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We proposed in this paper a new method, which we named the W4 method, to solve nonlinear equation systems. It may be regarded as an extension of the Newton-Raphson~(NR) method to be used when the method fails. Indeed our method can be…
A rational positive-definite quadratic form is perfect if it can be reconstructed from the knowledge of its minimal nonzero value m and the finite set of integral vectors v such that f(v) = m. This concept was introduced by Voronoi and…
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…
In the mid-eighties Tardos proposed a strongly polynomial algorithm for solving linear programming problems for which the size of the coefficient matrix is polynomially bounded by the dimension. Combining Orlin's primal-based modification…
In this paper we present an algorithm for computing all algebraic intermediate subfields in a separably generated unirational field extension (which in particular includes the zero characteristic case). One of the main tools is Groebner…
The criterion of differentiability of functions of quaternion variable is used as the basis of some algebraic field theory. Its necessary consequences are free Maxwell and Yang-Mills equations. The differentiability equations may be…
A new one-parameter family of iterative method for solving nonlinear equations is constructed and studied. Two variants, both with cubic convergence, are developed, one for finding simple zeros and other for multiple zeros of known…
We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is deterministic and the proof is elementary. In some cases, the…
We introduce techniques to analyze unitary operations in terms of quadratic form expansions, a form similar to a sum over paths in the computational basis when the phase contributed by each path is described by a quadratic form over…
In this work, a novel quantum Fourier ordinary differential equation (ODE) solver is proposed to solve both linear and nonlinear partial differential equations (PDEs). Traditional quantum ODE solvers transform a PDE into an ODE system via…
We present an algorithm which allows to solve analytically linear systems of differential equations which factorize to first order. The solution is given in terms of iterated integrals over an alphabet where its structure is implied by the…
Quantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation"…
In this note we study the number of quantum queries required to identify an unknown multilinear polynomial of degree d in n variables over a finite field F_q. Any bounded-error classical algorithm for this task requires Omega(n^d) queries…
The paper describes two iterative algorithms for solving general systems of M simultaneous linear algebraic equations (SLAE) with real matrices of coefficients. The system can be determined, underdetermined, and overdetermined. Linearly…
Fast algorithms for arithmetic on real or complex polynomials are well-known and have proven to be not only asymptotically efficient but also very practical. Based on Fast Fourier Transform (FFT), they for instance multiply two polynomials…
This article generalizes a recently introduced procedure to solve nonlinear systems of equations, radically departing from the conventional Newton-Raphson scheme. The original nonlinear system is first unfolded into three simpler…
Solving combinatorial optimization problems on current noisy quantum devices is currently being advocated for (and restricted to) binary polynomial optimization with equality constraints via quantum heuristic approaches. This is achieved…
A classical question of propositional logic is one of the shortest proof of a tautology. A related fundamental problem is to determine the relative efficiency of standard proof systems, where the relative complexity is measured using the…
Polynomial factorization and root finding are among the most standard themes of computational mathematics. Yet still, little has been done for polynomials over quaternion algebras, with the single exception of Hamiltonian quaternions for…
In this paper it is shown that the compact linearization approach, that has been previously proposed only for binary quadratic problems with assignment constraints, can be generalized to arbitrary linear equations with positive coefficients…