arXiv subjects

Hariprasad M.

Publications and source records attributed to Hariprasad M..

3 recordsLinked to original sources

arXiv2025

Efficient approximations of matrix multiplication using truncated decompositions

We exploit the truncated singular value decomposition and the recently proposed circulant decomposition for an efficient first-order approximation of the multiplication of large dense matrices. A decomposition of each matrix into a sum of a sparse matrix with relatively few dominant entries and a dense residue can also use the above approach, and we present methods for multiplication using a Fourier decomposition and a cycle decomposition-based sparsifications. The proposed methods scale as $\mathcal{O}(n^2 \log n)$ in arithmetic operations for $n \times n$ matrices for usable tolerances in relative error $\sim$ 1\%. We also present demonstrations of large gains in the efficiency and speed of end-to-end operations of Large Language Models (LLMs) as a motivation. Note that different decompositions for the two matrices $A$ and $B$ in the product $AB$ are also possible in this approach, using efficient a priori evaluations for suitability, to improve further on the error tolerances demonstrated here.

math.NA↗

arXiv2022

Circulant decomposition of a matrix and the eigenvalues of Toeplitz type matrices

We begin by showing that any $n \times n$ matrix can be decomposed into a sum of $n$ circulant matrices with periodic relaxations on the unit circle. This decomposition is orthogonal with respect to a Frobenius inner product, allowing recursive iterations for these circulant components. It is also shown that the dominance of a few circulant components in the matrix allows sparse similarity transformations using Fast-Fourier-transform (FFT) operations. This enables the evaluation of all eigenvalues of dense Toeplitz, block-Toeplitz, and other periodic or quasi-periodic matrices, to a reasonable approximation in $\mathcal{O}(n^2)$ arithmetic operations. The utility of the approximate similarity transformation in preconditioning linear solvers is also demonstrated.

math.NA↗

arXiv2020

Semi-analytical solutions for eigenvalue problems of chains and periodic graphs

We first show the existence and nature of convergence to a limiting set of roots for polynomials in a three-term recurrence of the form $p_{n+1}(z) = Q_k(z)p_{n}(z)+ γp_{n-1}(z)$ as $n$ $\rightarrow$ $\infty$, where the coefficient $Q_k(z)$ is a $k^{th}$ degree polynomial, and $z,γ\in \mathbb{C}$. We extend these results to relations for numerically approximating roots of such polynomials for any given $n$. General solutions for the evaluation are motivated by large computational efforts and errors in the iterative numerical methods. Later, we apply this solution to the eigenvalue problems represented by tridiagonal matrices with a periodicity $k$ in its entries, providing a more accurate numerical method for evaluation of spectra of chains and a reduction in computational effort from $\mathcal{O}(n^2)$ to $\mathcal{O}(n)$. We also show that these results along with the spectral rules of Kronecker products allow an efficient and accurate evaluation of spectra of many spatial lattices and other periodic graphs.

math.NA↗