The MATLAB team at MathWorks tested performance scaling of the backslash ("\") matrix division operator to solve for x in the equation A*x = b. In their testing, matrix A occupies far more memory (290 GB) than is available in a single high-end desktop machine—typically a quad core processor with 4-8 GB of RAM, supplying approximately 20 Gigaflops.
Therefore, they spread the calculation across machines. In order to solve linear systems of equations they need to be able to access all of the elements of the array even when the array is spread across multiple machines. This problem requires significant amounts of network communication, memory access, and CPU power. They scaled up to a cluster in EC2, giving them the ability to work with larger arrays and to perform calculations at up to 1.3 Teraflops, a 60X improvement. They were able to do this without making any changes to the application code.
Here's a graph showing the near-linear scalability of an EC2 cluster across a range of matrix sizes with corresponding increases in cluster size for MATLAB's parallel backslash operator:
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