arXiv subjects

Suvendu Kar

Publications and source records attributed to Suvendu Kar.

3 recordsLinked to original sources

arXiv2025

A fast solver for ill-conditioned linear systems using randomized stable solutions of its blocks

We present an enhanced version of the row-based randomized block-Kaczmarz method to solve a linear system of equations. This improvement makes use of a regularization during block updates in the solution, and a dynamic proposal distribution based on the current residual vector and effective mutual orthogonality between all blocks. The improved method provides significant gains in solving highly ill-conditioned linear systems that are either sparse, or dense least-squares problems that are significantly over/under determined. Considering the poor guarantees in effectively preconditioning iterative solutions for such ill-conditioned problems, it may also serve as a pre-solver for accelerating other iterative numerical methods, and as an inner iteration in certain types of GMRES solvers for linear systems.

math.NA↗

arXiv2025

An Optimal Least-Square Solver For Scaled Partial-Isometric Linear Systems

We present an $O(mn)$ direct least-squares solver for $m \times n$ linear systems with a scaled partial isometry. The proposed algorithm is also useful when the system is block diagonal and each block is a scaled partial isometry with distinct scaling factors. We also include numerical experiments as a demonstration.

math.NA↗

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↗