Related papers: Scaling a unitary matrix
Many iterative methods in applied mathematics can be thought of as fixed-point iterations, and such algorithms are usually analyzed analytically, with inequalities. In this paper, we present a geometric approach to analyzing contractive and…
We use differentiable programming and gradient descent to find unitary matrices that can be used in the period finding algorithm to extract period information from the state of a quantum computer post application of the oracle. The standard…
The power method (or iteration) is a well-known classical technique that can be used to find the dominant eigenpair of a matrix. Here, we present a variational quantum circuit method for the power iteration, which can be used to find the…
Nonlinear matrix equations arise in many practical contexts related to control theory, dynamical programming and finite element methods for solving some partial differential equations. In most of these applications, it is needed to compute…
We describe a dynamic programming algorithm for exact counting and exact uniform sampling of matrices with specified row and column sums. The algorithm runs in polynomial time when the column sums are bounded. Binary or non-negative integer…
We introduce a hierarchy of linear systems for showing that a given subspace of pure quantum states is entangled (i.e., contains no product states). This hierarchy outperforms known methods already at the first level, and it is complete in…
In this paper, a class of optimization problems with nonlinear inequality constraints is discussed. Based on the ideas of sequential quadratic programming algorithm and the method of strongly sub-feasible directions, a new superlinearly…
One of the key challenges in identifying nonlinear and possibly non-Gaussian state space models (SSMs) is the intractability of estimating the system state. Sequential Monte Carlo (SMC) methods, such as the particle filter (introduced more…
We consider the iterative solution of large linear systems of equations in which the coefficient matrix is the sum of two terms, a sparse matrix $A$ and a possibly dense, rank deficient matrix of the form $\gamma UU^T$, where $\gamma > 0$…
We define and solve classes of sparse matrix problems that arise in multilevel modeling and data analysis. The classes are indexed by the number of nested units, with two-level problems corresponding to the common situation in which data on…
We propose stochastic splitting algorithms for solving large-scale composite inclusion problems involving monotone and linear operators. They activate at each iteration blocks of randomly selected resolvents of monotone operators and,…
A cumbersome operation in numerical analysis and linear algebra, optimization, machine learning and engineering algorithms; is inverting large full-rank matrices which appears in various processes and applications. This has both numerical…
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…
We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades.…
Two methods to decompose block matrices analogous to Singular Matrix Decomposition are proposed, one yielding the so called economy decomposition, and other yielding the full decomposition. This method is devised to avoid handling matrices…
Diagonalization, or eigenvalue decomposition, is very useful in many areas of applied mathematics, including signal processing and quantum physics. Matrix decomposition is also a useful tool for approximating matrices as the product of a…
This paper concerns models and convergence principles for dealing with stochasticity in a wide range of algorithms arising in nonlinear analysis and optimization in Hilbert spaces. It proposes a flexible geometric framework within which…
We present further properties of a previously proposed recursive scheme for parameterisation of n-by-n unitary matrices. We show that the factors in the recursive formula may be introduced in any desired order. The method is used to study…
This paper addresses the numerical solution of the matrix square root problem. Two fixed point iterations are proposed by rearranging the nonlinear matrix equation $A - X^2 = 0$ and incorporating a positive scaling parameter. The proposals…
We present a successive constraint approach that makes it possible to cheaply solve large-scale linear matrix inequalities for a large number of parameter values. The efficiency of our method is made possible by an offline/online…