to validate the relevance of the new measures. We then illustrate the ideas on a large-scale voice call network, where key features are discovered that are not evident from snapshots or aggregates.

scholarly journals Computing the Fréchet Derivative of the Matrix Logarithm and Estimating the Condition Number

2013 ◽  
Vol 35 (4) ◽  
pp. C394-C410 ◽  
Author(s):  
Awad H. Al-Mohy ◽  
Nicholas J. Higham ◽  
Samuel D. Relton
Keyword(s):  
Condition Number ◽  
Fréchet Derivative ◽  
Frechet Derivative ◽  
Matrix Logarithm ◽  
The Matrix

scholarly journals Semidefinite Approximations of the Matrix Logarithm

2018 ◽  
Vol 19 (2) ◽  
pp. 259-296 ◽  
Author(s):  
Hamza Fawzi ◽  
James Saunderson ◽  
Pablo A. Parrilo
Keyword(s):  
Matrix Logarithm ◽  
The Matrix

scholarly journals An Improved Taylor Algorithm for Computing the Matrix Logarithm

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2018
Author(s):  
Javier Ibáñez ◽  
Jorge Sastre ◽  
Pedro Ruiz ◽  
José M. Alonso ◽  
Emilio Defez
Keyword(s):  
Matrix Polynomial ◽  
State Of The Art ◽  
Backward Error Analysis ◽  
Schur Decomposition ◽  
Backward Error ◽  
Popular Method ◽  
Numerical Tests ◽  
Taylor Approximation ◽  
Matrix Logarithm ◽  
The Matrix

The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. In this work, we present a Taylor series algorithm, based on the free-transformation approach of the inverse scaling and squaring technique, that uses recent matrix polynomial formulas for evaluating the Taylor approximation of the matrix logarithm more efficiently than the Paterson–Stockmeyer method. Two MATLAB implementations of this algorithm, related to relative forward or backward error analysis, were developed and compared with different state-of-the art MATLAB functions. Numerical tests showed that the new implementations are generally more accurate than the previously available codes, with an intermediate execution time among all the codes in comparison.

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