Applications of Dense Linear Algebra - linear-algebra

Question

What are the common real-world applications of Dense Linear Algebra?
Many problems can be easily described and efficiently computed using Linear Algebra as a common language between human and computer. More often than not though these systems require the solution of sparse matrices, not dense ones. What are common applications that defy this rule?
I'm curious if the community should invest further time to improve DLA packages like LAPACK. Who uses LAPACK in a computationally constrained application? Who uses LAPACK to solve large problems requiring parallelism?
Specifically, what are problems that can not be solved today due to insufficient dense linear algebra capabilities.

Answer 1

This depends on what you mean by real-world. Real-world for me is physics so I'll tell you ones in physics first and then branch out. In physics we often have to find the eigenvalues and eigenvectors of a matrix called the Hamiltonian (it basically contains information about the energy of a system). These matrices can be dense, at least in blocks. These blocks can be quite large. This brings up another point: sparse matrices can be dense in blocks and then it is best to use a dense linear algebra solver for each of the blocks.
There is also something called the density matrix of a system. It can be found using the eigenvectors of the Hamiltonian. In one algorithm that I use we often are finding the eigenvectors/values of these density matrices and the density matrices are dense, at least in blocks.
Dense linear algebra is used in material science and hydrodynamics as well, as mentioned in this article. This also relates to quantum chemistry, which is another area in which they are used.
Dense linear algebra routines have also been used to solve quantum scattering of charged particles(it doesn't say so in the linked article, but it was used) and to analyze the Cosmic Microwave Background. More broadly, it is used in solving an array of electromagnetic problems relating to real-world things like antenna design, medical equipment design, and determining/reducing the radar signature of a plane.
Another very real world application is that of curve fitting. However, there are other ways of doing it than using linear algebra that have broader scope.
In summary, dense linear algebra is used in a variety of applications, most of which are science- or engineering-related.
As a side note, many people have previously and are presently putting a great deal of effort into dense linear algebra libraries including ones that use graphics cards to do the computations.

Answer 2

Many methods for linear regression require heavy lifting on big, dense data matrices. The most straightforward example I can think of is linear least squares using the Moore-Penrose pseudoinverse.

Answer 3

Sparse solvers might be more useful in the long run, but dense linear algebra is crucial to the development of sparse solvers, and can't really be neglected:
Dense systems are often an easier domain in which to do algorithmic development, because there's one less thing to worry about.
The size at which sparse solvers become faster than the best dense solvers (even for very sparse matrices) is much larger than most people think it is.
The fastest sparse solvers are generally built on the fastest dense linear algebra operations.

Answer 4

In some sense a special case of Andrew Cone's example, but Kalman Filters eg here typically have a dense state error covariance matrix, though the observation model matrix and transition matrices may be sparse.