This balanced and comprehensive study presents the theory, methods and applications of matrix analysis in a new theoretical framework, allowing readers to understand second-order and higher-order matrix analysis in a completely new light Alongside the core subjects in matrix analysis, such as singular value analysis, the solution of matrix equations and eigenanalysis, the author introduces new applications and perspectives that are unique to this book The very topical subjects of gradient analysis and optimization play a central role here Also included are subspace analysis, projection analysis and tensor analysis, subjects which are often neglected in other books Having provided a solid foundation to the subject, the author goes on to place particular emphasis on the many applications matrix analysis has in science and engineering, making this book suitable for scientists, engineers and graduate students alike

Matrix Analysis and Applications

Given the square matrices $A, B, D, E$ and the matrix $C$ of conforming dimensions, we consider the linear matrix equation $A{\mathbf X} E+D{\mathbf X} B = C$ in the unknown matrix ${\mathbf X}$. Our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems, which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing.

/pdf/computational-methods-for-linear-matrix-equations-29c8rmvfvt.pdf

Computational Methods for Linear Matrix Equations

We present a fast distributed solution to the convex network flow optimization problem. Our approach uses a family of dual descent algorithms that approximate the Newton direction to achieve faster convergence rates than existing distributed methods. The approximate Newton directions are obtained through matrix splitting techniques and sparse Taylor approximations of the inverse Hessian. We couple this descent direction with a distributed line search algorithm which requires the same information as our descent direction to compute. We show that, similarly to conventional Newton methods, the proposed algorithm exhibits super-linear convergence within a neighborhood of the optimal value. Numerical experiments corroborate that convergence times are between one to two orders of magnitude faster than existing distributed optimization methods. A connection with recent developments that use consensus to compute approximate Newton directions is also presented.

Accelerated Dual Descent for Network Flow Optimization

Dimensional reduction and screening masses in pure gauge theories at finite temperature

A novel variant of the parallel QR algorithm for solving dense nonsymmetric eigenvalue problems on hybrid distributed high performance computing systems is presented. For this purpose, we introduce the concept of multiwindow bulge chain chasing and parallelize aggressive early deflation. The multiwindow approach ensures that most computations when chasing chains of bulges are performed in level 3 BLAS operations, while the aim of aggressive early deflation is to speed up the convergence of the QR algorithm. Mixed MPI-OpenMP coding techniques are utilized for porting the codes to distributed memory platforms with multithreaded nodes, such as multicore processors. Numerous numerical experiments confirm the superior performance of our parallel QR algorithm in comparison with the existing ScaLAPACK code, leading to an implementation that is one to two orders of magnitude faster for sufficiently large problems, including a number of examples from applications.

http://sma.epfl.ch/~anchpcommon/publications/newpqr.pdf

A Novel Parallel QR Algorithm for Hybrid Distributed Memory HPC Systems

Find scalars λ and vectors x = 0 for which Ax = λx The form of the matrix affects the way in which we solve this problem, and we also have variety as to what is to be found. • A symmetric and real (or Hermitian and complex). This is the most common case. In some cases we want only the eigenvalues (and perhaps only some of them); and in other cases, we also want the eigenvectors. There are special classes of such A, e.g. banded, positive definite, sparse, and others. • A non-symmetric, but with a diagonal Jordan canonical form. This means there is a nonsin-gular matrix P for which P −1 AP = D = diag[λ 1 , ..., λ n ]

G. Efroymson

Papers

A Matrix Eigenvalue Problem