Related papers: Fast Algorithms for the Multi-dimensional Jacobi P…
In this short letter, I present a powerful application in dimensionality reduction of the lesser-used Jacobi-Davidson algorithm for the generalized eigenvalue decomposition. When combined with matrix-free implementations of relevant…
A rapid transformation is derived between spherical harmonic expansions and their analogues in a bivariate Fourier series. The change of basis is described in two steps: firstly, expansions in normalized associated Legendre functions of all…
We consider the quantum complexity of estimating matrix elements of unitary irreducible representations of groups. For several finite groups including the symmetric group, quantum Fourier transforms yield efficient solutions to this…
The problem of fast computation of multivariate kernel density estimation (KDE) is still an open research problem. In our view, the existing solutions do not resolve this matter in a satisfactory way. One of the most elegant and efficient…
Discrete cosine transform (DCT) and other Fourier-related transforms have broad applications in scientific computing. However, off-the-shelf high-performance multi-dimensional DCT (MD DCT) libraries are not readily available in parallel…
Multipoint polynomial evaluation and interpolation are fundamental for modern symbolic and numerical computing. The known algorithms solve both problems over any field of constants in nearly linear arithmetic time, but the cost grows to…
Over the last two decades, several fast, robust, and high-order accurate methods have been developed for solving the Poisson equation in complicated geometry using potential theory. In this approach, rather than discretizing the partial…
In many processes, the variations in underlying characteristics can be approximated by noisy multi-periodic patterns. If large-scale patterns are superimposed by a noise with long-range correlations, the detection of multi-periodic patterns…
The discrete Fourier transform (DFT) is an important operator which acts on the Hilbert space of complex valued functions on the ring Z/NZ. In the case where N=p is an odd prime number, we exhibit a canonical basis of eigenvectors for the…
We present a novel algorithm, named the 2D-FFAST, to compute a sparse 2D-Discrete Fourier Transform (2D-DFT) featuring both low sample complexity and low computational complexity. The proposed algorithm is based on mixed concepts from…
The efficient implementation of matrix arithmetic operations underpins the speedups of many quantum algorithms. We develop a suite of methods to perform matrix arithmetics -- with the result encoded in the off-diagonal blocks of a…
A real persymmetric Jacobi matrix of order $n$ whose eigenvalues are $2k^2$ $(k=0, ..., n-1)$ is presented, with entries given as explicit functions of $n$. Besides the possible use for testing forward and inverse numerical algorithms, such…
This paper introduces an efficient algorithm for computing the general oscillatory matrix functions. These computations are crucial for solving second-order semi-linear initial value problems. The method is exploited using the scaling and…
The matrix expressions for every parts of a transformer are firstly described. Based on semi-tensor product (STP) of matrices the hypervectors are reconsidered and the linear transformation over hypervectors is constructed by using…
We give a new method to prove in a uniform and easy way various transformation formulas for Gauss hypergeometric functions. The key is Jacobi's canonical form of the hypergeometric differential equation. Analogy for $q$-hypergeometric…
We consider matrix orthogonal polynomials related to Jacobi type matrices of weights that can be defined in terms of a given matrix Pearson equation. Stating a Riemann-Hilbert problem we can derive first and second order differential…
Efficient algorithms for computing linear convolutions based on the fast Fourier transform are developed. A hybrid approach is described that combines the conventional practice of explicit dealiasing (explicitly padding the input data with…
A polynomial transform is the multiplication of an input vector $x\in\C^n$ by a matrix $\PT_{b,\alpha}\in\C^{n\times n},$ whose $(k,\ell)$-th element is defined as $p_\ell(\alpha_k)$ for polynomials $p_\ell(x)\in\C[x]$ from a list…
The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal solution to large classes of control problems. Unfortunately, this generality comes at a price, the calculation of such solutions is typically intractible for systems…
We devise the fast adjoint response algorithm for the gradient of physical measures (long-time-average statistics) of discrete-time hyperbolic chaos with respect to many system parameters. Its cost is independent of the number of…