Related papers: Kaltofen's division-free determinant algorithm dif…
We present a general and automated approach for computing model gradients for PDE solvers built on sparse spectral methods, and implement this capability in the widely used open-source Dedalus framework. We apply reverse-mode automatic…
In appropriate frameworks, automatic differentiation is transparent to the user at the cost of being a significant computational burden when the number of operations is large. For iterative algorithms, implicit differentiation alleviates…
Given a square, nonsingular matrix of univariate polynomials $\mathbf{F}\in\mathbb{K}[x]^{n\times n}$ over a field $\mathbb{K}$, we give a deterministic algorithm for finding the determinant of $\mathbf{F}$. The complexity of the algorithm…
It is known that the multiplication of an $N \times M$ matrix with an $M \times P$ matrix can be performed using fewer multiplications than what the naive $NMP$ approach suggests. The most famous instance of this is Strassen's algorithm for…
Metasurfaces are an emerging technology that may supplant many of the conventional optics found in imaging devices, displays, and precision scientific instruments. Here, we develop a method for designing optical systems composed of multiple…
In this paper, we construct a derivative-free multi-step iterative scheme based on Steffensen's method. To avoid excessively increasing the number of functional evaluations and, at the same time, to increase the order of convergence, we…
In recent years, a number of fast algorithms for computing the determinant of a Toeplitz matrix were developed. The fastest algorithm we know so far is of order $k^2\log{n}+k^3$, where $n$ is the number of rows of the Toeplitz matrix and…
One of the most significant challenges in Computing Determinant of Rectangular Matrices is high time complexity of its algorithm. Among all definitions of determinant of rectangular matrices, used definition has special features which make…
Automatic differentiation, as implemented today, does not have a simple mathematical model adapted to the needs of modern machine learning. In this work we articulate the relationships between differentiation of programs as implemented in…
The adjoint method is an efficient way to numerically compute gradients in optimization problems with constraints, but is only formulated to differentiable cost and constraint functions on real variables. With the introduction of complex…
Let $\K$ be an algebraic number field of degree $d$ and discriminant $\Delta$ over $\Q$. Let $\A$ be an associative algebra over $\K$ given by structure constants such that $\A\cong M_n(\K)$ holds for some positive integer $n$. Suppose that…
This paper is a sequel to "Computing diagonal form and Jacobson normal form of a matrix using Groebner bases", J. of Symb. Computation, 46 (5), 2011. We present a new fraction-free algorithm for the computation of a diagonal form of a…
We derive a formula for the adjoint $\overline{A}$ of a square-matrix operation of the form $C=f(A)$, where $f$ is holomorphic in the neighborhood of each eigenvalue. We then apply the formula to derive closed-form expressions in particular…
The quest for non-commutative matrix multiplication algorithms in small dimensions has seen a lot of recent improvements recently. In particular, the number of scalar multiplications required to multiply two $4\times4$ matrices was first…
Based on the Bezout approach we propose a simple algorithm to determine the {\tt gcd} of two polynomials which doesn't need division, like the Euclidean algorithm, or determinant calculations, like the Sylvester matrix algorithm. The…
In contrast with many other convex optimization classes, state-of-the-art semidefinite programming solvers are yet unable to efficiently solve large scale instances. This work aims to reduce this scalability gap by proposing a novel…
A modification of Newton's method for solving systems of $n$ nonlinear equations is presented. The new matrix-free method relies on a given decomposition of the invertible Jacobian of the residual into invertible sparse local Jacobians…
We develop a new algorithm to compute determinants of all possible Hankel matrices made up from a given finite length sequence over a finite field. Our algorithm fits within the dynamic programming paradigm by exploiting new recursive…
A new algorithm for smooth constrained optimization is proposed that never computes the value of the problem's objective function and that handles both equality and inequality constraints. The algorithm uses an adaptive switching strategy…
The problem of finding a nontrivial factor of a polynomial f(x) over a finite field F_q has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the…