Fast Diagonalization of Dense Matrices: A Cost-Effective Approach

In this article, we explore a new method for efficiently diagonalizing dense matrices using a block-diagonal approach. The traditional method for solving this problem is computationally expensive, but our proposed technique significantly reduces the computational cost and memory requirements.
To understand how we achieve this efficiency, let’s break down the process into steps. First, we factor the matrix into two smaller blocks, one of which is diagonal. This is done using a technique called "Fast Diagonalization" (FD), which has already been shown to be effective in solving similar problems.
Next, we invert each block separately, resulting in a total computational cost of O(Ndof). While this may seem like a lot, it’s important to note that the block-diagonal structure of the matrix allows for parallel implementation on distributed memory machines, significantly reducing the time required for computation.
In addition, we only require a modest amount of additional memory to store the block matrices, resulting in a total storage cost of O(pdNs + Ndof). This is much more efficient than traditional methods, which often require a large amount of memory to discretize separately in space and time.
Overall, our proposed method offers significant improvements over traditional techniques for diagonalizing dense matrices, making it an important tool for a wide range of applications in science and engineering. By leveraging the block-diagonal structure of the matrix, we are able to achieve efficient parallel implementation on distributed memory machines, significantly reducing the computational cost and memory requirements of the method.