arXiv subjects

Françoise Tisseur

Publications and source records attributed to Françoise Tisseur.

6 recordsLinked to original sources

arXiv2026

Computing accurate singular values using a mixed-precision one-sided Jacobi algorithm

We present a relative forward error analysis of a mixed-precision preconditioned one-sided Jacobi algorithm, analogous to a two-sided version introduced in [N. J. Higham, F. Tisseur, M. Webb and Z. Zhou, SIAM J. Matrix Anal. Appl. 46 (2025), pp. 2423-2448], which uses low precision to compute the preconditioner, applies it in high precision, and computes the singular value decomposition using the one-sided Jacobi algorithm at working precision. Our analysis yields relative forward error bounds involving the scaled condition number of the preconditioned matrix rather than that of the input matrix, potentially improving accuracy when the former is smaller. We present and analyze two approaches for constructing effective preconditioners. Our numerical experiments demonstrate that our algorithm achieves smaller relative forward errors than the LAPACK routines $\texttt{DGESVJ}$ and $\texttt{DGEJSV}$, as well as the MATLAB function $\texttt{svd}$, particularly for ill-conditioned matrices. Timing tests show that our approach accelerates the convergence of the Jacobi iterations and that the dominant cost arises from a single high-precision matrix-matrix multiplication. With improved software or hardware support for this bottleneck, our algorithm would be faster than the LAPACK one-sided Jacobi algorithm $\texttt{DGESVJ}$ and comparable in speed to the state-of-the-art preconditioned one-sided Jacobi algorithm $\texttt{DGEJSV}$, but much more accurate.

math.NA↗

arXiv2026

Computing accurate singular vectors and eigenvectors using mixed-precision Jacobi algorithms

Mixed-precision variants of the Jacobi algorithm for symmetric positive definite eigenproblems and the one-sided Jacobi algorithm for singular value decompositions have recently been shown to compute eigenvalues and singular values to high relative accuracy. However, these analyses do not address the accuracy of the computed eigenvectors and singular vectors. In this paper, we prove error bounds for the computed eigenvectors and singular vectors, where the error is measured by the sine of the angle between the vector and its computed counterpart. The obtained bounds preserve the relative gap structure of the bounds for Jacobi algorithms proved by Demmel and Veselić, but involve the scaled condition number of the preconditioned matrix rather than that of the original matrix (the former of which is typically much smaller). Numerical experiments support our theoretical bounds and demonstrate that the mixed-precision preconditioned Jacobi algorithms are especially effective for ill-conditioned matrices with small absolute gaps and moderate relative gaps between eigenvalues or singular values.

math.NA↗

arXiv2026

Analysis of Floating-Point Matrix Multiplication Computed via Integer Arithmetic

Ootomo, Ozaki, and Yokota [Int. J. High Perform. Comput. Appl., 38 (2024), p. 297-313] have proposed a strategy to recast a floating-point matrix multiplication in terms of integer matrix products. The factors A and B are split into integer slices, the product of these slices is computed exactly, and AB is approximated by accumulating these integer products in floating-point arithmetic. This technique is particularly well suited to mixed-precision matrix multiply-accumulate units with integer support, such as the NVIDIA tensor cores or the AMD matrix cores. The number of slices allows for performance-accuracy tradeoffs: more slices yield better accuracy but require more multiplications, which in turn reduce performance. We propose an inexpensive way to estimate the minimum number of multiplications needed to achieve a prescribed level of accuracy. Our error analysis shows that the algorithm may become inaccurate (or inefficient) if rows of A or columns of B are badly scaled. We perform a range of numerical experiments, both in simulation and on the latest NVIDIA GPUs, that confirm the analysis and illustrate strengths and weaknesses of the algorithm.

math.NA↗

arXiv2026

Fast Algorithms for Optimal Damping in Mechanical Systems

Optimal damping aims at determining a vector of damping coefficients $ν$ that maximizes the decay rate of a mechanical system's response. This problem can be formulated as the minimization of the trace of the solution of a Lyapunov equation whose coefficient matrix depends on $ν$. For physical relevance, the damping coefficients must be nonnegative and the resulting system must be asymptotically stable. We identify conditions under which the system is never stable or may lose stability for certain choices of $ν$. In the latter case, we propose replacing the constraint $ν\ge 0$ with $ν\ge d$, where $d$ is a nonzero nonnegative vector chosen to ensure stability. We derive explicit expressions for the gradient and Hessian of the objective function and show that the Karush--Kuhn--Tucker conditions are equivalent to the vanishing of a nonlinear residual function at an optimal solution. To compute such a solution, we propose a Barzilai--Borwein residual minimization algorithm (BBRMA), which is simple and efficient but not globally convergent, and a spectral projected gradient (SPG) method, which is globally convergent. By exploiting the structure of the problem, we show how to efficiently compute the objective function and its gradient, with eigenvalue decompositions constituting the dominant cost in terms of execution time. Numerical experiments show that both methods require fewer eigenvalue decompositions than the fast optimal damping algorithm (FODA), and that, although SPG may incur additional decompositions due to line search, it often converges faster than BBRMA, leading to a lower overall computational effort.

math.NA↗

arXiv2025

Computing accurate eigenvalues using a mixed-precision Jacobi algorithm

We provide a rounding error analysis of a mixed-precision preconditioned Jacobi algorithm, which uses low precision to compute the preconditioner, applies it at high precision (amounting to two matrix-matrix multiplications) and solves the eigenproblem using the Jacobi algorithm at working precision. Our analysis yields meaningfully smaller relative forward error bounds for the computed eigenvalues compared with those of the Jacobi algorithm. We further prove that, after preconditioning, if the off-diagonal entries of the preconditioned matrix are sufficiently small relative to its smallest diagonal entry, the relative forward error bound is independent of the condition number of the original matrix. We present two constructions for the preconditioner that exploit low precision, along with their error analyses. Our numerical experiments confirm our theoretical results and compare the relative forward error of the proposed algorithm with the Jacobi algorithm, a preconditioned Jacobi algorithm, and MATLAB's $\texttt{eig}$ function. Timings using Julia suggest that the dominant cost of obtaining this level of accuracy comes from the high precision matrix-matrix multiplies; if support in software or hardware for this were improved, then this would become a negligible cost.

math.NA↗

arXiv2020

The role of topology and mechanics in uniaxially growing cell networks

In biological systems, the growth of cells, tissues, and organs is influenced by mechanical cues. Locally, cell growth leads to a mechanically heterogeneous environment as cells pull and push their neighbors in a cell network. Despite this local heterogeneity, at the tissue level, the cell network is remarkably robust, as it is not easily perturbed by changes in the mechanical environment or the network connectivity. Through a network model, we relate global tissue structure (i.e. the cell network topology) and local growth mechanisms (growth laws) to the overall tissue response. Within this framework, we investigate the two main mechanical growth laws that have been proposed: stress-driven or strain-driven growth. We show that in order to create a robust and stable tissue environment, networks with predominantly series connections are naturally driven by stress-driven growth, whereas networks with predominantly parallel connections are associated with strain-driven growth.

physics.bio-ph↗