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Applications of Dense Linear Algebra

What are the common applications of the real world Dense linear algebra ?

Many problems can be easily described and effectively calculated using linear algebra as a common language between a person and a computer. Most often, although these systems require solving sparse matrices, not dense ones. What are the common applications that challenge this rule?

I am wondering if the community needs to spend extra time improving DLA packages like LAPACK . Who uses LAPACK in a limited computing application? Who uses LAPACK to solve big problems requiring parallelism?

In particular, what are the problems that today cannot be solved due to the insufficient capabilities of dense linear algebra.

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It depends on what you mean by the real world. The real world for me is physics, so first I will tell you about this in physics, and then branch out. In physics, we often have to find the eigenvalues ​​and eigenvectors of a matrix called the Hamiltonian (it mainly contains information about the energy of the system). These matrices can be dense, at least in blocks. These blocks can be quite large. This leads to another point: sparse matrices can be dense in blocks, and then it is best to use a dense linear algebra algorithm for each of the blocks.

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  • 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.
  • The fastest sparse solvers are usually based on operations with the fastest dense linear algebra.
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In a sense, it’s a special case of Andrew Cone’s example, but Kalman Filters, for example, here , as a rule, have a covariance matrix of errors with a dense state, although the observation matrix of models and transition matrices can be sparse.

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