Related papers: Computing Borcherds Products
The article presents a computationally effective algorithm for calculating the multiresolution discrete Fourier transform (MrDFT). The algorithm is based on the idea of reducing the computational complexity which was introduced by Wen and…
This is my talk on the Bourbaki seminar, November 1996. It contains an elementary introduction to Borcherds' product formulas.
An efficient quantum algorithm is proposed to solve in polynomial time the parity problem, one of the hardest problems both in conventional quantum computation and in classical computation, on NMR quantum computers. It is based on the…
We consider the problem of secure distributed matrix multiplication in which a user wishes to compute the product of two matrices with the assistance of honest but curious servers. We show how to construct polynomial schemes for the outer…
We propose a new algorithm for multiplying dense polynomials with integer coefficients in a parallel fashion, targeting multi-core processor architectures. Complexity estimates and experimental comparisons demonstrate the advantages of this…
We present a bounded probability algorithm for the computation of the Chow forms of the equidimensional components of an algebraic variety. Its complexity is polynomial in the length and in the geometric degree of the input equation system…
Polynomial multiplication is known to have quasi-linear complexity in both the dense and the sparse cases. Yet no truly linear algorithm has been given in any case for the problem, and it is not clear whether it is even possible. This…
We present a polynomial time algorithm to approximately scale tensors of any format to arbitrary prescribed marginals (whenever possible). This unifies and generalizes a sequence of past works on matrix, operator and tensor scaling. Our…
We consider the problem of preprocessing an $n\times n$ matrix $\mathbf{M}$, and supporting queries that, for any vector $v$, returns the matrix-vector product $\mathbf{M} v$. This problem has been extensively studied in both theory and…
Our goal is to find accurate and efficient algorithms, when they exist, for evaluating rational expressions containing floating point numbers, and for computing matrix factorizations (like LU and the SVD) of matrices with rational…
In this research, the Bernoulli polynomials are introduced. The properties of these polynomials are employed to construct the operational matrices of integration together with the derivative and product. These properties are then utilized…
We present RXTX, a new algorithm for computing the product of matrix by its transpose $XX^{t}$ for $X\in \mathbb{R}^{n\times m}$. RXTX uses $5\%$ fewer multiplications and $5\%$ fewer operations (additions and multiplications) than…
Determining the properties of molecules and materials is one of the premier applications of quantum computing. A major question in the field is: how might we use imperfect near-term quantum computers to solve problems of practical value? We…
We present a simple and fast algorithm for computing the $N$-th term of a given linearly recurrent sequence. Our new algorithm uses $O(\mathsf{M}(d) \log N)$ arithmetic operations, where $d$ is the order of the recurrence, and…
We review some recent developments in numerical algorithms to solve the time-dependent Maxwell equations for systems with spatially varying permittivity and permeability. We show that the Suzuki product-formula approach can be used to…
In this paper, we report on an implementation in the free software Mathemagix of lacunary factorization algorithms, distributed as a library called Lacunaryx. These algorithms take as input a polynomial in sparse representation, that is as…
We provide an integral representation for the (implied) copulas of dependent random variables in terms of their moment generating functions. The proof uses ideas from Fourier methods for option pricing. This representation can be used for a…
We propose a non-commutative algorithm for multiplying 2x2 matrices using 7 coefficient products. This algorithm reaches simultaneously a better accuracy in practice compared to previously known such fast algorithms, and a time complexity…
Based on the generation function of Laguerre polynomials, We proposed a new Laguerre polynomial expansion scheme in the calculation of evolution of time dependent Schr\"odinger equation. Theoretical analysis and numerical test show that the…
Additive Fourier Transform is sdudied. A fast multiplication algorithm for polynomials over the binary field is given. The bit complexity of the algorithm is $O(n(log n)(\log\log n)^2)$.